Arranging Ratios

We will learn how to solve different types of problems on arranging ratios in ascending order and descending order.

When a ratio is expressed in fraction or in decimal we first need to convert the ratio into whole number to compare the ratios.

The order of a ratio is important to compare two or more ratios. By reversing the antecedent and consequent of a ratio are different ratio is obtained.

Solved problems on comparing and arranging ratios in ascending and descending order:

1. Compare the ratios 1$$\frac{1}{3}$$ : 1$$\frac{1}{5}$$ and 1.6 : 1.2

Solution:

1$$\frac{1}{3}$$ : 1$$\frac{1}{5}$$ and 1.6 : 1.2

= $$\frac{4}{3}$$ : $$\frac{6}{5}$$ and $$\frac{16}{10}$$ : $$\frac{12}{10}$$

= $$\frac{4}{3}$$ × 15 : $$\frac{6}{5}$$ × 15 and $$\frac{16}{10}$$ × 10 : $$\frac{12}{10}$$ × 10

= 20 : 18 and 16 : 12

= $$\frac{20}{18}$$ and $$\frac{16}{12}$$

= $$\frac{10 × 2}{9 × 2}$$ and $$\frac{4 × 4}{3 × 4}$$

= $$\frac{10}{9}$$ and $$\frac{4}{3}$$

= 10 : 9 and 4 : 3

Now, $$\frac{10}{9}$$ and $$\frac{4}{3}$$ are to be compared. L.C.M. of 9 and 3 = 9

$$\frac{10}{9}$$ = $$\frac{10 × 1}{9 × 1}$$ and $$\frac{4}{3}$$ = $$\frac{4 × 3}{3 × 3}$$

= $$\frac{10}{9}$$ and $$\frac{12}{9}$$

Since, $$\frac{10}{9}$$ < $$\frac{12}{9}$$

Therefore, 10 : 9 < 4 : 3

Hence, 1$$\frac{1}{3}$$ : 1$$\frac{1}{5}$$ < 1.6 : 1.2

2. Compare the ratios 14 : 23, 5 : 12 and 61 : 92 in ascending order.

Solution:

Given ratios can be written as $$\frac{14}{23}$$, $$\frac{5}{12}$$ and $$\frac{61}{92}$$

L.C.M. of the denominators 23, 12 and 92 = 276

$$\frac{14}{23}$$ = $$\frac{14 × 12}{23 × 12}$$ = $$\frac{168}{276}$$

$$\frac{5}{12}$$ = $$\frac{5 × 23}{12 × 23}$$ = $$\frac{115}{276}$$

and

$$\frac{61}{92}$$ = $$\frac{61 × 3}{92 × 3}$$ = $$\frac{183}{276}$$

Since, $$\frac{115}{276}$$ < $$\frac{168}{276}$$ < $$\frac{183}{276}$$

Therefore, $$\frac{5}{12}$$ < $$\frac{14}{23}$$ < $$\frac{61}{92}$$

Hence, 5 : 12 < 14 : 23 < 61 : 92

3. Arrange the ratios 1 : 3, 5 : 12, 4 : 15 and 2 : 3 in descending order.

Solution:

Given ratios can be written as $$\frac{1}{3}$$, $$\frac{5}{12}$$, $$\frac{4}{15}$$ and $$\frac{2}{3}$$

L.C.M. of the denominators 3, 12, 15 and 3 = 60

$$\frac{1}{3}$$ = $$\frac{1 × 20}{3 × 20}$$ = $$\frac{20}{60}$$

$$\frac{5}{12}$$ = $$\frac{5 × 5}{12 × 5}$$ = $$\frac{25}{60}$$

$$\frac{4}{15}$$ = $$\frac{4 × 4}{15 × 4}$$ = $$\frac{16}{60}$$

and

$$\frac{2}{3}$$ = $$\frac{2 × 20}{3 × 20}$$ = $$\frac{40}{60}$$

Since, $$\frac{40}{60}$$ > $$\frac{25}{60}$$ > $$\frac{20}{60}$$ > $$\frac{16}{60}$$

Therefore, $$\frac{2}{3}$$ > $$\frac{5}{12}$$ > $$\frac{1}{3}$$ > $$\frac{4}{15}$$

Hence, 2 : 3 > 5 : 12 > 1 : 3 >  4 : 15.

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