2
Fractions and Decimals
introduction
In class VI, you have already learnt about fractions and decimals. The study of fractions included
type of fractions, equivalent fractions, comparison of fractions, representation of fractions on a
number line and their ordering, operations of addition, subtraction, multiplication and division on
fractions. The study of decimals included conversion of decimals to fraction and fractions to
decimals, comparison of decimals, representation of decimals on a number line and their ordering,
addition and subtraction of decimals.
In this chapter, we shall review, revise and strengthen what we have done about fractions and
decimals in our previous class. We shall learn multiplication and division of decimals, simplification
(by use of BODMAS) of numerical expressions involving fractions and decimals, uses of fractions
and decimals in real life problems.
Fractions
A fraction is a number which represents a part of whole. The whole may be a single object or a
group of objects.
p
q
A fraction is written as
11
7
where p, q are whole numbers and q ≠ 0. Numbers such as
are all fractions. In the fraction
p
q
1
2
,
2
3
,
4
5
,
where p, q are whole numbers and q ≠ 0, the horizontal
line is called the division line. The number below the division line i.e. q is called denominator
(or divisor) and it tells us into how many equal parts a whole is divided. The number above the
division line i.e. p is called numerator (or dividend) and it tells us how many equal parts are
taken.
classification of fractions
Proper fraction — a fraction whose numerator is greater than zero but less than its denominator
is called a proper fraction.
For example,
3
5
4
7
,
43
128
,
are all proper fractions.
Improper fraction — a fraction whose numerator is equal to or greater than its denominator is
called an improper fraction.
For example,
12
5
,
11
7
,
217
89
are all improper fractions.
Notice that every natural number can be written as a fraction.
For example, 7 =
7
1
, 39 =
39
1
.
So every natural number is an improper fraction.
Learning Mathematics–VII
28
Mixed fraction (or mixed number) — a number that consists of two parts, a natural number
and a proper fraction, is called a mixed fraction (or mixed number).
5
7
For example, 3 , 8
In fact, 3
5
7
=3+
9
11
5
7
are mixed fractions.
and 8
9
11
=8+
9
11
.
Every mixed fraction can be written as an improper fraction and every improper fraction can
be written as a mixed fraction.
5
7
5
7
For example, 3 = 3 +
51
11
=
=
4 × 11 + 7
11
3× 7 + 5
7
=
26
7
=4+
7
11
=4
and
7
11
.
11 51
– 44
7
4
Decimal fraction — a fraction whose denominator is 10, 100, 1000, .... is called a decimal
fraction.
For example,
3
10
,
327
100
,
21
1000
are decimal fractions.
Vulgar fraction — a fraction whose denominator is a number other than 10, 100, 1000, ... is
called a vulgar fraction.
For example,
7
11
,
5
214
,
17
80
,
231
130
are vulgar fractions.
Like fractions — two (or more) fractions having same denominator are called like fractions.
For example,
3
11
,
15
11
,
9
11
,
45
11
are like fractions.
Unlike fractions — two (or more) fractions having different denominators are called unlike
fractions.
For example,
3 7 9 39
, , ,
5 11 4 14
are unlike fractions.
Equivalent fractions — two (or more) fractions are called equivalent fractions if they have same
value.
Simplest (irreducible) form of a fraction — if the numerator and denominator of a fraction
have no common factor (except 1), then the fraction is said to be in its simplest (irreducible) form
or in lowest terms.
Basic property of fractions
The value of a fraction does not change if the numerator and the denominator of a fraction are
(i)multiplied by the same (non-zero) number
(ii)divided by the same (non-zero) number.
Dividing the numerator and denominator of a fraction by the same number is usually called
cancelling.
Example 1. Write the natural numbers from 3 to 19. What fraction of them are even
numbers?
Solution. Natural numbers from 3 to 19 are: 3, 4, 5, 6, …, 19 and their number = 17.
Out of these, the even numbers are: 4, 6, 8, 10, 12, 14, 16, 18 and their number = 8.
∴ The required fraction =
8
17
.
Fractions and Decimals
29
2
...
Example 2. Fill in the missing number in 5 = 45 .
Solution. The denominator in the second fraction is 45.
To get 45 from 5, we have to multiply 5 by 9.
∴ To make both fractions equal, we multiply the numerator and denominator of the first fraction
by 9.
So
2
5
=
2×9
5×9
=
18
45
.
Example 3. Reduce the following fractions in their lowest terms:
(i)
75
135
(ii)
660
168
.
Solution.
(i)H.C.F. of 75 and 135 is 15 (obtain it).
Divide the numerator and denominator of the given fraction by 15.
75
135
75 ÷ 15
135 ÷ 15
=
=
5
9
.
Alternative. Express each of the numerator and denominator of the given fraction as the product
of primes. Then cancel the factors that are common to both numerator and denominator.
660
(ii) 168 =
3× 5× 5
75
=
3× 3× 3× 5
135
2 × 2 × 3 × 5 × 11
2 × 2 × 2 × 3× 7
=
5 × 11
2×7
5
9
=
=
.
55
.
14
Conversion of given fractions to equivalent like fractions
To convert two (or more) fractions to equivalent like fractions, proceed as under:
(i)find the L.C.M. of the denominators of the given fractions.
(ii)multiply the numerator and denominator of each fraction by such a number so that the
denominator of each fraction is the L.C.M.
Example 4. Convert
7 17 41
, ,
8 12 45
to equivalent like fractions.
Solution. L.C.M. of 8, 12, 45= 2 × 2 × 3 × 2 × 15
= 360
7
8
17
12
41
45
=
=
=
2
2
3
7 × 45
315
=
,
8 × 45
360
8,
4,
2,
2,
12,
6,
3,
1,
45
45
45
15
17 × 30
510
=
,
12 × 30
360
41 × 8
45 × 8
=
328
360
.
Thus, the given fractions are equivalent to
Comparison of fractions
315 510 328
,
,
360 360 360
respectively.
To compare two (or more) fractions, proceed as under:
(i)convert all fractions into equivalent like fractions.
(ii)the fraction having the greater numerator is greater.
Example 5. Arrange the fractions
5 11 13
, ,
6 16 18
in ascending order.
Solution. L.C.M. of 6, 16, 18 = 2 × 3 × 8 × 3 = 144.
2
3
6,
3,
1,
16,
8,
8,
18
9
3
Learning MatheMatics–Vii
30
Write the given fractions as equivalent like fractions.
5
6
=
5 × 24
6 × 24
=
120 11
,
144 16
As 99 < 104 < 120
=
⇒
11 × 9
16 × 9
99
144
=
<
99 13
,
144 18
104
144
<
=
13 × 8
18 × 8
120
144
⇒
=
11
16
Hence, the given fractions in ascending order are
104
144
<
.
13
18
<
11 13 5
, ,
16 18 6
Ascending
means smaller
to greater
5.
6
.
exercise 2.1
1. What fraction of each of the following figure is shaded part?
(i)
(ii)
(iii)
(iv)
2. What fraction of an hour is 35 minutes?
3. Convert the following fractions into improper fractions:
7
(ii) 5 4
(iii) 7 5 .
(i) 2
9
11
14
4. Convert the following fractions into mixed fractions :
101
94
73
(ii)
(iii)
.
(i)
6
13
8
5. Fill in the missing numbers in the following equivalent fractions:
3
...
5
30
...
56
.
(i)
=
(ii)
=
(iii)
=
7
35
...
18
9
72
6. Reduce the following fractions to their simplest form:
48
276
72
(i)
(ii)
(iii)
72
115
336
7. Convert the following fractions into equivalent like fractions:
5 7
7
9
19
.
(i) 3 , ,
(ii)
,
,
4
6
7
8
8
25
10
40
8. Arrange the given fractions in descending order:
(ii) 1 , 3 , 7 .
(i) 2 , 2 , 8
9 3 21
5 7 10
9. Arrange the given fractions in ascending order:
5 3
13 8 17 7
(i) , , 9 , 20
(ii)
, , , .
14 21
18
15
24 12
addition/subtraction of fractions
To add/subtract two (or more) fractions, proceed as under:
(i) Convert the mixed fractions (if any) to improper fractions.
(ii) Convert all the fractions into equivalent like fractions.
(iii) Combine the numerators of all these like fractions with their proper signs ‘+ or –’ and place
it over the common denominator to obtain a single like fraction.
Fractions and Decimals
31
(iv)Reduce the fraction obtained in step (iii) to simplest form and then convert it into mixed
fraction (if need be).
Example 1. Simplify the following:
5
6
(i)2 3 + 5
4
3
8
+
(ii)1
2
3
5
8
+
–
7
12
4
15
+2
Solution.
(i)2
3
4
5
6
3
11
=
8
4
+5 +
=
Shortcut method
4
5
6
3
8
L.C.M. of 4, 6, 8 = 24
11
35
3
+
+
4
6
8
= 11 × 6 + 35 × 4 + 3 × 3
24
66 + 140 + 9
=
= 215
24
24
7
4
+2
12
15
+
+
–
=
5
3
+
=
5 × 40 + 5 × 15 − 7 × 10 + 34 × 8
120
=
200 + 75 − 70 + 272
120
=
159
40
5
8
+
140
24
+
9
24
L.C.M. of 4, 6, 8 = 24
=
5
8
(ii)1
2
3
3
8
+
11 × 6
35 × 4
3× 3
66
+
+
=
4×6
6× 4
8×3
24
66 + 140 + 9
= 215 = 8 23 .
24
24
24
=
2 3 + 5
35
6
+
–
7
12
= 8 23 .
24
34
15
+
=
3
2
2
477
120
3,
1,
1,
1,
8,
8,
4,
2,
12,
4,
2,
1,
15
5
5
5
L.C.M. = 3 × 2 × 2 × 2 × 5 = 120
= 3 39 .
40
Example 2. Kiran purchased 3 1 kg apples and 4 3 kg oranges. What is the total weight of
2
4
the fruits purchased by her?
Solution. Total weight of the fruits= c3 1 + 4 3 m kg =
2
4
=
33
4
c
same picture in
7
9
7
9
hours. Vaibhav finished colouring the
hours and Vaibhav coloured the same picture in
4
L.C.M. of 9 and 4 = 36
Write the given fraction as equivalent like fractions.
7#4
9#4
=
28
36
and
As 28 > 27,
28
36
>
27
36
=
kg
4
In order to find who worked longer, let us compare
7
9
7 # 2 + 19
4
hours. Who worked longer? By what fraction was it longer?
Solution. Mirdula coloured a picture in
3
hours.
kg =
kg = 8 1 kg
Example 3. Mirdula finished colouring a picture in
3
4
7
19
+ m
2
4
3
4
⇒
3#9
4#9
7
> 3
9
4
=
=
27
36
7
9
and
3
.
4
Learning Mathematics–VII
32
Therefore, Mirdula worked longer.
Further,
7
9
–
3
4
=
28
36
–
27
36
=
28 − 27
36
1
.
36
=
∴ Fraction of time by which Mirdula worked longer =
1
36
hours.
Exercise 2.2
1. Evaluate the following:
(i) 4 +
7
8
3
(ii)
(iv)2 2 + 3 1 3
3
5
–
2
7
(iii)
(v) 8 1 – 3 5 2
2
(vi)
8
9
11
7
10
–
+
4
15
2
5
+
3
.
2
2. Simplify the following:
(i)7 3 – 3 5 +
4
(iii)2 3
14
–
6
35
6
7
8
– 2 +
5
(ii) 3 – 1 1 –
7
15
(iv) 6 1 – 2
–5
6
21
2
8
1
12
1
10
3. Sahil wants to put a picture in a frame. The picture
picture cannot be more than
4. Ramesh solved
By how much?
5. Bitoo ate
3
5
2
7
73
10
7
.
25
is 7 3
5
+3
cm wide. To fit in the frame the
cm wide. How much should the picture be trimmed?
part of an exercise while Reshma solved
4
5
of it. Who solved lesser part?
part of an apple and the remaining part was eaten by his sister Reena. How
much part of the apple did Reena eat? Who had the larger share? By how much?
6. Jaishree studies for 5 2 hours daily. She devotes 2 4 hours of her time for science and
3
5
mathematics. How much time does she devote for other subjects?
Multiplication of fractions
Recall that multiplication is a repeated addition.
So 4 × 7 = 7 + 7 + 7 + 7 i.e. 7 added four times.
In a similar way, we can say that
4×
But
2
7
2
7
∴4×
=
+
2
7
2
7
2
7
=
+
+
2
7
2
7
8
.
7
+
+
2
7
2
7
+
=
2
i.e. 2 added four
7
7
2+2+2+2
= 8.
7
7
times.
If we write 4 as a fraction, then we can write 4 ×
2
7
as
4
1
×
2
.
7
Note that to get the answer 8 from 4 × 2 , we multiply the numerators together and multiply the
7
1
7
denominators together.
Thus, to multiply two fractions proceed as under:
(i)Convert the mixed fractions (if any) to improper fractions.
(ii)Multiply the numerators together and the denominators together. Place the product of the
numerators over the product of the denominators.
(iii)Reduce the fraction obtained in step (ii) to lowest terms. Still better is to cancel the common
factors (if any) from the numerators and denominators of the given fractions before multiplying
numerators together and denominators together.
(iv)Convert the fraction obtained in step (iii) to mixed fraction (if need be).