ICSE Class 8 Maths Chapter 04 Ratio and Proportion

Chapter 4 - Ratio And Proportion

We have already studied ratio and proportion in previous classes. In this chapter, we shall refresh that knowledge and solve a few tougher problems.

Ratio

A ratio is a comparison of the sizes of two or more quantities of the same kind by division.

If a and b are two quantities of the same kind (in same units), then the fraction \(\frac{a}{b}\) is called the ratio of a to b. Thus, the ratio of a to b = \(\frac{a}{b}\). It is written as a : b.

The quantities a and b are called the terms of the ratio. a is called the first term (or antecedent) and b is called the second term (or consequent).

Remarks

Both terms of a ratio can be multiplied or divided by the same (non-zero) number. Usually, a ratio is expressed in lowest terms (or simplest form).

The order of the terms in a ratio is important.

Ratio exists only between quantities of the same kind.

Quantities to be compared (by division) must be in the same units.

A ratio is a number, so it has no units.

If the terms of a ratio are in fractions, convert them into natural numbers by multiplying each term by the L.C.M. of their denominators.

To compare two ratios, convert them into equivalent like fractions.

If a quantity increases or decreases in the ratio a : b, then new quantity = \(\frac{b}{a}\) of the original quantity.

Ratio a : b : c

Three quantities of the same kind (in same units) are said to be in the ratio a : b : c if the quantities are ak, bk and ck respectively, where k is any positive real number.

Similarly, four quantities of the same kind (in same units) are said to be in the ratio a : b : c : d if the quantities are ak, bk, ck and dk respectively, where k is any positive real number.

Teacher's Note

Understanding ratios helps in cooking recipes where ingredients must maintain specific proportions, or in mixing paints to achieve the desired color consistency.

Examples

Example 1

Find the ratio of each of the following in simplest form:

(i) 3 score to 4 dozen

(ii) 350 m to \(1\frac{2}{5}\) km

Solution

(i) 3 score = 3 \(\times\) 20 = 60

4 dozen = 4 \(\times\) 12 = 48

Therefore, Ratio of 3 score to 4 dozen = \(\frac{60}{48}\) = \(\frac{5}{4}\) = 5 : 4.

(ii) \(1\frac{2}{5}\) km = \(\frac{7}{5}\) km = \(\left(\frac{7}{5} \times 1000\right)\) m = 1400 m

Therefore, Ratio of 350 m to \(1\frac{2}{5}\) km = \(\frac{350}{1400}\) = \(\frac{1}{4}\) = 1 : 4.

Example 2

Simplify the following ratios:

(i) 2.4 : \(2\frac{2}{3}\)

(ii) \(1\frac{1}{2}\) : \(\frac{1}{3}\) : \(\frac{1}{8}\)

Solution

(i) Given ratio = 2.4 : \(2\frac{2}{3}\) = \(\frac{24}{10}\) : \(\frac{8}{3}\) = \(\frac{24}{10} \times \frac{3}{8}\) = \(\frac{9}{10}\) = 9 : 10.

(ii) Given ratio = \(\frac{3}{2}\) : \(\frac{1}{3}\) : \(\frac{1}{8}\)

Multiply each term by the L.C.M. of their denominators = \(\frac{3}{2} \times 24\) : \(\frac{1}{3} \times 24\) : \(\frac{1}{8} \times 24\) = 36 : 8 : 3.

Example 3

Which ratio is greater \(2\frac{1}{3}\) : \(3\frac{1}{3}\) or 3 : 6 : 4 : 8?

Solution

\(2\frac{1}{3}\) : \(3\frac{1}{3}\) = \(\frac{7}{3}\) : \(\frac{10}{3}\) = 7 : 10 = \(\frac{7}{10}\)

and 3.6 : 4.8 = \(\frac{3.6}{4.8}\) = \(\frac{36}{48}\) = \(\frac{3}{4}\)

L.C.M. of 10 and 4 is 20

\(\frac{7}{10}\) = \(\frac{7 \times 2}{10 \times 2}\) = \(\frac{14}{20}\) and \(\frac{3}{4}\) = \(\frac{3 \times 5}{4 \times 5}\) = \(\frac{15}{20}\)

As 15 > 14, \(\frac{15}{20}\) > \(\frac{14}{20}\) - therefore \(\frac{3}{4}\) > \(\frac{7}{10}\)

Hence, 3.6 : 4.8 is the greater ratio.

Example 4

Arrange the following ratios in ascending order of magnitude: 2 : 3, 8 : 15, 11 : 12 and 7 : 16

Solution

Given ratios are \(\frac{2}{3}\), \(\frac{8}{15}\), \(\frac{11}{12}\) and \(\frac{7}{16}\)

L.C.M. of 3, 15, 12 and 16 = 3 \(\times\) 4 \(\times\) 5 \(\times\) 4 = 240

\(\frac{2}{3}\) = \(\frac{2 \times 80}{8 \times 80}\) = \(\frac{160}{240}\), \(\frac{8}{15}\) = \(\frac{8 \times 16}{15 \times 16}\) = \(\frac{128}{240}\)

\(\frac{11}{12}\) = \(\frac{11 \times 20}{12 \times 20}\) = \(\frac{220}{240}\), \(\frac{7}{16}\) = \(\frac{7 \times 15}{16 \times 15}\) = \(\frac{105}{240}\)

As 105 < 128 < 160 < 180, \(\frac{105}{240}\) < \(\frac{128}{240}\) < \(\frac{160}{240}\) < \(\frac{220}{240}\)

- therefore \(\frac{7}{16}\) < \(\frac{8}{15}\) < \(\frac{2}{3}\) < \(\frac{11}{12}\)

Hence, the given ratios in ascending order of magnitude are 7 : 16, 8 : 15, 2 : 3 and 11 : 12.

Teacher's Note

Comparing ratios is similar to comparing test scores from different subjects with different maximum marks - you need to convert them to a common scale first.

Example 5

Divide \(\text{Rs}\) 975 among Parul and Payush in the ratio \(2\frac{1}{3}\) : \(1\frac{5}{6}\).

Solution

Given ratio = \(2\frac{1}{3}\) : \(1\frac{5}{6}\) = \(\frac{7}{3}\) : \(\frac{11}{6}\) = \(\frac{7}{3} \times 6\) : \(\frac{11}{6} \times 6\) = 14 : 11

Sum of the terms of the ratio = 14 + 11 = 25

Share of Parul = \(\frac{14}{25}\) of \(\text{Rs}\) 975 = \(\text{Rs}\) \(\left(\frac{14}{25} \times 975\right)\) = \(\text{Rs}\) 546

Share of Payush = \(\frac{11}{25}\) of \(\text{Rs}\) 975 = \(\text{Rs}\) \(\left(\frac{11}{25} \times 975\right)\) = \(\text{Rs}\) 429.

Example 6

Divide \(\text{Rs}\) 2324 among three children in the ratio \(1\frac{1}{4}\) : \(1\frac{1}{3}\) : \(\frac{7}{8}\).

Solution

Given ratio = \(1\frac{1}{4}\) : \(1\frac{1}{3}\) : \(\frac{7}{8}\) = \(\frac{5}{4}\) : \(\frac{4}{3}\) : \(\frac{7}{8}\) = \(\frac{5}{4} \times 24\) : \(\frac{4}{3} \times 24\) : \(\frac{7}{8} \times 24\) = 30 : 32 : 21

Sum of the terms of the ratio = 30 + 32 + 21 = 83

Share of first child = \(\frac{30}{83}\) of \(\text{Rs}\) 2324 = \(\text{Rs}\) \(\left(\frac{30}{83} \times 2324\right)\) = \(\text{Rs}\) 840

Share of second child = \(\frac{32}{83}\) of \(\text{Rs}\) 2324 = \(\text{Rs}\) \(\left(\frac{32}{83} \times 2324\right)\) = \(\text{Rs}\) 896

Share of third child = \(\frac{21}{83}\) of \(\text{Rs}\) 2324 = \(\text{Rs}\) \(\left(\frac{21}{83} \times 2324\right)\) = \(\text{Rs}\) 588.

Teacher's Note

Dividing amounts in given ratios is used in business partnerships where profits are shared according to investment ratios.

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