# Comparison of Ratios

In comparison of ratios, we first need to convert them into like fractions by using the following steps and then compare them.

Step I: Obtain the given ratios.

Step II: Now we express each of the given ratios as a fraction in the simplest form.

Step III: Find the L.C.M (least common multiple) of the denominators of the fractions obtained in the above step (Step II).

Step IV: Obtain the first fraction and its denominator. Divide the L.C.M (least common multiple) obtained in the above step (Step III) by the denominator to get a number z (say).

Now, multiply the numerator and denominator of the fraction by the z (L.C.M). Similarly apply the same procedure to the all other fraction.

In other words convert each fraction to its equivalent fractions with denominator equal to the L.C.M (least common multiple).

Thus, the denominators of all the fractions are be same.

Step V: Compare the numerators of the equivalent fractions whose denominators are same.

Compare the numerators of the fractions obtained in the above step (Step IV). The fraction having larger numerator will be larger than the other fraction.

Two or more ratios can be compared by writing their equivalent fractions with common denominators.

Solved examples of comparison of ratios:

1. Compare the ratios 4 : 5 and 2 : 3.

Solution:

Express the given ratios as fraction

4 : 5 = 4/5  and 2 : 3 =2/3

Now find the L.C.M (least common multiple) of 5 and 3

The L.C.M (least common multiple) of 5 and 3 is 15.

Making the denominator of each fraction equal to 15, we have

4/5 = (4 ×3)/(5 ×3) = 12/15 and 2/3 = (2 ×5)/(3 ×5) = 10/15

Clearly, 12 > 10

Now, 12/15 > 10/15

Therefore, 4 : 5 > 2 : 3.

2. Compare the ratios 5 : 6 and 7 : 9.

Solution:

Express the given ratios as fraction

5 : 6 = 5/6  and 7 : 9 =7/9

Now find the L.C.M (least common multiple) of 6 and 9

The L.C.M (least common multiple) of 6 and 9 is 18.

Making the denominator of each fraction equal to 18, we have

5/6  = (5 ×3)/(6 ×3) = 15/18 and 7/9  = (7 ×2)/(9 ×2) = 14/18

Clearly, 15 > 14

Now, 15/18 > 14/18

Therefore, 5 : 6 > 7 : 9.

3. Compare the ratios 1.2 : 2.5 and 3.5 : 7.

Solution:

1.2 : 2.5 = 1.2/2.5 and 3.5 : 7 =3.5/7

1.2/2.5 = (1.2 ×10)/(2.5 ×10 ) = 12/25 and 3.5/7 = (3.5 ×10)/(7 ×10) = 35/70 = 1/2

[We removed the decimal point from the ratios now, we will compare the ratio]

Now find the L.C.M (least common multiple) of 25 and 2

The L.C.M (least common multiple) of 25 and 2 is 50.

Making the denominator of each fraction equal to 50, we have

= 12/25 = (12 ×2)/(25 ×2)  = 24/50 and 1/2 = (1 ×25)/(2 ×25)  = 25/50

Now, 25/50 >  24/50

Therefore,  3.5 : 7 > 1.2 : 2.5.

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