## Fractions and Decimals

### Operations on Fractions:

1. To Reduce a Fraction to its Lowest Terms:

A Fraction is said to be in its lowest terms when the H.C.F (highest common factor) of the numerator and denominator is 1.

Note: In other words, a fraction is in lowest terms, when the numerator and denominator are prime to each other. Two numbers that do not have any common factor other than 1 are said to be prime to each other i.e. the two numbers are relative primes. Eg: 8 and 9.

Note that relative primes are not primes themselves; only that the common factor between them is only 1.

Examples:

1. 24/30 = (4 × 6)/ (5 × 6) = 4/5

{The HCF of 24 and 30 is 6}

2. 18/27 = (9 × 2)/ (9 × 3) = 2/3

The HCF of 18 and 27 is 9.

3. 40/45 = (8 × 5)/ (9 × 5) = 8/9

2. How to Compare Fractions:

The best way to compare fractions is to find cross product of the given fractions.

Consider two fractions a/b and c/d.

Step 1: First find the cross products of a/b and c/d.

the cross products of a/b and c/d are

a × d and b × c, where

a × d is the cross product of numerator in a/b and denominator in c/d, and

b × c is the cross product of denominator in b/c and numerator in c/d

Step 2: If ad > bc, then a/b > c/d or
if ad < bc, then a/b < c/d

i.e., the fraction on that side is greater in which side the greater of the two products lies.

The following example will illustrate in detail all the necessary steps to compare two fractions.

Example 1:

Is 4/7 greater than 11/19 ?

Solution: Is 4/7 > 11/19

Step 1:

The cross products of the two fractions are

4 × 19 and 7 × 11

i.e. 76 and 77

Step 2: Now 76> < 77.

Since the greater product 77 lies on the right side, we conclude the fraction 11/19 on the right side is the greater one.

So, 4/7 is less than 11/19,

i.e. 4/7 < 11/19

Example 2:

Is 13/16 greater than 21/33 ?

Solution:

Step 1:  Cross products of 13/16 and 21/33 are:

13 × 33 = 429 and 21 × 16 = 336

Step 2: Since 429 > 336, so 13/16 > 21/33

Example 3:

Arrange the following fractions in ascending order
3/4, 7/9, 5/7

Solution:

First, find cross product of the fractions ¾ and 7/9. The cross products are:

3 × 9 (=27) and 4 × 7 (28). Since 27 < 28, therefore ¾ < 7/9

Next find the cross product of the fractions 7/9 and 5/7. They are:

7 × 7, i.e. 49 and 9 × 5 i.e., 45. Since, 49 > 45, so 7/9 > 5/7

Now, cross products of ¾ and 5/7 are:  3 × 7 = 21 and 4 × 5 = 20. Since, 21 > 20, so ¾ > 5/7

The ascending order of the given fractions is 5/7 < ¾ < 7/9

Short-cut:
Cross product method of comparing fractions is preferable when there are two fractions.

Since there are three fractions in this example, try this short cut:
¾ = 0.75, 7/9 = 0.77 and 5/7 = 0.71

Since 0.71 < 0.75 < 0.77, so 5/7 < ¾ < 7/9

Example 4:

Arrange the following fractions in descending order
3/5 , 4 /7 , 11/15

Solution:

Cross product of 3/5 and 4/7 is:  3 × 7 = 21 and 5 × 4 = 20.
Since 21 > 20, 3/5 > 4/7

Cross product of 4/7 and 11/15 is:  4 × 15 = 60 and 7 × 11 = 77. Since, 60 < 77, 4/7 < 11/15

Cross product of 3/5 and 11/15 is:  3 × 15 = 45 and 5 × 15 = 75. Since, 45 < 75, 3/5 < 11/15

The descending order of the fractions is 11/15 > 3/5 > 4/7

Short cut:
3/5 = 0.6, 4/7 = 0.57 and 11/15 = 0.7
Since 0.7 > 0.6 > 0.57, so, 11/15 > 3/5 > 4/7

3. How to Insert Fractions Between Two Given Fractions:

Suppose a/b and c/d are two fractions.

Then a fraction that lies between a/b and c/d is
(a + b)/(c + d)

Example:

Insert two fractions between 3/5 and 7/9

Solution:

using the formula above, one fraction that lies between 3/5 and 7/9 is

(3 + 7)/ (5 + 9) = 10/16 = 5/8

To find the other, now insert a fraction between 5/8 and 7/9. It is
(5 + 7)/ (8 + 9) = 12/17

Therefore, both the fractions 5/8 and 12/17 lie between 3/5 and 7/9

4. How to Add Two Fractions:

Like Fractions have same denominators. So, to add like fractions, just add numerators:

Example1:  5/7 + 2/7 = 7/7 = 1

Example 2: 4/9 + 3/9 = 7/9

Unlike Fractions have different denominators. To add them, find L.C.M of the denominators.

Example1:

Find:  5/7 + 3/8

Since 7 and 8 do not have any common factor other than 1, their L.C.M. is their product

i.e. L.C.M. of 7 and 8 is 7 × 8 = 56

So 5/7 + 3/8 = (5 × 6)/ (7 × 8) + (3 × 7)/ (8 × 7) = 30/56 + 21/56 = 51/56

NOTE: Subtracting fractions is same as adding fractions.

5. Multiplying Fractions:

Consider two fractions a/b and c/d. Then

a/b × c/d = (a × c)/ (b × d) = ac/bd

Note: Reduce the fractions to the lowest terms by cancelling any common factors in the numerator and denominators of the two fractions.

Example 1:

Multiply 4/6 and 8/9

Solution:

2 is a common factor of 4 and 6 in 4/6. Cancel it to reduce 4/6 to lowest terms, i.e. 2/3

Now, (2/3) × (8/9) = (2 × 8)/ (3 × 9) = 16/72

6. Dividing Fractions:

To divide fraction a/b by c/d, multiply a/b with the reciprocal of c/d i.e. d/c

Example 1:

Divide 4/7 by 3/5.

Solution:

reciprocal of 3/5 is 5/3.

Now, multiply 4/7 with 5/3.

(4/7) × (5/3) = 20/21

7. The word “Of” signifies Multiplication:

Half of 100 is (½) × 100 = 50

One third of 99 is (1/3) × 99 = 33

2/5 of 40 = (2/5) × 40 = 2 × 8 = 16