MULTIPLYING AND DIVIDING FRACTIONS WITHOUT
TEARS
Two or more fractions can evaluated by addition, subtraction,
multiplication, division or a combination of two or more of the above operations.
Fractions often
appear easy to evaluate from first sight but a careful attention often indicate
the contrary. This was revealed by a survey I carried out in a college form two
class some two months back. Most of the students acknowledged that fractions
are very easy to evaluate but were not able to explain why some still got some
of the exercises wrong.
This is one of the reasons why I have decided to treat
fractions and many other topics in Number Theory here.
This means before solving any fraction, it must be well understood in
order to apply the correct method. Some students give wrong solutions to
questions involving fractions because they often assume that it is easy hence
begin solving without interpreting the question.
In this write-up, you will
learn how to multiply, and divide fractions that: have the same denominators,
have different denominators, contain mixed fractions.
You will be given a
detail guide on what to do on how to simplify different fractions.
Each explanation ends with some practice exercises for you to solve and
evaluate your level of understanding.
There will also be short VIDEOS to
explain some of the exercises. This is to ensure that the concepts are well
understood. You can make request for further explanation if a concept is not
understood. Use the comment section for this.
Without any waste of
time, let us get into business right away. Follow the explanations with
attention and post your questions in the comments sections of this blog and be
patient for your answer.
MULTIPLICATION OF
FRACTIONS
To multiply fractions the
numerators are multiplied separately from the denominators. What I mean is that
the denominator is multiplied together and the numerators multiplied together.
EXAMPLE 1 : Evaluate the
following fractions simplifying your results as far as possible.
i)
2 × 1 ii) 3 × 4 iii)
2 × 4 iv) 6
× 1
3 3 5
5 3 11
7 8
Solution
i) 2
× 1 = 2×1
3 3
3×3
2 × 1 = 2
3
3 9 answer.
ii) 3 ×
4 = 3 × 4
5
5 5×5
3 × 4
= 12
5 5 25 answer.
iii) 2 × 4 = 2×4
3 11 3×11
2 × 4 = 8
3 11
33 answer.
iv) 6 × 1 = 6×1
7 8 7×8
6 × 1 = 6
7 8 56
6 × 1 = 3
7 8 28 answer.
MULTIPLICATION OF
MIXED FRACTIONS
To multiply mixed fractions,
convert first to improper fractions and solve as above.
EXAMPLE 2: Evaluate and
simplify the following.
i)
22 × 11 ii) 31 × 23 iii) 53 × 11
3 5 2 5 7 4
SOLUTION
i)
22 × 11 (convert to improper fractions and
solve as above.)
3 5
22 × 11 = 2+ (2×3) × 1+(1×5)
3 5 3 5
22 × 11 = 2
+6 × 1+5
3 5 3 5
22 × 11 = 8
×
6
3 5 3
5
22 × 11 = 48 or
33 or 31
3 5 15 15 5 . Answer
ii)
31 × 23 (
convert to improper fraction)
2
5
31 × 23 = 1+ ( 3×2) × 3+( 2× 5)
2 5 2 5
31 × 23 = 1+6 × 3+10
2 5 2 5
= 7 × 13
2 5
= 7×13
2×5
31 × 23 = 91
0r 91
2 5 10 10
EXERCISE FOR PRACTICE
i)
53 × 11 ii)
21 × 41 iii)
41 × 21 iv) 51 × 21
7 4 3 2 2 4 7 9
ANSWERS
i)
3 ii)
63 iii) 81 iv 1054
6 8 63
DIVISION OF FRACTIONS
To divide two fractions, invert the divisor and change the operation
from division to multiplication. Proceed as though it was multiplication as
done above.
EXAMPLE 1: Evaluate each of the following simplifying your result as
far as possible.
i)
1 ÷ 4
ii) 5 ÷ 1
ii) 3 ÷
2 iv) 6 ÷ 1
3 5 6 2 4 3 7 8
Solution
i)
1 ÷ 4
( invert the divisor and change operation to multiplication)
3 5
1 ÷ 4 = 1
× 5
3 5 3
4
1 ÷ 4
= 5
3 5
12
ii)
5 ÷
1 = 5 × 2
6 2 6
1
5 ÷ 1 = 10
or 5 or 12
6 2 6
3 3
iii)
3
÷ 2 = 3 × 3
4 3
4 2
3 ÷ 2
= 9 or 3
4 3
12 4
iv)
6
÷ 1 = 6 ×8
7 8 7
1
6 ÷ 1
= 48
or 66
7 8
7 7
EXERCISE FOR PRACTICE
Evaluate and simplify each of the following
i)
3 ÷ 2 ii) 3
÷ 5 iii) 3 ÷ 11
8 9 10
9 4 12
DIVISION OF MIXED FRACTIONS
To divide mixed fractions, convert all to improper fractions, invert
the divisor then change the operation from division to multiplication.
Example 1: Evaluate and simplify each of the following
i)
33 ÷ 21 ,
ii) 23 ÷ 15 ,
iii) 31 ÷ 23 .
4 2 4 11 2 5
Solution
i)
33 ÷21 (
convert to improper fractions)
4 2
33 ÷ 21 = 3+(3×4) ÷ 1+(
2×2)
4 2 4 2
33 ÷ 21 = 3 + 12 ÷
1+4
4 2 4 2
33 ÷ 21 = 15 ÷ 5 (invert the divisor and change operation to
multiplication)
4 2 4 2
= 15 × 2
4 5
33 ÷ 21 = 30 or 6 or 11
4 2 20 5 5
answer
ii)
23 ÷ 15 ( convert to improper fractions)
4
11
23 ÷ 15 = 3+
(2×4) ÷ 5+(
1×11)
4
11 4 11
23 ÷ 15 = 3 + 8 ÷ 5+11
4 11 4 11
23 ÷15 = 11 ÷ 16
4 11 4 11
= 11 ×
11
4 16
23 ÷ 15 = 121
4 11 64
. answer
iii)
31 ÷ 23 = 1+( 3×2) ÷ 3+( 2×5)
2
5 2 5
31 ÷ 23 = 1+6 ÷ 3+10
2 5 2 5
= 7 ÷ 13
2 5
= 7
× 5
2 13
31 ÷ 23 = 35 or 1 9
2
5 26 26 answer
EXERCISE FOR PRACTICE
Evaluate and simplify each of the following.
i)
22 ÷ 41 ii)
51 ÷ 21 iii)
35 ÷ 52 iv) 62 ÷ 41
3
2 7 9 7 3 5 4
ANSWERS i) 16
, ii) 324
, iii) 324 , iv)
143
27 133 133 85
To
simplify fractions involving many operations, apply BODMAS and move step by
step. In applying BODMAS start from “B” and end with “S”. Note the following:
B= brackets; evaluate fractions in brackets first.
O = “of” means multiplication
D= divide
M= Multiply
A= Add
S= Subtract.
The operations are performed starting from “B” and ending with “S”.
Example1: Evaluate the following and simplify your result as far as
possible
i)
( 4 - 1) ÷ 2 , ii) ( 3 + 1) × 5 iii)
(32 – 21) × 1 1 iv) (21 + 13 ) ÷ 32
7 3 5
5 3 7 3
2 7 2 4 5
Solution
i)
(4
- 1 ) ÷ 2 ( using
BODMAS evaluate the fractions in the brackets first)
7 3
5
(4 – 1) ÷
2 = 3(4)-7(1) ÷ 2
7 3
5 (7)(3) 5
= 12 – 7 ÷ 2
21 5
= 5 × 5
21 2
( 4- 1
) ÷ 2 = 25
7 3
5 42 answer.
ii)
14
, iii) 35 .
75 42
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