Fractions - Mathematics Form 1 Notes

• Introduction
• Proper Fraction
• Improper Fraction
• Mixed Fraction
• Changing a Mixed Number to an Improper Fraction
• Changing an Improper Fraction to a Mixed Number
• Comparing Fractions
  • Equivalent Fractions
• Operation on Fractions
  • Addition and Subtraction
  • Multiplying Simple Fractions
  • Multiplying Mixed Numbers
  • Dividing Simple Fractions
  • Dividing Mixed Numbers
  • Order of Operations on Fractions
• Past KCSE Questions on the Topic

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Introduction

• A fraction is written in the form ^a/_b 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.
  a→numerator
  b→denominator

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Proper Fraction

• In a proper fraction the numerator is smaller than the denominator. E.g.
  2, 1
  3  4

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Improper Fraction

• The numerator is bigger than or equal to denominator. E.g.
  7, 15, 9
  3   6   2

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Mixed Fraction

• An improper fraction written as the sum of an integer and a proper fraction. For example
  7 = 2 + 1
  3          3
  = 2¹/₃

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Changing a Mixed Number to an Improper Fraction

• Mixed number – 4²/₃ (contains a whole number and a fraction)
• Improper fraction - ¹⁴/₃ (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

3¹/₈= 3 × 8 + 1 = 25
= 25
    8

Example

4⁴/₉ = 4 × 9 + 4 = 40
= 40
    9

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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 = 47 ÷ 5 or

 

Example

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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^x6 1^x3 5^x2 2^x4
2^x6 4^x3 6^x2 3^x4

Step 3: 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

 5 × 100 = 41.67%
12

7 × 100 = 233.3%
3
11 × 100 = 220%
 5

9 × 100 = 225%
4

⁷/₃, ⁹/₄, ¹¹ /₅, ⁵/₁₂

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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 (simplify the answer)

Example

¹/₂ + ⁷/₈ = Common denominator is 8 because both 2 and 8 will go into 8
1 + 7 = 4 + 7
2    8       8

¹¹/₈ which simplifies to 1³/₈

Example

4³/₅ – ¹/₄ = Common denominator is 20 because both 4 and 5 will go into 20

4³/₅ = 4¹²/₂₀

–¹/₄ = ⁵/₂₀

4¹²/₂₀ − ⁵/₂₀ = 4⁷/₂₀

Or

4³/₅ – ¹/₄= 4 ^{(12 - 5)/₂₀} = 4⁷/₂

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

Examples

5²/₃ + 1⁴/₅

Solution

5²/₃ as an improper fraction is ¹⁷/₃

1⁴/₅ as an improper fraction is ⁹/₅

adding the improper fraction 

17 + 9 =  85 + 27 = 112
 3     5       15            15

converting ¹¹²/₁₅ to a mixed fraction we get 7⁷/₁₅

Or

we can add the whole numbers and fractions separately

5²/₃ + 1⁴/₅ = 5 + ²/₃ + 1 + ⁴/₅
= (5+1) + ²/₃ + ⁴/₅
= 6 + 10+12
             15
= 6 + 22
         15
= 6 + 1⁷/₁₅ = 7⁷/₁₅

Example

Evaluate -²/₃ + ^-1/₅

 

 

Solution

−2 + −1 = −16−3 = −19
  3       5        24         24

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

¹/₇ × ⁴/₁₂ = ⁴/₁₂ which reduces to ²/₂₁

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

2¹/₃ × 1¹/₂ = ⁷/₃ × ³/₂ = ²¹/₆
Which then reduces to 3¹/₂

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:
  ⁷/₃ × ³/₂ = ⁷/₂

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

¹/₈ ÷ ²/₃= becomes ¹/₈ × ³/₂ which when solved is ³/₁₆

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

3³/₄÷ 2⁵/₆ = becomes ¹⁵/₄ ÷ ¹⁷/₆ becomes ¹⁵/₄ × ⁶/₁₇
Which when solved is
15 ×  6 ³ = 45  which simplifies to 1¹¹/₃₄
 4 ²   17      34

Order of Operations on Fractions

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

Example

15 ÷ ¹/₄ of 12 = 15 ÷ (¹/₄ x 12)(we start with of then division)
= 15 ÷ 3
= 5

Example

¹/₆ + ¹/₂ x {³/₈ + (¹/₃ − ¹/₄)}

Solution

1 − 1= 4 − 1 = 1 (we start with bracket)
3    4      12      12

{³/₈ + ¹/₁₂} = ¹¹/24 (We then work out the outer bracket)
¹/₆ + ¹/₂ x ¹¹/₂₄ = ¹/₆ + ¹¹/₄₈ (We then work out the multiplication)
¹/₆ + ¹¹/₄₈ = ¹⁹/₄₈ (Addition comes last here)

Example

Evaluate

Solution

We first work out this first

1+ 1= 3+2 = 5
2   3      6      6
¹/₇ of (²/₅ − ¹/₆) = ¹/₇ x ⁷/₃₀ = ¹/₃₀
⁵/₆ ×30 = 25

Therefore
 

= 25 + ½ = 25½

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: ¹/₂{³/₅ + ¹/₄(⁷/₃ − ³/₇)of 1¹/₂ ÷5}

Solution

=¹/₂{³/₅ + ¹/₄(⁴⁰/₂₁) of 1¹/₂ ÷5}
=¹/₂{³/₅ + ¹/₄ x ⁴⁰/₂₁ x ³/₂ ÷5}
=¹/₂(³/₅ + ¹⁰/₂₁ x ³/₂ ÷5)
=¹/₂(³/₅ + ⁵/₃₅)
=¹/₂(21+5) = ¹/₂ x ²⁶/₃₅ = ¹³/₃₅
          35

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 + 1 − 1 = 20 + 12 − 1 5
3    5     4           60
= ¹⁷/₆₀
¹⁷/₆₀=1 hr
1 = 1 x ⁶⁰/₁₇
⁶⁰/₁₇ = 3.52941 1 8
= 3.529 hrs.

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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 ³/₈ of {7³/₅ − ¹/₃(1¹/₄ + 3¹/₃) × 2²/₅}
                                       . .
4. Convert the recurring 2.18 decimal into fraction
5. Simplify (0.00243)^-2/₅ × (0.0009)^½ 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 
   
   
10. Write the recurring decimal 0.3 as a fraction
11. Evaluate without using a calculator.
    
    
12. Without using tables or calculators evaluate.
    
    
13. Without using tables or calculator, evaluate the following.
    
                      .        . .
14. Express 1.93 + 0.25 as a single fraction
15. Simplify 
    
    
16. Evaluate:
    
17. Without using a calculator or table, work out the following leaving the answer as a mixed number in its simplest form:-
    
18. Work out the following, giving the answer as a mixed number in its simplest form.
    
19. Without using a calculator, evaluate:
    
    leaving the answer as a fraction in its simplest form
20. 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

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