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                                             CHAPTER ONE

Specific Objectives

By the end of the topic the learner should be able to:

• Identify, read and write natural numbers in symbols and words;
• Round off numbers to the nearest tens, hundreds, thousands, millions and billions;
• Classify natural numbers as even, odd or prime;
• Solve word problems involving natural numbers.

Content

• Place values of numbers
• Rounding off numbers to the nearest tens, hundreds, thousands, millions and billions
• Odd numbers
• Even numbers
• Prime numbers
• Word problems involving natural numbers

Introduction

Place value

A digit have a different value in a number because of its position in a number. The position of a digit in a number is called its place value.

Total value

This is the product of the digit and its place value.

Example

A place value chart can be used to identify both place value and  total value of a digit in a number. The place value chart is also used in writing numbers in words.

Example

• Three hundred and forty five million, six hundred and seventy eight thousand, nine hundred and one.
• Seven hundred and sixty nine million, Three hundred and one thousand, eight hundred and fifty four.

Billions

A billion is one thousands million, written as 1, 000, 000,000.There are ten places in a billion.

Example

What is the place value and total value of the digits below?

• 47,397,263,402 (place value 7 and 8).
• 389,410 ,000,245 ( place 3 and 9)

Solution

• The place value for 6 is ten thousands. Its total value is 60,000.
• The place value of 3 is hundred billions. Its total value is 300,000,000,000.

Rounding off

When rounding off to the nearest ten, the ones digit determines the ten i.e. if the ones digit is 1, 2, 3, or 4 the nearest ten is the ten number being considered. If the ones digit is 5 or more the nearest ten is the next ten or rounded up.

Thus 641 to the nearest ten is 640, 3189 to the nearest is 3190.

When rounding off to the nearest 100, then the last two digits or numbers end with 1 to 49 round off downwards. Number ending with 50 to 99 are rounded up.

Thus 641 to the nearest hundred is 600, 3189 is 3200.

Example

Rounding off each of the following numbers to the nearest number indicated in the bracket:

• 473,678 ( 100)
• 524,239 (1000)
• 2,499 (10)

Solution

• 473,678 is 473,700 to the nearest 100.
• 524,239 is 524,000 to the nearest 1000
• 2,499 is 2500 to the nearest 10.

Operations on whole Numbers

Addition

Example

Find out:

• 98 + 6734 + 348
• 6349 + 259 +7954

Solution

Arrange the numbers in vertical forms

98

6734

+   348

   7180

6349

259

+ 79542

                           86150

Subtracting

Example

Find: 73469 – 8971

Solution

73469

• 8971

   64498

Multiplication

The product is the result of two or more numbers.

Example

Work out: 469 x 63

Solution

469

    X 63

1407

+ 28140

29547

Division

When a number is divided by the result is called the quotient. The number being divided is the divided and the number dividing is the divisor.

Example

Find: 6493  14

Solution

We get 463 and 11 is the remainder

Note:

6493 = (463 x 14) + 11

In general, dividend = quotient x division + remainder.

Word problem

In working the word problems, the information given must be read and understood well before attempting the question.

The problem should be broken down into steps and identify each other operations required.

Example

Otego had 3469 bags of maize, each weighing 90 kg. He sold 2654 of them.

• How many kilogram of maize was he left with?
• If he added 468 more bags of maize, how many bags did he end up with?

Solution

• One bag weighs 90 kg.

3469 bags weigh 3469 x 90 = 312,210 kg

2654 bags weigh 2654 x 90 = 238,860 kg

Amount of maize left          = 312,210 – 238,860

= 73,350 kg.

• Number of bags = 815 + 468

=1283

Even Number

A number which can be divided by 2 without a remainder E.g. 0,2,4,6 0 or 8

3600, 7800, 806, 78346

Odd Number

Any number that when divided by 2 gives a remainder. E.g. 471,123, 1197,7129.The numbers ends with the following digits 1, 3, 5,7 or 9.

Prime Number

A prime number is a number that has only two factors one and the number itself.

For example, 2, 3, 5, 7, 11, 13, 17 and 19.

Note:

• 1 is not a prime number.
• 2 is the only even number which is a prime number.

End of topic

Past KCSE Questions on the topic

• Write 27707807 in words
• All prime numbers less than ten are arranged in descending order to form a number
• Write down the number formed
• What is the total value of the second digit?
• Write the number formed in words.

CHAPTER TWO

Specific Objectives

By the end of the topic the learner should be able to:

• Express composite numbers in factor form;
• Express composite numbers as product of prime factors;
• Express factors in power form.

Content

• Factors of composite numbers.
• Prime factors.
• Factors in power form

Introduction

Definition

A factor is a number that divides another number without a remainder.

A natural number with only two factors, one and itself is a prime number. Or any number that only can be divided by 1 and itself.  Prime numbers have exactly 2 different factors.

Prime Numbers up to 100.

Composite numbers

Any number that has more factors than just itself and 1.They can be said to be natural number other than 1 which are not prime numbers.They can be expressed as a product of two or more prime factors.

9 = 3 x 3

12 = 2 x 2 x 3

105 = 3 x 5 x 7

The same number can be repeated several times in some situations.

32 = 2 x 2 x 2 x 2 x 2 =

72 =2 x 2 x 2 x 3 x 3 =

To express a number in terms of prime factors ,it is best to take the numbers from the smallest and divide by each of them as many times as possible before going to the next.

Example

Express the following numbers in terms of their prime factors

• 300
• 196

Solution

300 = 2 x 2 x 3 x 5 x 5

=

196 = 2 x 2 x 7 x 7

=

Exceptions

The numbers 1 and 0 are neither prime nor composite.  1 cannot be prime or composite because it only has one factor, itself.  0 is neither a prime nor a composite number because it has infinite factors.  All other numbers, whether prime or composite, have a set number of factors.  0 does not follow the rules.

End of topic

Past KCSE Questions on the topic

• Express the numbers 1470 and 7056, each as a product of its prime factors.

Hence evaluate:            1470²

7056

Leaving the answer in prime factor form

• All prime numbers less than ten are arranged in descending order to form a number

(a) Write down the number formed

(b) What is the total value of the second digit?

                         CHAPTER THREE

Specific Objectives

By the end of the topic the learner should be able to:

The learner should be able to test the divisibility of numbers by 2, 3, 4, 5, 6, 8, 9, 10 and 11.

48

Content

Divisibility test of numbers by 2, 3, 4, 5, 6, 8, 9, 10 and 11

Introduction

Divisibility test makes computation on numbers easier. The following is a table for divisibility test.

End of topic

Past KCSE Questions on the topic

CHAPTER FOUR

Specific Objectives

By the end of the topic the learner should be able to:

• Find the GCD/HCF of a set of numbers.
• Apply GCD to real life situations.

Content

• GCD of a set of numbers
• Application of GCD/HCF to real life situations

Introduction

A Greatest Common Divisor is the largest number that is a factor of two or more numbers.

When looking for the Greatest Common Factor, you are only looking for the COMMON factors contained in both numbers. To find the G.C.D of two or more numbers, you first list the factors of the given numbers, identify common factors and state the greatest among them.

The G.C.D can also be obtained by first expressing each number as a product of its prime factors. The factors which are common are determined and their product obtained.

Example

Find the Greatest Common Factor/GCD for 36 and 54 is 18.

Solution

The prime factorization for 36 is 2 x 2 x 3 x 3.

The prime factorization for 54 is 2 x 3 x 3 x 3.

They both have in common the factors 2, 3, 3 and their product is 18.

That is why the greatest common factor for 36 and 54 is 18.

Example

Find the G.C.D of 72, 96, and 300

Solution

End of topic

Past KCSE Questions on the topic

• Find the greatest common divisor of the term. 144x³y² and 81xy⁴

1. b) Hence factorize completely this expression 144x³y²-81xy⁴ (2 marks)

• The GCD of two numbers is 7and their LCM is 140. if one of the numbers is 20, find the other number

• The LCM of three numbers is 7920 and their GCD is 12. Two of the numbers are 48 and 264. Using factor notation find the third number if one of its factors is 9

CHAPTER FIVE

Specific Objectives

By the end of the topic the learner should be able to:

• List multiples of numbers.
• Find the L CM of a set of numbers.
• Apply knowledge of L CM in real life situations.

Content

• Multiples of a number
• L CM of a set of numbers
• Application of L CM in real life situations.

Introduction

Definition

LCM or LCF is the smallest multiple that two or more numbers divide into evenly i.e. without a remainder. A multiple of a number is the product of the original number with another number.

Some multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56 …

Some multiples of 7 are 7, 14, 21, 28, 35, 42, 49, 56 …

A Common Multiple is a number that is divisible by two or more numbers. Some common multiples of 4 and 7 are 28, 56, 84, and 112.

When looking for the Least Common Multiple, you are looking for the smallest multiple that they both divide into evenly.  The least common multiple of 4 and 7 is 28.

Example

Find the L.C.M of 8, 12, 18 and 20.

Solution

The L.C.M is the product of all the divisions used.

Therefore, L.C.M. of 8, 12, 18 and 20 = 2 x 2 x 2 x 3 x 3 x5

= x 5

= 360

Note;

Unlike the G.C.D tables, if the divisor /factor does not divide a number exactly, then the number is retained, e.g., 2 does not divide 9 exactly, therefore 9 is retained .The last row must have all values 1.

End of topic

Past KCSE Questions on the topic

• Find the L.C.M of x² + x, x² – 1 and x² – x .
• Find the least number of sweets that can be packed into polythene bags which contain either 9 or 15 or 20 or 24 sweets with none left over.
• A number n is such that when it is divided by 27, 30, or 45, the remainder is always 3. Find the smallest value of n.
• A piece of land is to be divided into 20 acres or 24 acres or 28 acres for farming and

Leave 7 acres for grazing. Determine the smallest size of such land.

• When a certain number x is divided by 30, 45 or 54, there is always a remainder of 21. Find the least value of the number x .

A number m is such that when it is divided by 30, 36, and 45, the remainder is always 7. Find the smallest possible value of m.

CHAPTER SIX

Specific Objectives

By the end of the topic the learner should be able to:

• Define integers
• Identify integers on a number line
• Perform the four basic operations on integers using the number line.
• Work out combined operations on integers in the correct order
• Apply knowledge of integers to real life situations.

Content

• Integers
• The number line
• Operation on integers
• Order of operations
• Application to real life situations

Introduction

The Number Line

Integers are whole numbers, negative whole numbers and zero. Integers are always represented on the number line at equal intervals which are equal to one unit.

Operations on Integers

Addition of Integers

Addition of integers can be represented on a number line .For example, to add

+3 to 0 , we begin at 0 and move 3 units to the right as shown below in red to get +3, Also  to add + 4 to +3 we move 4 units to the right as shown in blue to get +7.

To add -3 to zero we move 3 units to the left as shown in red below to get -3 while to add -2 to -3 we move 2 steps to the left as shown in blue to get -5.

Example

(+7) – (0) = (+7)

To subtract +7 from 0 ,we find a number n which when added to get 0 we get +7 and in this case n = +7 as shown above in red.

Example

(+2) – (+7) = (-5)

Start at +7 and move to +2. 5 steps will be made towards the left. The answer is therefore -5.

Example

-3 – (+6) = -9

|__|__|__|__|__|__|__|__|__|__|__|__|__|__|

-4 -3 -2   -1     0    1    2    3 4   5    6    7   8    9   10

We start at +6 and moves to -3. 9 steps to the left, the answer is -9.

Note:

• In general positives signs can be ignored when writing positive numbers i.e. +2 can be written as 2 but negative signs cannot be ignored when writing negative numbers -4 can only be written s -4.

4 – (+3) = 4 -3

= 1

-3- (+6) =3 – 6

= -3

• Positive integers are also referred to as natural numbers. The result of subtracting the negative of a number is the same as adding that number.

2 – (- 4) = 2 + 4

= 6

(-5) – (- 1) = -5 + 2

= -3

• In mathematics it is assumed that that the number with no sign before it has appositive sign.

Multiplication

In general

• ( a negative number ) x ( appositive number ) = ( a negative number)
• (a positive number ) x ( a negative number ) = ( a negative number)
• ( a negative number ) x ( a negative number ) = ( a positive number)

Examples

-6 x 5 = -30

7 x -4 = – 28

-3 x -3 = 9

-2 x -9 = 18

Division

Division is the inverse of multiplication. In general

• (a positive number )  ( a positive number ) = ( a positive number)
• (a positive number ) ( a negative number ) = ( a negative  number)
• ( a negative number ) ( a negative number ) = ( a positive number)
• ( a negative number ) ( appositive number ) = ( a negative number)

Note;

For multiplication and division of integer:

• Two like signs gives positive sign.
• Two unlike signs gives negative sign
• Multiplication by zero is always zero and division by zero is always zero.

Order of operations

BODMAS is always used to show as the order of operations.

B – Bracket first.

O – Of is second.

D – Division is third.

M – Multiplication is fourth.

A – Addition is fifth.

S – Subtraction is considered last.

Example

6 x 3 – 4

Solution

Use BODMAS

(2 – 1) = 1 we solve brackets first

(4) = 2 we then solve division

(6 x 3) = 18 next is multiplication

Bring them together

18 – 2 +5 +1 = 22 we solve addition first and lastly subtraction

18 + 6 – 2 = 22

End of topic

Past KCSE Questions on the topic

1.) The sum of two numbers exceeds their product by one. Their difference is equal to their product less five. Find the two numbers.                                                (3mks)

2.)         3x – 1 > -4

2x + 1≤ 7

3.)        Evaluate                       -12 ÷ (-3) x 4 – (-15)

-5 x 6 ÷ 2 + (-5)

4.) Without using a calculator/mathematical tables, evaluate leaving your answer as a simple fraction

(-4)(-2) + (-12) ¸ (+3)     +    -20 + (+4) + -6)

-9 – (15)                 46- (8+2)-3

5.)        Evaluate -8 ¸ 2 + 12 x 9 – 4 x 6

• ¸7 x 2

6.) Evaluate without using mathematical tables or the calculator

1.9 x 0.032

20 x 0.0038

CHAPTER SEVEN

Specific Objectives

By the end of the topic the learner should be able to:

• Identify proper and improper fractions and mixed number.
• Convert mixed numbers to improper fractions and vice versa.
• Compare fractions;
• Perform the four basic operations on fractions.
• Carry out combined operations on fractions in the correct order.
• Apply the knowledge of fractions to real life situations.

Content

• Fractions
• Proper, improper fractions and mixed numbers.
• Conversion of improper fractions to mixed numbers and vice versa.
• Comparing fractions.
• Operations on fractions.
• Order of operations on fractions
• Word problems involving fractions in real life situations.

Introduction

A fraction is written in the form  where a and b are numbers and b is not equal to 0.The upper number is called the numerator and the lower number is the denominator.

Proper fraction

In proper fraction the numerator is smaller than the denominator. E.g.

Improper fraction

The numerator is bigger than or equal to denominator. E.g.

Mixed fraction

An improper fraction written as the sum of an integer and a proper fraction. For example

=

Changing a Mixed Number to an Improper Fraction

Mixed number –   4 (contains a whole number and a fraction)

Improper fraction – (numerator is larger than denominator)

Step 1 – Multiply the denominator and the whole number

Step 2 – Add this answer to the numerator; this becomes the new numerator

Step 3 – Carry the original denominator over

Example

3      =    3 × 8 + 1 = 25

=

Example

• =    4 × 9 + 4 = 40

=

Changing an Improper Fraction to a Mixed Number

Step 1– Divide the numerator by the denominator

Step 2– The answer from step 1 becomes the whole number

Step 3– The remainder becomes the new numerator

Step 4– The original denominator carries over

Example

=    47 ÷ 5    or

5  =   5    =    9

  2                                 

Example

=      2     =      2       = 4 ½

Comparing Fractions

When comparing fractions, they are first converted into their equivalent forms using the same denominator.

Equivalent Fractions

To get the equivalent fractions, we multiply or divide the numerator and denominator of a given fraction by the same number. When the fraction has no factor in common other than 1, the fraction is said to be in its simplest form.

Example

Arrange the following fractions in ascending order (from the smallest to the biggest):

1/2   1 /4   5/6 2/3

Step 1: Change all the fractions to the same denominator.

Step 2: In this case we will use 12 because 2, 4, 6, and3 all go into i.e. We get 12 by finding the L.C.M of the denominators. To get the equivalent fractions divide the denominator by the L.C.M and then multiply both the numerator and denominator by the answer,

For ½ we divide 12  2 = 6, then multiply both the numerator and denominator by 6 as shown below.

1 ^{x 6}       1^x3        5 ^x2       2 ^x4

2 _x6       4 _x3       6 _x2       3 _x4

Step3: The fractions will now be:

6/12         3/12           10/12        8/12

Step 4: Now put your fractions in order (smallest to biggest.)

3/12      6/12       8/12     10/12

Step 5: Change back, keeping them in order.

1/4   1/2   2/3   5/6

You can also use percentages to compare fractions as shown below.

Example

Arrange the following in descending order (from the biggest)

5/12   7 /3   11/5   9 /4

Solution

X 100 = 41.67%

X 100 = 233.3%

X 100 = 220%

X 100 = 225%

7/3, 9/4, 11/5, 5/12

Operation on Fractions

Addition and Subtraction

The numerators of fractions whose denominators are equal can be added or subtracted directly.

Example

2/7 + 3/7 = 5/7

6/8 – 5/8 = 1/8

When adding or subtracting numbers with different denominators like:

5/4 + 3 /6=?

2/5 – 2/7 =?

Step 1– Find a common denominator (a number that both denominators will go into or L.C.M)

Step 2– Divide the denominator of each fraction by the common denominator or L.C.M and then multiply the answers by the numerator of each fraction

Step 3– Add or subtract the numerators as indicated by the operation sign

Step 4 – Change the answer to lowest terms

Example

+   =    Common denominator is 8 because both 2 and 8 will go into 8

+  =

Which simplifies to   1

Example

4 –   =     Common denominator is 20 because both 4 and 5 will go into 20

4    =      4

–      =

4

Or

4 –   = =

Mixed   numbers can be added or subtracted easily by first expressing them as improper fractions.

Examples

5

Solution

5 = 5 +

Example

Evaluate

Solution

Multiplying Simple Fractions

Step 1– Multiply the numerators

Step 2– Multiply the denominators

Step 3– Reduce the answer to lowest terms by dividing by common divisors

Example

×   =    which reduces to

Multiplying Mixed Numbers

Step 1– Convert the mixed numbers to improper fractions first

Step 2– Multiply the numerators

Step 3– Multiply the denominators

Step 4– Reduce the answer to lowest terms

Example

2 × 1  =    ×   =

Which then reduces to 3

Note:

When opposing numerators and denominators are divisible by a common number, you may reduce the numerator and denominator before multiplying.   In the above example, after converting the mixed numbers to improper fractions, you will see that the 3 in the numerator and the opposing 3 in the denominator could have been reduced by dividing both numbers by 3, resulting in the following reduced fraction:

×    =    =   3

Dividing Simple Fractions

Step 1– Change division sign to multiplication

Step 2– Change the fraction following the multiplication sign to its reciprocal (rotate the fraction around so the old denominator is the new numerator and the old numerator is the new denominator)

Step 3– Multiply the numerators

Step 4– Multiply the denominators

Step 5– simplify the answer to lowest terms

Example

÷    =     becomes     ×      which when solved is

Dividing Mixed Numbers

Step 1 – Convert the mixed number or numbers to improper fraction.

Step 2 – Change the division sign to multiplication.

Step 3– Change the fraction following the multiplication sign to its reciprocal (flip the fraction around so the old denominator is the new numerator and the old numerator is the new denominator)

Step 4– Multiply the numerators.

Step 5– Multiply the denominators.

Step 6– Simplify the answer to lowest form.

Example

3 ÷ 2 =    becomes     ÷    becomes     ×   =

Which when solved is     ×   =   which simplifies to 1

Order of operations on Fractions

The same rules that apply on integers are the same for fractions

BODMAS

Example

15 (we start with of then division)

= 15

= 5

Example

=

Solution

1/3 – 1/4 = (we start with bracket)

(We then work out the outer bracket)

(We then work out the multiplication)

(Addition comes last here)

Example

Evaluate      + ½

Solution

We first work out this first

Therefore + ½ = 25 + ½

= 25 ½

Note:

Operations on fractions are performed in the following order.

• Perform the operation enclosed within the bracket first.
• If (of) appears, perform that operation before any other.

Example

Evaluate: =

Solution

=

=

=

=

Example

Two pipes A and B can fill an empty tank in 3hrs and 5hrs respectively. Pipe C can empty the tank in 4hrs. If the three pipes A, B and C are opened at the same time find how long it will take for the tank to be full.

Solution

1/3 +1/5 -1/4 = 20+12-15

60

= 17/60

17/60=1hr

1= 1 x 60/17

60/17 = 3.5294118

= 3.529 hrs.

End of topic

Past KCSE Questions on the topic

1. Evaluate without using a calculator.

2. A two digit number is such that the sum of the ones and the tens digit is ten. If the digits are reversed, the new number formed exceeds the original number by 54.

Find the number.

3. Evaluate
4. Convert the recurring decimal linto fraction
5. Simplify without using tables or calculator
6. Evaluate without using tables or calculators

7. Mr. Saidi keeps turkeys and chickens. The number of turkeys exceeds the number of chickens by 6. During an outbreak of a disease, ¼ of the chicken and ¹/₃ of the turkeys died. If he lost total of 30 birds, how many birds did he have altogether?
8. Work out
9. Evaluate          -4 of (-4 + -5 ¸15) + -3 – 4 ¸2)

84 ¸ -7 + 3 – -5

11. Write the recurring decimal 0.Can as Fraction
12. Evaluate without using a calculator.

13. Without using tables or calculators evaluate.

14. Without using tables or calculator, evaluate the following.

^–8 + (^–13) x 3 – (^–5)

^–1 + (^–6) ÷ 2 x 2

15. Express  as a single fraction

16. Simplify ½ of    3½ + 1½ (2½ – ²/₃)

¾ of 2½ ¸ ½

17. Evaluate:

²/₅¸ ½ of ^4/₉ – 1¹/₁₀

        ¹/₈ – ¹/₆ of 3/₈  

18. Without using a calculator or table, work out the following leaving the answer as a mixed number in its simplest form:-

¾ + 1²/₇ ÷ ³/₇ of 2¹/₃

(9/₇–³/₈) x ²/₃

19. Work out the following, giving the answer as a mixed number in its simplest form.

²/₅¸ ½ of 4/₉ – 1 ¹/₁₀

¹/₈ – ¹/₁₆ x 3/₈

20. Evaluate;

23. Without using a calculator, evaluate:

1⁴/₅ of 25/₁₈¸1²/₃ x 24

2¹/₃ – ¼ of 12 ¸⁵/₃              leaving the answer as a fraction in its simplest form

24. There was a fund-raising in Matisse high school. One seventh of the money that was raised was used to construct a teacher’s house and two thirds of the remaining money was used to construct classrooms. If shs.300, 000 remained, how much money was raised

CHAPTER EIGHT

Specific Objectives

By the end of the topic the learner should be able to:

• Convert fractions into decimals and vice versa
• Identify recurring decimals
• Convert recurring decimals into fractions
• Round off a decimal number to the required number of decimal places
• Write numbers in standard form
• Perform the four basic operations on decimals
• Carry our operations in the correct order
• Apply the knowledge of decimals to real life situations.

Content

• Fractions and decimals
• Recurring decimals
• Recurring decimals and fractions
• Decimal places
• Standard form
• Operations on decimals
• Order of operations
• Real life problems involving decimals.

Introduction

A fraction whose denominator can be written as the power of 10 is called a decimal fraction or a decimal. E.g. 1/10, 1/100, 50/1000.

A decimal is always written as follows 1/10 is written as 0.1 while 5/100 is written as 0.05.The dot is called the decimal point.

Numbers after the decimal points are read as single digits e.g. 5.875 is read as five point eight seven five. A decimal fraction such 8.3 means 8 + 3/10.A decimal fraction which represents the sum of a whole number and a proper fraction is called a mixed fraction.

Place value chart

Decimal to Fractions

To convert a number from fraction form to decimal form, simply divide the numerator (the top number) by the denominator (the bottom number) of the fraction.

Example:

5/8

8¬   Add as many zeros as needed.

48

20

16

40

40

0

Converting a decimal to a fraction

To change a decimal to a fraction, determine the place value of the last number in the decimal.  This becomes the denominator.  The decimal number becomes the numerator.  Then reduce your answer.

Example:

.625 – the 5 is in the thousandths column, therefore,

.625 =  = reduces to

Note:

Your denominator will have the same number of zeros as there are decimal digits in the decimal number you started with  –  .625 has three decimal digits so the denominator will have three zero.

Recurring Decimals

These are decimal fractions in which a digit or a group of digits repeat continuously without ending.

We cannot write all the numbers, we therefore place a dot above a digit that is recurring. If more than one digit recurs in a pattern, we place a dot above the first and the last digit in the pattern.

E.g.

0.3333……………………….is written as 0.

0.4545………………………is written as 0.

0.324324………………….is written as 0.

Any division whose divisor has prime factors other than 2 or 5 forms a recurring decimal or non-terminating decimal.

Example

Express each as a fraction

• 7

Solution

• Let r = 0.66666 ——-( I )

10r = 6.6666 ————— (II)

Subtracting I from II

9r = 6

• Let r = 0.73333——— (I)

10 r=7.3333333——— (II)

100r = 73.33333——— (III)

Subtracting (II) from (III)

90r = 66

• Let r = 0.151515 ———( I)

100r = 15.1515 ———– (II)

99r= 15

Decimal places

When the process of carrying out division goes over and over again without ending we may round off the digits to any number of required digits to the right of decimal points which are called decimal places.

Example

Round 2.832 to the nearest hundredth.

Solution

Step 1 – Determine the place to which the number is to be rounded is.

2.832

Step 2 – If the digit to the right of the number to be rounded is less than 5, replace it and all the digits to the right of it by zeros.  If the digit to the right of the underlined number is 5 or higher, increase the underlined number by 1 and replace all numbers to the right by zeros. If the zeros are decimal digits, you may eliminate them.

2.832   =   2.830   =   2.83

Example

Round 43.5648 to the nearest thousandth.

Solution

                        43.5648   =   43.5650 =   43.565

Example

Round 5,897,000 to the nearest hundred thousand.

Solution

            5,897,000    = 5,900,000

Standard Form

A number is said to be in standard form if it is expressed in form A X, Where 1 < A <10 and n is an integer.

Example

Write the following numbers in standard form.

• 36 ) 576 c.) 0.052

Solution

• 36/10 x 10 = 3.6 x
• 576/100 x 100 = 5.76 x
• 052 = 0.052 x 100/100

Operation on Decimals

Addition and Subtraction

The key point with addition and subtraction is to line up the decimal points!

Example

2.64 + 11.2 =       2.64

+11.20®  in this case, it helps to write 11.2 as 11.20

13.84

Example

14.73 – 12.155 =      14.730  ® again adding this 0 helps

• 155

2.575

Example

127.5 + 0.127 =       327.500

+      0.127

327.627

Multiplication

When multiplying decimals, do the sum as if the decimal points were not there, and then calculate how many numbers were to the right of the decimal point in both the original numbers – next, place the decimal point in your answer so that there are this number of digits to the right of your decimal point?

Example

2.1 x 1.2.

Calculate 21 x 12 = 252. There is one number to the right of the decimal in each of the original numbers, making a total of two. We therefore place our decimal so that there are two digits to the right of the decimal point in our answer.

Hence 2.1 x 1.2 =2.52.

Always look at your answer to see if it is sensible. 2 x 1 = 2, so our answer should be close to 2 rather than 20 or 0.2 which could be the answers obtained by putting the decimal in the wrong place.

Example

1.4 x 6

Calculate 14 x 6 = 84. There is one digit to the right of the decimal in our original numbers so our answer is 8.4

Check 1 x 6 = 6 so our answer should be closer to 6 than 60 or 0.6

Division

When dividing decimals, the first step is to write your numbers as a fraction. Note that the symbol / is used to denote division in these notes.

Hence 2.14 / 1.2 = 2.14

1.2

Next, move the decimal point to the right until both numbers are no longer decimals. Do this the same number of places on the top and bottom, putting in zeros as required.

Hence   becomes

This can then be calculated as a normal division.

Always check your answer from the original to make sure that things haven’t gone wrong along the way. You would expect 2.14/1.2 to be somewhere between 1 and 2. In fact, the answer is 1.78.

If this method seems strange, try using a calculator to calculate 2.14/1.2,   21.4/12, 214/120 and   2140 / 1200. The answer should always be the same.

Example

4.36 / 0.14 = 4.36 = 436 = 31.14

• 14

Example

27.93 / 1.2 = 27.93 = 2793 = 23.28

• 120

Rounding Up

Some decimal numbers go on forever! To simplify their use, we decide on a cutoff point and “round” them up or down.

If we want to round 2.734216 to two decimal places, we look at the number in the third place after the decimal, in this case, 4. If the number is 0, 1, 2, 3 or 4, we leave the last figure before the cut off as it is. If the number is 5, 6, 7, 8 or 9 we “round up” the last figure before the cut off by one. 2.734216 therefore becomes 2.73 when rounded to 2 decimal places.

If we are rounding to 2 decimal places, we leave 2 numbers to the right of the decimal.

If we are rounding to 2 significant figures, we leave two numbers, whether they are decimals or not.

Example

243.7684 = 243.77 (2 decimal places)

= 240 (2 significant figures)

1973.285 = 1973.29 (2 decimal places)

= 2000 (2 significant figures)

• = 2.47 (2 decimal places)

= 2.5 (2 significant figures)

0.99879 = 1.00 (2 decimal places)

= 1.0 (2 significant figures)

Order of operation

The same rules on operations is always the same even for decimals.

Examples

Evaluate

0.02 + 3.5 x 2.6 – 0.1 (6.2 -3.4)

Solution

0.02 + 3.5 x 2.6 – 0.1 x 2.8 = 0.02 + 0.91 -0.28

= 8.84

End of topic

Past KCSE Questions on the topic

• Without using logarithm tables or a calculator evaluate.

384.16 x 0.0625

96.04

• Evaluate without using mathematical table

1000           0.0128

200

• Express the numbers 1470 and 7056, each as a product of its prime factors.

Hence evaluate:            1470²

7056

Leaving the answer in prime factor form

• Without using mathematical tables or calculators, evaluate

• Evaluate without using mathematical tables  or the calculator

CHAPTER NINE

Specific Objectives

By the end of the topic the learner should be able to:

• Find squares of numbers by multiplication
• Find squares from tables
• Find square root by factor method
• Find square root from tables.

Content

• Squares by multiplication
• Squares from tables
• Square roots by factorization
• Square roots from tables.

Introduction

Squares

The square of a number is simply the umber multiplied by itself once. For example the square of 15 is 225.That is 15 x 15 = 225.

Square from tables

The squares of numbers can be read directly from table of squares. This tables give only approximate values of the squares to 4 figures. The squares of numbers from 1.000 to 9.999 can be read directly from the tables.

The use of tables is illustrated below

Example

Find the square of:

a.) 4.25

b.) 42.5

c.) 0.425

Tables

• To read the square of 4.25, look for 4.2 down the column headed x. Move to the right along this row, up to where it intersects with the column headed 5.The number in this position is the square of 4.25

So 4.2 = 18.06 to 4 figures

• The square of 4.25 lies between 4and 5  between 1600 and 2500.

4= (

=

= 18.06 x 100

= 1806

• 4= (

=

=18.06 x 1/100

= 0.1806

The square tables have extra columns labeled 1 to 9 to the right of the thick line. The numbers under these columns are called mean differences. To find 3.162, read 3.16 to get 9.986.Then read the number in the position where the row containing 9.986 intersects with the differences column headed 2.The difference is 13 and this should be added to the last digits of 9.986

9.986

+    13

9.999

56.129 has 5 significant figures and in order to use 4 figures tables, we must first round it off to four figures.

56.129 = 56.13 to 4 figures

= (5.613 x

= 31.50 x

= 3150

Square Roots

Square roots are the opposite of squares. For example 5 x 5 = 25, we say that 5 is a square root of 25.

Any positive number has two square roots, one positive and the other negative .The symbol for the square root of a number is.

A number whose square root is an integer is called a perfect square. For example 1, 4,9,25 and 36 are perfect squares.

Square roots by Factorization.

The square root of a number can also be obtained using factorization method.

Example

Find the square root of 81 by factorization method.

Solution

81 =   (Find the prime factor of 81)

= (3×3) (3 x 3) (Group the prime factors into two identical numbers)

= 3 x 3             (Out of the two identical prime factors, choose one and find their product)

= 9

Note:

Pair the prime factors into two identical numbers. For every pair, pick only one number then obtain the product.

Example

Find  by factorization.

Solution

= 2 x 3 x 7

= 42

Example

Find  by factorization

Solution

= 3 x 7

=21

Square Root from tables

Square roots of numbers from 1.0 to 99.99 are given in the tables and can be read directly.

Examples

Use tables to find the square root of:

• 86 b.)  42.57   c.) 359    d.) 0.8236

Solution

• To read the square root of 1.86, look for 1.8 in the column headed x, move to the right along this row to where it intersects with the column headed 6.The number in this position is the square root of 1.86.Thus =1.364 to 4 figures.
• Look for 42 in the column headed x and move along the row containing 42 to where it intersects with the column headed 5.Read the number in this position, which is 6.519. The difference for 7 from the difference column along this row is 6.The difference is added to 6.519 as shown below:

6.519

+ 0.006

6.525

Thus,  to 4 figures.

For any number outside this range, it is necessary to first express it as the product of a number in this range and an even power of 10.

• 359 = 3.59 x

= 1.895 x 10

= 18.95 (four figures)

• 8236 = 82.36 x

=

= (9.072 + 0.004) x

= 0.9076 (4 figures)

End of topic

Past KCSE Questions on the topic

• Evaluate without using tables or calculators

• Evaluate using reciprocals, square and square root tables only.

• Using a calculator, evaluate

(Show your working at each stage)

• Use tables of reciprocals and square roots to evaluate

5.) Use tables to find;

4. a) I) 4.978²
5. ii) The reciprocal of 31.65
6. b) Hence evaluate to 4.S.F the value of

4.978² – ¹/_31.65

6.) Use tables of squares, square roots and reciprocals to evaluate correct to 4 S.

7. Without using mathematical tables or calculator, evaluate: 153 x 1.8     giving your answer in standard form                                                         0.68 x 0.32

CHAPTER TEN

Specific Objectives

By the end of the topic the learner should be able to:

• Use letters to represent numbers
• Write statements in algebraic form
• Simplify algebraic expressions
• Factorize an algebraic expressions by grouping
• Remove brackets from algebraic expressions
• Evaluate algebraic expressions by substituting numerical values
• Apply algebra in real life situations.

Content

• Letters for numbers
• Algebraic fractions
• Simplification of algebraic expressions
• Factorization by grouping
• Removal of brackets
• Substitution and evaluation
• Problem solving in real life situations.

Introduction

An algebraic expression is a mathematical expression that consists of variables, numbers and operations. The value of this expression can change Clarify the definitions and have students take notes on their graphic organizer.

Note:

• Algebraic Expression—contains at least one variable, one number and one operation. An example of an algebraic expression is n + 9.
• Variable—a letter that is used in place of a number. Sometimes, the variable will be given a value.  This value will replace the variable in order to solve the equation.  Other times, the variable is not assigned a value and the student is to solve the equation to determine the value of the variable.
• Constant—a number that stands by itself. The 9 in our previous vocabulary is an example of a constant.
• Coefficient—a number in front of and attached to a variable. For example, in the expression 5x + 3, the 5 is the coefficient.
• Term—each part of an expression that is separated by an operation. For instance, in our earlier example n + 9, the terms are n and 7.

Examples

Write each phrase as an algebraic expression.

Nine increased by a number r       9 + r

Fourteen decreased by a number x     14 – x

Six less than a number t     t – 6

The product of 5 and a number n 5 X n   or   5n

Thirty-two divided by a number y      31 ÷ x   or

Example

An electrician charges sh 450 per hour and spends sh 200 a day on gasoline. Write an algebraic expression to represent his earnings for one day.

Solution:    Let x represent the number of hours the electrician works in one day. The electrician’s earnings can be represented by the following algebraic expression:

Solution

450x – 200

Simplification of Algebraic Expressions

Note:

Basic steps to follow when simplify an algebraic expression:

• Remove parentheses by multiplying factors.
• Use exponent rules to remove parentheses in terms with exponents.
• Combine like terms by adding coefficients.
• Combine the constants.

Like and unlike terms

Like terms have the same variable /letters raised to the same power i.e. 3 b + 2b = 5b or a + 5a = 6a and they can be simplified further into 5b and 6a respectively.While unlike terms have different variables i.e. 3b + 2 c or 4b + 2x and they cannot be simplified further.

Example

3a +12b +4a  -2b   =   7a + 10b  ( collect the lick terms )

2x – 5y + 3x – 7y + 3w  =  5x – 12y + 3w

Example

Simplify: 2x – 6y – 4x + 5z – y

Solution

2x – 6y – 4x + 5z –y = 2x – 4x – 6y –y + 5z

= (2x – 4x) – (6y + y) + 5z

= – 2x – 7y + 5z

Note:

-6y – y = -( 6y + y)

Example

Simplify:

Solution

The L.C.M of 2, 3 and 4 is 12.

Therefore

Example

Simplify:

Solution

Example

5x² – 2x² = 3x²

4a²bc – 2a²bc = 2a²bc

a²b – 2c + 3a²b +c = 4a²b – c

Note:

Capital letter and small letters are not like terms.

Brackets

Brackets serve the same purpose as they do in arithmetic.

Example

Remove the brackets and simplify:

• 3 ( a + b ) – 2(a – b)
• 1/3a + 3 (5a + b – c )
• 2b + 3 (3 – 2 (a – 5)

Solution

• 3 ( a + b ) – 2(a – b) = 3a + 3b – 2a + 2b

= 3a – 2a + 3b + 2b

= a + 5b

• 1/3a + 3 (5a + b – c ) = 1/3a + 15 a + 3b – 3c

=

• 2b + a { 3 – 2 (a – 5 )} = 2b + a {3 – 2a + 10}

=2b + 3a – 2 + 10a

= 2b + 3a + 10 a – 2

= 2b + 13a – 2

The process of remaining the brackets is called expansion while the reverse process of inserting the brackets is called factorization.

Example

Factorize the following:

• 3m + 3n = 3 (m + n ) ( the common term is 3 so we put it outside the bracket)

Solution

• (is common)

(Is common)

=

Factorization by grouping

When the terms of an expression which do not have a common factor are taken pairwise, a common factor can be found. This method is known as factorization by grouping.

Example

Factorize:

• 3ab + 2b + 3ca + 2c
• ab + bx – a – x

Solution

• 3ab + 2b + 3ca + 2c = b ( 3a + 2 ) + c (3a +2 )

= (3a + 2) (b + c)

• ab + bx – a – x = b ( a + x ) – 1 ( a + x)

= (a + x) (b – 1)

Algebraic fractions

In algebra, fractions can be added and subtracted by finding the L.C.M of the denominators.

Examples

Express each of the following as a single fraction:

Solution

=     (10x -10 + 5x + 10 + 4x)

=

=

=

Simplification by factorization

Factorization is used to simplify expressions

Examples

Simplify   p² – 2pq + q²

2p² -3pq + q²

Solution

Numerator is solved first.

Then solve the denominator

2p²-2pq – pq – q²

(2p-q) (p-q)

Example

Simplify

Solution

Num. (4m – 3n) (4m + 3n)

Den. 4m² – 4mn + 3mn – 3n²

(4m + 3n) (m – n)

(4m – 3n) (4m + 3n)

(4m + 3n) (m – n)

4m – 3n

m – n

Example

Simplify the expression.

Solution

Numerator

18x (y – r)

Denominator

9x (r – y)

Therefore

=

Example

Simplify

Solution

=

Example

Simplify the expression completely.

Solution

=              =

Note:

Substitution

This is the process of giving variables specific values in an expression

Example

Evaluate the expression  if x =2 and y = 1

Solution

=

End of topic

Past KCSE Questions on the topic

1. Given that y = 2x – z

x + 3z    express x in terms of y and z

2. Simplify the expression

x – 1     –   2x + 1

x                3x

Hence solve the equation

x – 1     –    2x + 1   = 2

x                 3x         3

3. Factorize a² – b²

Hence find the exact value of 2557² – 2547²

4. Simplify p² – 2pq + q²

P³ – pq² + p² q –q³

5. Given that y = 2x – z, express x in terms of y and z.

Four farmers took their goats to a market. Mohammed had two more goats as Koech had 3 times as many goats as Mohammed, whereas Odupoy had 10 goats less than both Mohammed and Koech.

(i)         Write a simplified algebraic expression with one variable, representing the total number of goats.

(ii)        Three butchers bought all the goats and shared them equally. If each butcher got 17 goats, how many did odupoy sell to the butchers?

6. Solve the equation

1  =5  -7

4x    6x

7. Simplify

  a        +             b

2(a+ b)             2(a-b)

8. Three years ago, Juma was three times as old. As Ali in two years time, the sum of their ages will be 62. Determine their ages

                                    CHAPTER ELEVEN

Specific Objectives

By the end of the topic the learner should be able to:

• Define rates
• Solve problems involving rates
• Define ratio
• Compare two or more quantities using ratios
• Change quantities in a given ratio
• Compare two or more ratios
• Represent and interpret proportional parts
• Recognize direct and inverse proportions
• Solve problems involving direct and inverse proportions
• Convert fractions and decimals to percentages and vice-versa
• Calculate percentage change in a given quantity
• Apply rates, ratios, percentages to real life situations and proportion.

Content

• Rates
• Solving problems involving rates
• Ratio
• Comparing quantities using ratio
• Increase and decrease in a given ratio
• Comparing ratios
• Proportion: direct and inverse
• Solve problems on direct and inverse proportions
• Fractions and decimals as percentages
• Percentage increase and decrease
• Application of rates, ratios, percentages and proportion to real life situations.

Introduction

Rates

A rate is a measure of quantity, and comparing one quantity with another of different kind.

Example

If a car takes two hours to travel a distance of 160 km. then we will say that it is travelling at an average rate of 80 km per hour.  If two kilograms of maize meal is sold for sh. 38.00, then we say that maize meal is selling at the rate of sh.19.00 per kilogram.

Example

What is the rate of consumption per day if twelve bags of beans are consumed in 120 days?

Solution.

Rate of consumption = number of bags/number of days

=

=1/10 bags per day

Example

A laborer’s wage is sh.240 per eight hours working day. What is the rate of payment per hour?

Solution

Rate = amount of money paid/number of hours

=

=sh.30 per hour

Ratio

A ratio is a way of comparing two similar quantities. For example, if alias is 10 years old and his brother basher is 14 years old. Then alias age is 10/14 of Bashir’s age, and their ages are said to be in the ratio of 10 to 14. Written, 10:14.

Alias age: Bashir’s age =10:14

Bashir’s age: alias age =14:10

In stating a ratio, the units must be the same. If on a map 2cm rep 5km on the actual ground, then the ratio of map distance to map distance is 2cm: 5×1 00 000cm, which is 2:500 000.

A ratio is expressed in its simplest form in the same way as a fraction,

E.g. 10/14 = 5/7, hence 10:14= 5:7.

Similarly, 2:500 000 = 1: 250 000,

A proportion is a comparison of two or more ratios. If, example, a, b and c are three numbers such that a: b: c=2:3:5, then a, b, c are said to be proportional to 2, 3, 5 and the relationship should be interpreted to mean a/2 = b/3=c/5.

Similarly, we can say that a: b =2:3, b: c=3:5 a: c=2:5

Example 3

If a: b = 3: 4 and b: c = 5: 7 find a: c

Solution

a: b =3 : 4…………………(i)

b: c=5 : 7………………….(ii)

Consider the right hand side;

Multiply (i) by 5 and (ii) by 4 to get, a: b=15: 20 and b: c=20: 28

Thus, a: b: c = 15: 20: 28 and a: c=15: 28

Increase and decrease in a given ratio

To increase or decrease a quantity in a given ratio, we express the ratio as a fraction and multiply it by the quantity.

Example

Increase 20 in the ratio 4: 5

Solution

New value =5/4×20

=5×5

=25

Example

Decrease 45 in the ratio 7:9

Solution

New value    =7/9 x45

=7×5

=35

Example

The price of a pen is adjusted in the ratio 6:5. If the original price was sh.50. What is the new price?

Solution

New price: old price = 6:5

New price /old price = 6/5

New price    = 6/5×50

= sh. 60

Note:

When a ratio expresses a change in a quantity an increase or decrease   , it is usually put in the form of new value: old value

Comparing ratios

In order to compare ratios, they have to be expressed as fractions first,  ie., a:b = a/b . the resultant fraction can then be compared.

Example

Which ratio is greater, 2: 3 or 4: 5?

Solution 2:3 = 2/3, 4:5 = 4/5

2/3 = 10/15, 4/5 = 12/15 = 4/5 > 2/3

Thus, 4: 5 > 2: 3

Distributing a quantity in a given ratio

If a quantity is to be divided in the ratio a: b: c, the fraction of the quantity represented by:

• A will be
• B will be
• C will be

Example

A 72-hactare farm is to be shared among three sons in the ratio 2:3:4. What will be the sizes in hectares of the three shares?

Solution

Total number of parts is 2+3+4= 9

The she shares are: 2/9 x72ha =16ha

3/9 x 72ha= 24ha

4/9 x72ha = 36ha

Direct and inverse proportion

Direct proportion

The table below shows the cost of various numbers of cups at sh. 20 per cup.

The ratio of the numbers of cups in the fourth column to the number of cups in the second column is 4:2=2:1. The ratio of the corresponding costs is 80:40=2:1. By considering the ratio of costs in any two columns and the corresponding ratio and the number of cups, you should notice that they are always the same.

If two quantities are such that when the one increases (decreases) in particular ratio, the other one also increases (decreases) in the ratio,

Example

A car travels 40km on 5 litres of petrol. How far does it travel on 12 litres of petrol?

Solution

Petrol is increased in the ratio 12: 5

Distance= 40x 12/5 km

Example

A train takes 3 hours to travel between two stations at an average speed of 40km per hour. A t what average speed would it need to travel to cover the same distance in 2hours?

Solution

Time is decreased in the ratio 2:3 Speed must be increased in the ratio 3:2 average speed is 40 x 3/2 km = 60 km/h

Example

Ten men working six hours a day take 12 days to complete a job. How long will it take eight men working 12 hours a day to complete the same job?

Solution

Number of men decreases in the ratio 8:10

Therefore, the number of days taken increases in the ratio 10:8.

Number of hours increased in the ratio 12:6.

Therefore, number of days decreases in the ratio 6:12.

Number of days taken =12 x x

=7 ½ days

Percentages

A percentage (%) is a fraction whose denominator is 100. For example, 27% means 27/100.

Converting fractions and decimals into percentages

To write a decimal or fraction as a %: multiply by 100 .For example

0.125    = 0.125 ´ 100 = 12.5%

=      ´ 100   (i.e.      Of 100%) = 40%

Or                 = 2 ¸ 5 ´ 100 = 40%

Example

Change 2/5 into percentage.

Solution

=

x =  x 100

= 40%

Example

Convert 0.67 into a percentage: solution

0.67 =

As a percentage, 0.67=  x 100

=67%

Percentage increase and decrease

A quantity can be expressed as a percentage of another by first writing it as a fraction of the given quantity.

Example

A farmer harvested 250 bags of maize in a season. If he sold 200 bags, what percentage of his crops does this represent?

Let x be the percentage sold.

Then, x/100 =

So, x    = =   x 100

= 80%

Example

A man earning sh. 4 800 per month was given a 25% pay rise. What was his new salary?

Solution

New salary = 25/100 x 4800 + 4 800

= 1 200 + 4 800

= sh. 6 000

Example

A dress which was costing sh. 1 200 now goes for sh. 960. What is the percentage decrease?

Solution

Decrease in cost is 1 200- 960= sh. 240

Percentage decrease = 240/1 200 x 100

= 20%

Example

The ratio of john’s earnings to muse’s earnings is 5:3. If john’s earnings increase by 12%, his new figure becomes sh. 5 600. Find the corresponding percentage change in muse’s earnings if the sum of their new earnings is sh.9 600

Solution

John’s earnings before the increase is 100/112 x 5600 = sh. 5 000

John’s earnings/muse’s earnings = 5/3

Musa’s earnings before the increase = 3/5 x5000

= sh. 3 000

Musa’s new earnings                              =9 600 – 5 600

=sh. 4 000

Musa’s change in earnings                      =4 000-1 000

=sh. 3 000

Percentage change in muse’s earnings = 1000/3000 x100

=33  %

End of topic

Past KCSE Questions on the topic

1. Akinyi bought and beans from a wholesaler. She then mixed the maize and beans in the ratio 4:3 she bought the maize at Kshs 21 per kg and the beans 42 per kg. If she was to make a profit of 30%. What should be the selling price of 1 kg of the mixture?
2. Water flows from a tap at the rate of 27 cm³ per second into a rectangular container of length 60 cm, breadth 30 cm and height 40 cm. If at 6.00 PM the container was half full, what will be the height of water at 6.04 pm?
3. Two businessmen jointly bought a minibus which could ferry 25 paying passengers when full. The fare between two towns A and B was Kshs 80 per passenger for one way. The minibus made three round trips between the two towns daily. The cost of fuel was Kshs 1500 per day. The driver and the conductor were paid daily allowances of Kshs 200 and Kshs 150 respectively.

A further Kshs 4000 per day was set aside for maintenance, insurance and loan repayment.

(a)        (i)         How much  money was collected from the passengers that day?

(ii)        How much was the net profit?

(b)        On another day, the minibus was 80% full on the average for the three round trips, how much did each businessman get if the day’s profit was shared in the ratio 2:3?

1. Wainaina has two dairy farms, A and B. Farm A produces milk with 3 ¼ percent fat and farm B produces milk with 4 ¼ percent fat.

(a)        Determine

(i)         The total mass of milk fat in 50 kg of milk from farm A and 30 kg of milk from farm B

(ii)        The percentage of fat in a mixture of 50kg of milk A and 30kg of milk from B

(b)        The range of values of mass of milk from farm B that must be used  in a 50kg mixture so that the mixture may  have at least 4 percent fat.

1. In the year 2001, the price of a sofa set in a shop was Kshs 12,000

(a)        Calculate the amount of money received from the sales of 240 sofa sets that year.

(b)        (i)         In the year 2002 the price of each sofa set increased by 25% while

the number  of sets sold decreased by 10%. Calculate the percentage increase in the amount received from the sales

(ii)       If the end of year 2002, the price of each sofa set changed in the ratio 16: 15, calculate the price of each sofa set in the year 2003.

(c)        The number of sofa sets sold in the year 2003 was P% less than the number sold in the year 2001.

Calculate the value of P, given that the amounts received from sales if the two years were equal.

1. A solution whose volume is 80 litres is made up of 40% of water and 60% of alcohol. When x litres of water is added, the percentage of alcohol drops to 40%.

(a)        Find the value of x

(b)        Thirty litres of water is added to the new solution. Calculate the percentage of alcohol in the resulting solution

(c)        If 5 litres of the solution in (b) above is added to 2 litres of the original solution, calculate in the simplest form, the ratio of water to that of alcohol in the resulting solution.

1. Three business partners, Asha, Nangila and Cherop contributed Kshs 60,000, Kshs 85,000 and Kshs 105, 000 respectively. They agreed to put 25% of the profit back into business each year. They also agreed to put aside 40% of the remaining profit to cater for taxes and insurance. The rest of the profit would then be shared among the partners in the ratio of their contributions. At the end of the first year, the business realized a gross profit of Kshs 225, 000.

(a)        Calculate the amount of money Cherop received more than Asha at the end of the first year.

(b)        Nangila further invested Kshs 25,000 into the business at the beginning of the second year.  Given that the gross profit at the end of the second year increased in the ratio 10:9, calculate Nangila’s share of the profit at the end of the second year.

1. Kipketer can cultivate a piece of land in 7 hrs while Wanjiku can do the same work in 5 hours. Find the time they would take to cultivate the piece of land when working together.

1. Mogaka and Ondiso working together can do a piece of work in 6 days. Mogaka working alone, takes 5 days longer than Onduso. How many days does it take Onduso to do the work alone.

1. A certain amount of money was shared among 3 children in the ratio 7:5:3 the largest share was Kshs 91. Find the

(a)        Total amount of money

(b)        Difference in the money received as the largest share and the smallest share.

CHAPTER TWELVE

Specific Objectives

By the end of the topic the learner should be able to:

• State the units of measuring length
• Convert units of length from one form to another
• Express numbers to required number of significant figures
• Find the perimeter of a plane figure and circumference of a circle.

Content

• Units of length (mm, cm, m, km)
• Conversion of units of length from one form to another
• Significant figures
• Perimeter
• Circumference (include length of arcs)

Introduction

Length is the distance between two points. The SI unit of length is metres. Conversion of units of length.

1 kilometer (km) = 1000metres

1 hectometer (hm) =100metres

1 decameter (Dm) =10 metres

1 decimeter (dm) = 1/10 metres

1 centimeter (cm) = 1/100 metres

1 millimeter (mm) = 1/1000 metres

The following prefixes are often used when referring to length:

Mega – 1 000 000

Kilo – 1 000

Hecto – 100

Deca -10

Deci-1/10

Centi-1/100

Milli-1/1000

Micro-1/1 000 000

Significant figures

The accuracy with which we state or write a measurement may depend on its relative size. It would be unrealistic to state the distance between towns A and B as 158.27 km. a more reasonable figure is 158 km.158.27km is the distance expressed to 5 significant figures and 158 km to 3 significant figures.

Example

Express each of the following numbers to 5, 4, 3, 2, and 1 significant figures:

• 906 315
• 08564
• 0089
• 156 000

Solution

The above example show how we would round off a measurement to a given number of significant figures

Zero may not be a significant. For example:

• 085 – zero is not significant therefore, 0.085 is a two- significant figure.
• 30 – zero is significant. Therefore 2.30 is a three-significant figure.
• 5 000 –zero may or may not be significant figure. Therefore, 5 000 to three significant figure is 5 00 (zero after 5 is significant). To one significant figure is 5 000. Zero after 5 is not significant.
• 805 Or 305 – zero is significant, therefore 31.805 is five significant figure. 305 is three significant figure.

Perimeter

The perimeter of a plane is the total length of its boundaries. Perimeter is a length and is therefore expressed in the same units as length.

Square shapes

5cm

Its perimeter is 5+5+5+5= 2(5+5)

=2(10)

=20cm

Hence                          5×4 = 20

So perimeter of a square = Sides x 4

Rectangular shapes

Figure12.2 is a rectangle of length 5cm and breadth 3cm.

5cm

3 cm

Its perimeter is 5+3+5+3 =2(5+3)cm

=2×8

= 16cm

Hence    perimeter of a rectangle    p=2(L+ W)

Triangular shapes

To find the perimeter of a triangle add all the three sides.

c

a

b

Perimeter = (a + b + c) units, where a, b and c are the lengths of the sides of the triangle.

The circle

The circumference of a circle = 2 r or

Example

• Find the circumference of a circle of a radius 7cm.
• The circumference of a bicycle wheel is 140 cm. find its radius.

Solution

• C= πd

=22/7 x 7

=44 cm

• C=πd

=2πr

=2×22/7xr

=140 ÷44/7

=22.27 cm

Length of an arc

An arc of a circle is part of its circumference. Figure 12.10 (a) shows two arcs AMB and ANB.  Arc AMB, which is less than half the circumference of the circle, is called the minor arc, while arc ANB, which is greater half of the circumference is called the major arc. An arc which is half the circumference of the circle is called a semicircle.

Example

An arc of a circle subtends an angle 60 at the centre of the circle. Find the length of the arc if the radius of the circle is 42 cm. (π=22/7).

Solution

The length, l, of the arc is given by:

L =θ/360 x 2πr.

Θ=60, r=42 cm

Therefore, l =60/360 x2 x 22/7 x 42

= 44 cm

Example

The length of an arc of a circle is 62.8 cm. find the radius of the circle if the arc subtends an angle 144 at the centre, (take π=3.142).

Solution

L =θ/360 x 2πr = 62.8 and   θ= 144

Therefore,

R=

=24.98 cm

Example

Find the angle subtended at the centre of a circle by an arc of length 11cm if the radius of the circle is 21cm.

Solution

L=θ/360 x 2 xπr =11 cm and r =21m

L=θ/360 x2x22/7x 21=11

Thus, θ=11x360x7 / 2x22x21

End of topic

Past KCSE Questions on the topic

1.)      Two coils which are made by winding copper wire of different gauges and length have the same mass. The first coil is made by winding 270 metres of wire with cross sectional diameter 2.8mm while the second coil is made by winding a certain length of wire with cross-sectional diameter 2.1mm. Find the length of wire in the second coil .

2. The figure below represents a model of a hut with HG = GF = 10cm and FB = 6cm. The four slanting edges of the roof are each 12cm long.

Calculate

Length DF.

Angle VHF

The length of the projection of line VH on the plane EFGH.

The height of the model hut.

The length VH.

The angle DF makes with the plane ABCD.

3. A square floor is fitted with rectangular tiles of perimeters 220 cm. each row (tile length wise) carries 20 less tiles than each column (tiles breadth wise). If the length of the floor is 9.6 m.

Calculate:

1. The dimensions of the tiles
2. The number of tiles needed
3. The cost of fitting the tiles, if tiles are sold in dozens at sh. 1500 per dozen and the labour cost is sh. 3000

                   CHAPTER THIRTEEN

Specific Objectives

By the end of the topic the learner should be able to:

• State units of area
• Convert units of area from one form to another
• Calculate the area of a regular plane figure including circles
• Estimate the area of irregular plane figures by counting squares
• Calculate the surface area of cubes, cuboids and cylinders.

Content

• Units of area (cm2 , m2 , km2 , Ares, ha)
• Conversion of units of area
• Area of regular plane figures
• Area of irregular plane shapes
• Surface area of cubes, cuboids and cylinders

Introduction

Units of Areas

The area of a plane shape is the amount of the surface enclosed within its boundaries. It is normally measured in square units. For example, a square of sides 5 cm has an area of

5 x 5 = 25 cm

A square of sides 1m has an area of 1m, while a square of side 1km has an area of 1km

Conversion of units of area

1 m² =1mx 1m

= 100 cm x 100 cm

= 10 000 cm²

1 km ²   = 1 km x 1 km

= 1 000 m x 1 000 m

=1 000 000 m²

1 are    = 10 m  x 10 m

=100 m²

1 hectare (ha) = 100 Ares

=10 000 m²

Area of a regular plane figures

Areas of rectangle

5cm

3 cm

Area, A =5×3 cm

=15

Hence, the area of the rectangle, A =L X W square units, where l is the length and b breadth.

Area of a triangle

H

Base

Area of a triangle

A =1/2bh square units

Area of parallelogram

Area =1/2bh +1/2bh

=bh square units

Note:

This formulae is also used for a rhombus

Area of a trapezium

The figure below shows a trapezium in which the parallel sides are a units and b units, long. The perpendicular distance between the two parallel sides is h units.

Area of a triangle ABD =1/2  ah square units

Area of triangle DBC = ½ bh square units

Therefore area of trapezium ABCD =1/2 ah +1/2 bh

= 1/2h (a + b) square units.

Thus, the area of a trapezium is given by a half the sum of the length of parallel sides multiplied by the perpendicular distance between them.

That is, area of trapezium =

Area of a circle

The area A of a circle of radius r is given by: A =

The area of a sector

A sector is a region bounded by two radii and an arc.

Suppose we want to find the area of the shaded part in the figure below

The area of the whole circle is πr²

The whole circle subtends 360ͦat the centre.

Therefore, 360ͦ corresponds to πr²

1ͦ corresponds to 1/360 ͦx πr²

60 ͦ corresponds to 60 ͦ/360ͦ x πr²

Hence, the area of a sector subtending an angle θ at the centre of the circle is given by

Example

Find the area of the sector of a circle of radius 3cm if the angle subtended at the centre is 140 ͦ(take π=22/7)

Solution

Area A of a sector is given by

Here, r =3 cm and θ =

Therefore, A=

= 11 cm²

Example

The area of a sector of a circle is 38.5 cm². Find the radius of the circle if the angle subtended at the centre is  (Take π=22/7)

Solution

From the formula a = θ/360 x πr², we get 90/360 x 22/7 x r² = 38.5

Therefore, r² =

Thus, r = 7

Example

The area of a circle radius 63 cm is 4158 cm². Calculate the angle subtended at the centre of the circle. (Take π =22/7)

Using a =θ/360 x πr²,

Θ =

=

Surface area of solids

Consider a cuboid ABCDEFGH shown in the figure below. If the cuboid is cut through a plane parallel to the ends, the cut surface has the same shape and size as the end faces. PQRS is a plane. The plane is called the cross-section of the cuboid

A solid with uniform cross-section is called a prism. The following are some of the prisms. The following are some of the prisms.

The surface area of a prism is given by the sum of the area of the surfaces.

The figure below shows a cuboid of length l, breath b and height h. its area is given by;

A=2lb+2bh+2hl

=2(lb. + bh +hl)

For a cube offside 2cm;

A =2(3×2²)

=24 cm²

Example

Find the surface area of a triangular prism shown below.

Area of the triangular surfaces = ½ x5x12 x2cm²

=60 cm²

Area of the rectangular surfaces=20 x13 +5 x 20 +12 x20

=260 + 100 + 240  = 600cm²

Therefore, the total surface area= (60+600) cm²

=660 cm²

Cylinder

A prism with a circular cross-section is called a cylinder, see the figure below.

If you roll a piece of paper around the curved surface of a cylinder and open it out, you will get a rectangle whose breath is the circumference and length is the height of the cylinder. The ends are two circles. The surface area S of a cylinder with base and height h is therefore given by;

S=2πrh + 2πr²

Example

Find the surface area of a cylinder whose radius is 7.7 cm and height 12 cm.

Solution

S =2 π (7.7) x 12 + 2 π (7.7) cm²

=2 π (7.7) x 12 + (7.7) cm²

=2 x 7.7 π (12 + 7.7) cm²

=2 x 7.7 x π (19.7) cm²

=15.4π (19.7) cm²

=953.48 cm²

Area of irregular shapes

The area of irregular shape cannot be found accurately, but it can be estimated. As follows;

• Draw a grid of unit squares on the figure or copy the figure on such a grid, see the figure below

• Count all the unit squares fully enclosed within the figure.
• Count all partially enclosed unit squares and divide the total by two, i.e.., treat each one of them as half of a unit square.
• The sum of the numbers in (ii) and (ii) gives an estimate of the areas of the figure.

From the figure, the number of full squares is 9

Number of partial squares= 18

Total number of squares = 9 + 18/2

=18

Approximate area = 18 sq. units.

End of topic

Past KCSE Questions on the topic

• Calculate the area of the shaded region below, given that AC is an arc of a circle centre B. AB=BC=14cm CD=8cm and angle ABD = 75⁰        (4 mks)

2.)        The scale of a map is 1:50000. A lake on the map is 6.16cm². find the actual area of the lake in hactares.                                                                                                                                    (3mks)

3.)        The figure below is a rhombus ABCD of sides 4cm. BD is an arc of circle centre C. Given that ÐABC = 138⁰. Find the area of shaded region.                                                                    (3mks)

4.)        The figure below sows the shape of Kamau’s farm with dimensions shown in meters

Find the area of Kamau’s farm in hectares                                                                (3mks)

5.)        In the figure below AB and AC are tangents to the circle centre O at B and C respectively,

the angle AOC = 60⁰

Calculate

(a)  The length of AC

6.)        The figure below shows the floor of a hall. A part of this floor is in the shape of a rectangle of         length 20m and width 16m and the rest is a segment of a circle of radius 12m. Use the figure to           find:-

(a) The size of angle COD                                                                                                         (2mks)

(b) The area of figure DABCO                                                                                       (4mks)

(c) Area of sector ODC                                                                                                  (2mks)

(d) Area of the floor of the house.                                                                                             (2mks)

7.)        The circle below whose area is 18.05cm² circumscribes a triangle ABC where AB = 6.3cm, BC = 5.7cm and AC = 4.8cm. Find the area of the shaded part

8.)        In the figure below, PQRS is a rectangle in which PS=10k cm and PQ = 6k cm. M and N are midpoints of QR and RS respectively

1. Find the are of the shaded region (4 marks)
2. Given that the area of the triangle MNR = 30 cm². find the dimensions of the rectangle (2 marks)
3. Calculate the sizes of angles and            giving your answer to 2 decimal places                                                            (4 marks)

9.)        The figure below shows two circles each of radius 10.5 cm with centres A and B. the circles touch each other at T

Given that angle XAD =angle YBC = 160⁰ and lines XY, ATB and DC are parallel, calculate the area of:

1. The minor sector AXTD (2 marks)
2. Figure AXYBCD (6marks)
3. The shaded region (2 marks)

10.)      The floor of a room is in the shape of a rectangle 10.5 m long by 6 m wide. Square tiles of

length 30 cm are to be fitted onto the floor.

(a) Calculate the number of tiles needed for the floor.

(b) A dealer wishes to buy enough tiles for fifteen such rooms. The tiles are packed in cartons

each containing 20 tiles. The cost of each carton is Kshs. 800. Calculate

(i) the total cost of the tiles.

(ii) If in addition, the dealer spends Kshs. 2,000 and Kshs. 600 on transport and subsistence

respectively, at what price should he sell each carton in order to make a profit of 12.5%

(Give your answer to the nearest Kshs.)

11.)      The figure below is a circle of radius 5cm. Points A, B and C are the vertices of the triangle

ABC in which ÐABC = 60^o and ÐACB=50^o which is in the circle. Calculate the area of DABC )

12.)      Mr.Wanyama has a plot that is in a triangular form. The plot measures 170m, 190m

and 210m, but the altitudes of the plot as well as the angles are not known. Find the area

of the plot in hectares

13.)      Three sirens wail at intervals of thirty minutes, fifty minutes and thirty five minutes.

If they wail together at 7.18a.m on Monday, what time and day will they next wail together?

14.)      A farmer decides to put two-thirds of his farm under crops. Of this, he put a quarter under maize and four-fifths of the remainder under beans. The rest is planted with carrots.

If 0.9acres are under carrots, find the total area of the farm

CHAPTER FOURTEEN

Specific Objectives

By the end of the topic the learner should be able to:

• State units of volume
• Convert units of volume from one form to another
• Calculate volume of cubes, cuboids and cylinders
• State units of capacity
• Convert units of capacity from one form to another
• Relate volume to capacity
• Solve problems involving volume and capacity.

Content

• Units of volume
• Conversion of units of volume
• Volume of cubes, cuboids and cylinders
• Units of capacity
• Conversion of units of capacity
• Relationship between volume and capacity
• Solving problems involving volume and capacity

Introduction

Volume is the amount of space occupied by a solid object. The unit of volume is cubic units.

A cube of edge 1 cm has a volume of 1 cm x 1 cm x 1 cm =1 cm³.

Conversion of units of volume

A cube of side 1 m has a volume of 1 m³

But 1 m = 100 cm

1 m x 1 m x 1 m = 100 cm x 100 cm x 100 cm

Thus, 1 m = (0.01 x 0.01 x 0.01) m³

=0.000001m³

=1 x 10¯⁶m³

A cube side 1 cm has a volume of 1 cm³.

But 1 cm=10mm

1 cm x 1 cm x 1 cm = 10 mm x 10 mm x 10 mm

Thus, 1 cm³= 1000mm³

Volume of cubes, cuboids and cylinders

Cube

A cube is a solid having six plane square faces in which the angle between two adjacent faces is a right-angle.

Volume of a cube= area of base x height

=l ²x l

=l³

Cuboid

A cuboid is a solid with six faces which are not necessarily square.

Volume of a cuboid = length x width x height

=a sq. units x h

= ah cubic units.

Cylinder

This is a solid with a circular base.

Volume of a cylinder = area of base x height

=πr² x h

=πr²h cubic units

Example

Find the volume of a cuboid of length 5 cm, breadth 3 cm and  height 4 cm.

Solution

Area of its base = 5×4 cm²

Volume     =5x4x3 cm³

= 60 cm³

Example

Find the volume of a solid whose cross-section is a right- angled triangle of base 4 cm, height 5 cm and length 12 cm.

Solution

Area of cross-section =1/2 x4 x 5

=10 cm²

Therefore volume       =10 x 12

=120 cm³

Example

Find the volume of a cylinder with radius 1.4 m and height  13 m.

Solution

Area of cross-section= 22/7 x 1.4 x 1.4

=6.16 m²

Volume = 6.16 x 13

=80.08 m³

In general, volume v of a cylinder of radius r and length l given by v=πr²l

Capacity

Capacity is the ability of a container to hold fluids. The SI unit of capacity is litre (l)

Conversion of units to capacity

1 centiliter (cl)=10 millilitre (ml)

1 decilitre dl = 10 centilitre (cl)

1 litre (l) =10 decilitres (dl)

1 Decalitre (Dl) = 10 litres (l)

1 hectolitre (Hl) =10 decalitre(Dl)

1 kilolitre (kl) =10 hectolitres (Hl)

1 kilolitre (kl)= 1000 litres (l)

1 litre (l) =1000 millilitres (ml)

Relationship between  volume and capacity

A cubed of an edge 10 cm holds 1 litre of liquid.

1 litre =10 cm x 10cm x 10cm

= 1 000 cm³

1 m³ =10⁶ cm³

1 m³ =10³ litres.

End of topic

Past KCSE Questions on the topic

1. ) All the water is poured into a cylindrical container of circular radius 12cm. If the cylinder has height 45cm, calculate the surface area of the cylinder which is not in contact with water.

• The British government hired two planes to airlift football fans to South Africa for the World cup tournament. Each plane took 10 ½ hours to reach the destination.

Boeng 747 has carrying capacity of 300 people and consumes fuel at 120 litres per minute. It makes 5 trips at full capacity. Boeng 740 has carrying capacity of 140 people and consumes fuel at 200 litres per minute. It makes 8 trips at full capacity. If the government sponsored the fans one way at the cost of 800 dollars per fan, calculate:

(a) The total number of fans airlifted to South Africa.                                                    (2mks)

(b) The total cost of fuel used if one litre costs 0.3 dollars.                                             (4mks)

(c) The total collection in dollars made by each plane.                                                    (2mks)

(d) The net profit made by each plane.                                                                           (2mks)

• A rectangular water tank measures 2.6m by 4.8m at the base and has water to a height

of 3.2m.  Find the volume of water in litres that is in the tank

• Three litres of water (density1g/cm³) is added to twelve litres of alcohol (density 0.8g/cm³.

What is the density of the mixture?

• A rectangular tank whose internal dimensions are 2.2m by 1.4m by 1.7m is three fifth full

of milk.

(a) Calculate the volume of milk in litres

(b) The milk is packed in small packets in the shape of a right pyramid with an equilateral base triangle of sides 10cm. The vertical height of each packet is 13.6cm. Full packets obtained are sold at shs.30 per packet. Calculate:

(i) The volume in cm³ of each packet to the nearest whole number

(ii) The number of full packets of milk

(iii) The amount of money realized from the sale of milk

• An 890kg culvert is made of a hollow cylindrical material with outer radius of 76cm and an inner radius of 64cm. It crosses a road of width 3m, determine the density of the material ssused in its construction in Kg/m³ correct to 1 decimal place.

                                    CHAPTER FIFTEEN

Specific Objectives

By the end of the topic the learner should be able to:

• Define mass
• State units of mass
• Convert units of mass from one form to another
• Define weight
• State units of weight
• Distinguish mass and weight
• Relate volume, mass and density.

Content

• Mass and units of mass
• Weight and units of weight
• Density
• Problem solving involving real life experiences on mass, volume, density and weight.

Introduction

Mass

The mass of an object is the quantity of matter in it. Mass is constant quantity, wherever the object is, and matter is anything that occupies space. The three states of matter are solid, liquid and gas.

The SI unit of mass is the kilogram. Other common units are tone, gram and milligram.

The following table shows units of mass and their equivalent in kilograms.

Weight

The weight of an object on earth is the pull of the earth on it. The weight of any object varies from one place on the earth’s surface to the other. This is because the closure the object is to the centre of the earth, the more the gravitational pull, hence the more its weight. For example, an object weighs more at sea level then on top of a mountain.

Units of weight

The SI unit of weight is newton.  The pull of the earth, sun and the moon on an object is called the force of gravity due to the earth, sun and moon respectively. The force of gravity due to the earth on an object of mass 1kg is approximately equal to 9.8N. The strength of the earth’s gravitational pull (symbol ‘g’) on an object on the surface of the earth is about 9.8N/Kg.

Weight of  an object = mass of an object x gravitation

Weight N =mass kg x g N/kg

Density

The density of a substance is the mass of a unit cube of the substance.  A body of mass (m)kg and volume (v) m³ has:

• Density (d) = mass (m)/ density (d)
• Mass (m)= density (d) x volume (v)
• Volume (v) = mass (m) / density (d)

Units of density

The SI units of density is kg/m³. the other common unit is g/cm³

1 g/cm³ = 1 000kg/m³

Example

Find the mass of an ice cube of side 6 cm, if the density of the ice is 0.92 g/ cm³.

Solution

Volume of cube = 6x6x6 = 216 cm³

Mass = density x volume

=216 x 0.92

=198.72 g

Example

Find the volume of cork of mass 48 g. given that density of cork is 0.24 g/cm³

Solution

Volume = mass/density

=48/0.24

=200cm³

Example

The density of iron is 7.9 g/cm³. what is this density in kg/m³

Solution

1 g/cm³ = 1 000 kg/m³

7.9 g/cm³ =7.9 x 1000/1

= 7 900kg/m³

Example

A rectangular slab of glass measures 8 cm by 2 cm by 14 cm and has a mass of 610g. calculate the density of the glass in kg/m³

Solution

Volume of the slab = 8x 2×14

=224 cm³

Mass of the slab = 610 g

Density     = 610/244

= 2.5 x 1 000 kg  = 25 000kg/m³

End of topic

Past KCSE Questions on the topic

1.) A squared brass plate is 2mm thick and has a mass of 1.05kg. The density of brass is 8.4g/cm. Calculate the length of the plate in centimeters.                               (3mks)

2.) A sphere has a surface area 18cm². find its density if the sphere has a mass of 100g. (3mks)

3.) Nyahururu Municipal Council is to construct a floor of an open wholesale market whose area is 800m². the floor is to be covered with a slab of uniform thickness of 200mm. in order to make the slab, sand, cement and ballast are to be mixed such that their masses are in the ratio 3:2:3. The             mass of dry slab of volume 1m³ is 2000kg. Calculate

(a) (i) The volume of the slab                                                                                                    (2mks)

(ii) The mass of the dry slab.                                                                                                     (2mks)

(iii) The mass of cement to be used.                                                                                          (2mks)

(b) If one bag of the cement is 50kg, find the number of bags to be purchased.               (1mk)

(c) If a lorry carries 10 tonnes of ballast, calculate the number of lorries of ballast to be             purchased.                                                                                                                                (3mks)

4.)  A sphere has a surface area of 18.0cm². Find its density if the sphere has a mass of 100 grammes.

(3 mks)

• A piece of metal has a volume of 20 cm³ and a mass of 300g. Calculate the density of the metal

in kg/m³.

• 5 litres of water density 1g/cm³ is added to 8 litres of alcohol density 0.8g/cm³. Calculate the density of the mixture

CHAPTER TEN

Specific Objectives

By the end of the topic the learner should be able to:

• Convert units of time from one form to another
• Relate the 12 hour and 24 hour clock systems
• Read and interpret travel time-tables
• Solve problems involving travel time tables.

Content

• Units of time
• 12 hour and 24 hour clock systems
• Travel time-tables
• Problems involving travel time tables

Introduction

Units of time

1 week = 7 days

1 day = 24 hours

1 hour = 60 minutes

1 minutes = 60 seconds

Example

How many hours are there in one week?

Solution

1 week = 7 days

1 day =24 hours

1 week = (7 x24) hours

=168 hours

Example

Covert 3h 45 min into minutes

Solution

1 h = 60 min

3 h = (3×60) min

3h 45min =((60×3) + 45) min

=(180+45) min

=225 min

Example

Express 4h 15 min in sec

Solution

1 hour = 60 min

1 min= 60 sec

4h 15 min=(4×60+15 ) min

=240+15 min

=255 min

=255 x 60 sec

=15 300 sec.

The 12 and the 24 hour systems

In the 12 hour system, time is counted from midnight. The time from midnight to midday is written as am .while that from midday to midnight is written as pm.

In the 24 hour system, time is counted from midnight and expressed in hours.

Travel time table

Travel timetables shows the expected arrival and departure time for vehicles. Ships, aeroplanes, trains.

Example

The table below shows a timetable for a public service vehicle plying between two towns A and D via towns B and C.

• What time does the vehicle leave town A?
• At what time does it arrive in town D?
• How long does it take to travel from town A to
• What time does the vehicle takes to travel from town C to  D?

Solution

• 20 A.M
• 00 P.M
• Arrival time in town D was 4.00 p.m. it departure from town A was 8.20 a.m.

Time taken= (12.00-8.20 +4 h)

=3 h 40 min +4 h

= 7 h 40 min

• The vehicle arrived in town D at 4.00 p.m. it departed from town C at 2.50 p.m.

Time taken = 4.00-2.50

=1 h  10 min

End of topic

Past KCSE Questions on the topic

1.)       A van travelled from Kitale to Kisumu a distance of 160km. The average speed of the van for the first 100km was 40km/h and the remaining part of the journey its average speed was 30km/h. Calculate the average speed for the whole journey.                                                                   (3 mks)

2.)        A watch which looses a half-minute every hour was set to read the correct time at 0545h on Monday. Determine the time, in the 12 hour system, the watch will show on the following Friday at 1945h.

3.)        The timetable below shows the departure and arrival time for a bus plying between two towns   M and R, 300km apart    0710982617

(a) How long does the bus take to travel from town M to R?

(b) What is the average speed for the whole journey?

                     CHAPTER SEVENTEEN

Specific Objectives

By the end of the topic the learner should be able to:

• solve linear equations in one unknown
• solve simultaneous linear equations by substitution and elimination
• Linear equations in one and two unknown.

Content

• Linear equations in one unknown
• Simultaneous linear equations
• Linear equations in one and two unknowns from given real life situations

Introduction

Linear equations are straight line equations involving one or two unknowns. In this chapter, we will deal with the formation and solving of such equations consider the following cases.

Example

Solve for the unknowns in each of the following equations

3x + 4 = 10

– 2 = 4

= 5/4

Solution

3x + 4 =10

3x + 4 – 4 = 10 – 4   (to make x the subject subtract 4 on both sides)

3x = 6

X = 2

– 2 = 4

– 2 + 2 = 4 + 2     (to make x the subject add 2 to both sides)

= 6

X = 18

=

3 x ( =  x 3

P+ 5 =

4(p + 5) =

4p + 20 = 15

4p = -5

P =

= – 1 ¼

Solving an equation with fractions or decimals, there is an option of clearing the fractions or decimals in order to create a simpler equation involving whole numbers.

1. To clear fractions, multiply both sides of the equation (distributing to all terms) by the LCD of all the fractions.
2. To clear decimals, multiply both sides of the equation (distributing to all terms) by the lowest power of 10 that will make all decimals whole numbers.

Steps for Solving a Linear Equation in One Variable:

1. Simplify both sides of the equation.
2. Use the addition or subtraction properties of equality to collect the variable terms on one side of the equation and the constant terms on the other.
3. Use the multiplication or division properties of equality to make the coefficient of the variable term equal to 1.
4. Check your answer by substituting your solution into the original equation.

Note:

All other linear equations which have only one solution are called conditional.

Example

Solve for the unknown in each in of the following equations

• –   = 1/8
• =

Solution

• –   = 1/8 x 24  ( multiply both sides by the L.C.M of 2 ,3 and 8)

−8 (x – 2) = 3

− 8x – 16 =3

4x = -25

• =(multiply both sides by the L.C.M of 2 5 and 4)

30y- 4 (14y -3) = 5 (y – 4)

-26 + 12 = 5y -20

31 y = 32

Problems leading to Linear equations

Equations are very useful in solving problems. The basic technique is to determine what quantity it is that we are trying to find and make that the unknown. We then translate the problem into an equation and solve it. You should always try to minimize the number of unknowns. For example, if we are told that a piece of rope 8 metres long is cut in two and one piece is x metres, then we can write the remaining piece as (8 – x) metres, rather than introducing a second unknown.

Word problems

Equations arise in everyday life. For example Mary bought a number of oranges from Anita’s kiosk. She then went to Marks kiosk and bought the same number of oranges. Mark them gave her three more oranges. The oranges from the two kiosks were wrapped in different paper bags. On reaching her house, she found that a quarter of the first Lot oranges and a fifth of the second were bad. If in total six oranges were bad, find how many oranges she bought from Anita’s kiosk.

Solution

Let the number of oranges bought at Anita’s kiosk be x.

Then, the number of oranges obtained from Marks kiosk will be x +3.

Number of Bad oranges from Marks kiosk was.

Total number of Bad oranges is equal to =

Thus, =  +

Multiply each term of the equation by 20 (L.C.M of 4 and 5) to get rid of the denominator.

=20 x

5x + 4 (x + 3) = 120

5x + 4x + 12 =120 (Removing brackets)

Subtracting 12 from both sides.

9x =108

X = 12

Thus, the number of oranges bought from Anita’s kiosk was 12.

Note:

If any operation is performed on one side of an equation,it must also be performed on the other side.

Example

Solve for x in the equation:

Solution

Eliminate the fractions by multiplying each term by 6 (L.C.M, of 2 and 3 ).

6x (

3(x + 3) – 2 (x – 4) = 24

(note the change in sigh when the bracket are removed)

Linear Equations in Two Unknowns

Many problems involve finding values of two or more unknowns. These are often linked via a number of linear equations. For example, if I tell you that the sum of two numbers is 89 and their difference is 33, we can let the larger number be x and the smaller one y and write the given information as a pair of equations:

x + y = 89 (1)

x – Y = 33. (2)

These are called simultaneous equations since we seek values of x and y that makes both equations true simultaneously. In this case, if we add the equations we obtain 2x = 122, so x = 61. We can then substitute this value back into either equation, say the first, then 61 + y = 89 giving y = 28.

Example

The cost of two skirts and three blouses is sh 600.If the cost of one skirt and two blouses of the same quality sh 350,find the cost of each item.

Solution

Let the cost of one skirt be x shillings and that of one blouse be y shillings. The cost of two skirts and three blouses is 2x + 3y shillings.

The cost of one skirt and two blouses is x + 2y shillings.

So, 2x + 3y = 600……………….. (I)

X + 2y = 350 ……………………….. (II)

Multiplying equation (II) by 2 to get equation (III).

2x + 4y = 700 ……………… (III).

2x + 3y = 600…………………(I)

Subtracting equation (I) from (II), y = 100.

From equation (II),

X + 2y = 350 but y = 100

X + 200 = 350

X = 150

Thus the cost of one skirt is 150 shillings and that of a blouse is 100 shillings.

In solving the problem above, we reduced the equations from two unknowns to a single unknown in y by eliminating. This is the elimination method of solving simultaneous equations.

Examples

Solutions

Adding I to II

2a = 12

Subtracting II from I ;

2b = 2

.

Find the value of b

To eliminate (I) by 5 and (II) by 3 to get (III) and (IV) respectively and subtracting (IV) from (III);

2y = 6

Therefore y = 3

Substituting y = 3 in (I);

3x + 12 = 18

Therefore x =2

Note that the L.C.M of 3 and 5 is 15.

To eliminate y;

Multiplying (I) by 3, (II) by 2 to get (V) and (VI) and Subtracting (V) from (VI);

Subtracting x = 2 in (ii);

10 + 6y = 28

6y = 18

Therefore y = 3.

Note;

• It is advisable to study the equation and decide which variable is easier to eliminate.
• It is necessary to check your solution by substituting into the original equations.

Solution by substitution

Taking equation (II) alone;

Subtracting 2y from both sides;

Substituting this value of x in equation;

2(350 – 2y) + 3y = 600

700 – 4y + 3y = 600

Y = 100

Substituting this value of y in equation;

This method of solving simultaneous equations is called the substitution method

End of topic

Past KCSE Questions on the topic

1. A cloth dealer sold 3 shirts and 2 trousers for Kshs 840 and 4 shirts and 5 trousers for Kshs 1680 find the cost of 1 shirt and the cost of 1 trouser
2. Solve the simultaneous equations

2x – y = 3

x² – xy = -4

3. The cost of 5 skirts and blouses is Kshs 1750. Mueni bought three of the skirts and one of the blouses for Kshs 850. Find the cost of each item.
4. Akinyi bought three cups and four spoons for Kshs 324. Wanjiru bought five cups and Fatuma bought two spoons of the same type as those bought by Akinyi, Wanjiku paid Kshs 228 more than Fatuma. Find the price of each cup and each spoon.
5. Mary has 21 coins whose total value is Kshs. 72. There are twice as many five shillings coins as there are ten shilling coins. The rest one shillings coins. Find the number of ten shillings coins that Mary has. ( 4 mks)
6. The mass of 6 similar art books and 4 similar biology books is 7.2 kg. The mass of 2 such art books and 3 such biology books is 3.4 kg. Find the mass of one art book  and the mass of one  biology  book
7. Karani bought 4 pencils and 6 biros – pens for Kshs 66 and Tachora bought 2 pencils and 5 biro pens for Kshs 51.

(a)        Find the price of each item

(b)        Musoma spent Kshs. 228 to buy the same type of pencils and biro – pens if the number of biro pens he bought were 4 more than the number of pencils, find the number of pencils bought.

8. Solve the simultaneous equations below

2x – 3y = 5

-x + 2y = -3

9. The length of a room is 4 metres longer than its width. Find the length of the room if its area is 32m²
10. Hadija and Kagendo bought the same types of pens and exercise books from the same types of pens and exercise books from the same shop. Hadija bought 2 pens and 3 exercise books for Kshs 78. Kagendo bought 3 pens and 4 exercise books for Kshs 108.

Calculate the cost of each item

11. In fourteen years time, a mother will be twice as old as her son. Four years ago, the sum of their ages was 30 years. Find how old the mother was, when the son was born.
12. Three years ago Juma was three times as old as Ali. In two years time the sum of their ages will be 62. Determine their ages.
13. Two pairs of trousers and three shirts costs a total of Kshs 390. Five such pairs of trousers and two shirts cost a total of Kshs 810. Find the price of a pair of trousers and a shirt.
14. A shopkeeper sells two- types of pangas type x and type y. Twelve  x pangas and five type y pangas cost Kshs 1260, while nine type x pangas and fifteen type y pangas cost  Mugala bought eighteen type y pangas. How much did he pay for them?

CHAPTER EIGHTEEN

Specific Objectives

By the end of the topic the learner should be able to:

• State the currencies of different countries
• Convert currency from one form into another given the exchange rates
• Calculate profit and loss
• Express profit and loss as percentages
• Calculate discount and commission
• Express discount and commission as percentage.

Content

• Currency
• Current currency exchange rates
• Currency conversion
• Profit and loss
• Percentage profit and loss
• Discounts and commissions
• Percentage discounts and commissions

Introduction

In commercial arithmetic we deal with calculations involving business transaction. The medium of any business transactions is usually called the currency. The Kenya currency consist of a basic unit called a shilling.100 cents are equivalent to one Kenyan shillings, while a Kenyan pound is equivalent to twenty Kenya shillings.

Currency Exchange Rates

The Kenyan currency cannot be used for business transactions in other countries. To facilitate international trade, many currencies have been given different values relative to another. These are known as exchange rates.

The table below shows the exchange rates of major international currencies at the close of business on a certain day in the year 2015.The buying and selling column represents the rates at which banks buy and sell these currencies.

Note

The rates are not always fixed and they keep on charging. When changing the Kenyan currency to foreign currency, the bank sells to you. Therefore, we use the selling column rate. Conversely when changing foreign currency to Kenyan Currency, the bank buys from you, so we use the buying column rate.

Currency        Buying             Selling

DOLLAR                              102.1472           102.3324

STG POUND                        154.0278         154.3617

EURO                                  109.6072         109.8522

SA RAND                            7.3332             7.3486

KES / USHS                        33.0785           33.2363

KES / TSHS                        20.9123            21.0481

KES / RWF                          7.2313             7.3423

AE DIRHAM                         27.8073         27.8653

CAN $                                  77.6018          77.7661

JAPNESE YEN                     84.0234         84.1964

SAUDI RIYAL                     27.2284          27.2959

CHINESE YUAN                  16.0778          16.1082

AUSTRALIAN $                  71.8606          72.0420

Example

Convert each of the following currencies to its stated equivalent

• Us $305 to Ksh
• 530 Dirham to euro

Solution

• The bank buys Us 1 at Ksh 1472

Therefore US $ 305 = Ksh (102.1472 x 305)

= Ksh 31,154.896

= Ksh 31,154.00 (To the nearest shillings)

The bank buys 1 Dirham at Ksh 27.8073

Therefore 530 Dirham = Ksh (21.8073 x 530)

= Ksh 11, 557.00   (To the nearest shillings)

The bank sells 1 Euro at 109.8522

Therefore 530 Dirham = 11, 557/109.8522

= 105.170 Euros

Example

During a certain month, the exchange rates in a bank were as follows;