Chapter 5.4 - Cross products to find Equivalent fractions

In the previous section we learned a method for determining equivalent fractions of any given fraction. In this section we will discuss another method.

2/3 and 4/6 are two equivalent fractions. Using them, we are going to find two products:
(1) The product of numerator of first and denominator of second
(2) The product of numerator of second and denominator of first

This can be pictorially represented as:

The upper part of first is multiplied with the lower part of the second and vice versa. So these products are called cross products. The cross products of the above two equivalent fractions are:
(1) 2 × 6 = 12 and (2) 4 × 3 =12. Three more pairs are taken and cross products are calculated for them also. The calculation steps and results are tabulated below:

In the above table, we can observe a peculiar property: For any pair that we take, it’s cross products will be equal. This gives us another convenient method:
• To determine an equivalent fraction (with a specified numerator or denominator) for a given fraction.
• It also helps us to check whether two given fractions are equivalent or not. 
The following solved examples will demonstrate the method.
Solved example 5.8
Find the equivalent fraction of 3/7 such that, the numerator of the equivalent fraction is 12
Solution:
It is specified that, the numerator of the equivalent fraction must be 12. We will use the cross products method. The steps are given below:
• We have ³ ⁄ ₇ = ¹² ⁄ _x  Where x is the unknown denominator
•Taking cross products we can write:
3 × x = 12 × 7 [But 12 × 7 = 84]
• ∴ 3 × x = 84
• So x = 84 ÷ 3 = 28
Thus the required fraction is ¹² ⁄ ₂₈ 

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This problem can be solved by the previous method also:

• The numerator of the given fraction is 3.

• The numerator of the new equivalent fraction should be 12.

• 12 is greater than 3. So we have to multiply the original numerator by a number to obtain the new numerator.

• 3 ×  ? = 12. The answer is 4. 

• We have to multiply the denominator also by 4. The result and the calculation steps are given below:

³ ⁄ ₇  =   ^{(3 × 4)} ⁄ _{(7 × 4)}  =  ¹² ⁄ ₂₈  

• So the required equivalent fraction is  ¹² ⁄ ₂₈ 

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Solved example 5.9

Replace the '□' in each of the following problems by the correct number

Solution:

We will use the cross products method.
(i) • We have ⁵ ⁄ ₉ = ¹⁵ ⁄ _x  
Where x is the unknown denominator put in the place of  '□'.
•Taking cross products we can write:
5 × x = 15 × 9 [But 15 × 9 = 135]
• ∴ 5 × x = 135
• So x = 135 ÷ 5 = 27
Thus the required fraction is ¹⁵ ⁄ ₂₇ 

(ii) • We have ² ⁄ ₇ = ⁸ ⁄ _x  
•Taking cross products we can write:
2 × x = 7 × 8 [But 7 × 8 = 56]
• ∴ 2 × x = 56
• So x = 56 ÷ 2 = 28
Thus the required fraction is ⁸ ⁄ ₂₈ 

(iii) • We have ²⁷ ⁄ ₄₈ = ^x ⁄ ₁₆  
•Taking cross products we can write:
27 × 16 = x × 48 [But 27 × 16 = 432]
• ∴ 432 = x × 48
• So x = 432 ÷ 48 = 9

Thus the required fraction is ⁹ ⁄ ₁₆ 

(iv) • We have ⁴ ⁄ ₁₅ = ²⁴ ⁄ _x  
•Taking cross products we can write:
4 × x = 15 × 24 [But 15 × 24 = 360]
• ∴ 4 × x = 360
• So x = 360 ÷ 4 = 90
Thus the required fraction is ²⁴ ⁄ ₉₀   

Solved example 5.10

Check whether the following pairs of fractions are equivalent

(i) ⁷ ⁄ ₈ , ⁶³ ⁄ ₇₂     (ii) ⁴ ⁄ ₁₁ , ²⁸ ⁄ ₇₇      (iii) ³ ⁄ ₁₀ , ¹² ⁄ ₅₀    (iv) ⁴² ⁄ 91 , ⁶ ⁄ ₁₃

Solution:

We will use the cross products method.

(i) • First cross product = 7 × 72 = 504.  Second cross product = 63 × 8 = 504

    • The two cross products are equal. So they are equivalent fractions

(ii) • First cross product = 4 × 77 = 308.  Second cross product = 28 × 11 = 308

     • The two cross products are equal. So they are equivalent fractions

(iii) • First cross product = 3 × 50 = 150.  Second cross product = 12 × 10 = 120

      • The two cross products are not equal. So they are not equivalent fractions

(iv) • First cross product = 42 × 13 = 546.  Second cross product = 6 × 91 = 546

      • The two cross products are equal. So they are equivalent fractions
Solved example 5.11
Write the equivalent fraction of 11/14 having (i) Numerator 77 (b) Denominator 84

Solution:

We will use the cross products method.
(i) • We have ¹¹ ⁄ ₁₄ = ⁷⁷ ⁄ _x  
Where x is the unknown denominator.
•Taking cross products we can write:
11 × x = 77 × 14 [But 77 × 14 = 1078]
• ∴ 11 × x = 1078
• So x = 1078 ÷ 11 = 98
Thus the required fraction is ⁷⁷ ⁄ ₉₈ 

(iii) • We have ¹¹ ⁄ ₁₄ = ^x ⁄ ₈₄  
•Taking cross products we can write:
11 × 84 = x × 14 [But 11 × 84 = 924]
• ∴ 924 = x × 14
• So x = 924 ÷ 14 = 66

Thus the required fraction is ⁶⁶ ⁄ ₈₄ 

In the next section we will discuss about the 'Simplest form' of a fraction.

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