ICSE Class 8 Maths Arithmetical Problems Chapter 01 Ratio and Proporation

Ratio and Proportion

Ratio

We can compare the size of two quantities of the same kind in many different ways. One of them is ratio. The ratio of two quantities a and b of the same kind and in the same unit of measurement is the fraction \(\frac{a}{b}\). It is also denoted as a : b (read as "a to b" or "a is to b").

In the ratio a : b, a is the first term or the antecedent and b is the second term or the consequent.

A Few Points

1. A ratio is a number, it has no unit.

2. There can be no ratio between quantities of different kinds, for example, there can be no ratio between 3 kg and 4 m.

3. The quantities that are compared must be expressed in the same unit.

Example: To work out the ratio of Rs 2 and 75 p, we must convert Rs 2 into paise. Thus, the ratio of Rs 2 and 75 p is not \(\frac{2}{75}\), but \(\frac{200}{75} = \frac{8}{3}\).

4. A ratio does not change if its antecedent and consequent are multiplied or divided by the same number.

Example: 12 m : 3.6 m = \(\frac{12}{3.6} = \frac{12 \times 10}{3.6 \times 10} = \frac{120}{36} = \frac{120 \div 12}{36 \div 12} = \frac{10}{3}\) = 10 : 3.

5. To express a ratio in the simplest form, take these steps:

(i) If the antecedent and the consequent are integers then divide both by their HCF.

(ii) If the antecedent or the consequent or both are fractions then multiply both by the LCM of their denominators.

Example: Express the ratios (i) 36 g : 48 g and (ii) \(2\frac{1}{4}\) m : \(7\frac{2}{5}\) m in the simplest form.

Solution: (i) The HCF of 36 and 48 = 12.

\(\therefore\) 36 g : 48 g = 36 : 48 = (36 \div 12) : (48 \div 12) = 3 : 4.

(ii) The LCM of the denominators 4 and 5 = 20.

\(\therefore\) \(2\frac{1}{4}\) m : \(7\frac{2}{5}\) m = \(\frac{9}{4} : \frac{37}{5} = \left(\frac{9}{4} \times 20\right) : \left(\frac{37}{5} \times 20\right) = (9 \times 5) : (37 \times 4) = 45 : 148\).

Teacher's Note

When comparing distances on a map to actual distances, we use ratios. For instance, a map scale of 1:100,000 helps us understand how much real ground each inch on paper represents.

A Ratio Can Be Expressed As A Fraction

A ratio can be expressed as a fraction. So, the rules for comparing fractions apply to ratios as well.

Example: Arrange the given ratios in descending order.

(i) 4 : 7, 3 : 7 (ii) 5 : 8, 5 : 11 (iii) 3 : 5, 4 : 7 (iv) 3 : 4, 2 : 5, 7 : 8

Solution: (i) The ratios have the same consequent. So, the ratio with the greater antecedent will be greater. Thus, 4 : 7 > 3 : 7.

(ii) Here, the ratios have the same antecedent. So, the ratio with the smaller consequent will be greater. Thus, 5 : 8 > 5 : 11.

(iii) Since the ratios have different consequents and antecedents, we change them into like fractions.

The LCM of the consequents 5 and 7 = 35.

\(\therefore\) 3 : 5 = \(\frac{3}{5} = \frac{3 \times (\text{LCM} \div 5)}{5 \times (\text{LCM} \div 5)} = \frac{3 \times 7}{5 \times 7} = \frac{21}{35}\) = 21 : 35

and 4 : 7 = \(\frac{4}{7} = \frac{4 \times (\text{LCM} \div 7)}{7 \times (\text{LCM} \div 7)} = \frac{4 \times 5}{7 \times 5} = \frac{20}{35}\) = 20 : 35.

\(\therefore\) 21 > 20, 21 : 35 > 20 : 35, that is, 3 : 5 > 4 : 7.

(iv) The LCM of the consequents 4, 5 and 8 = 40.

\(\therefore\) 3 : 4 = \(\frac{3}{4} = \frac{3 \times (\text{LCM} \div 4)}{4 \times (\text{LCM} \div 4)} = \frac{3 \times 10}{4 \times 10} = \frac{30}{40}\) = 30 : 40,

2 : 5 = \(\frac{2}{5} = \frac{2 \times (\text{LCM} \div 5)}{5 \times (\text{LCM} \div 5)} = \frac{2 \times 8}{5 \times 8} = \frac{16}{40}\) = 16 : 40,

7 : 8 = \(\frac{7}{8} = \frac{7 \times (\text{LCM} \div 8)}{8 \times (\text{LCM} \div 8)} = \frac{7 \times 5}{8 \times 5} = \frac{35}{40}\) = 35 : 40.

\(\therefore\) 35 > 30 > 16, 35 : 40 > 30 : 40 > 16 : 40, that is, 7 : 8 > 3 : 4 > 2 : 5.

Teacher's Note

Comparing ratios is similar to ranking sports teams by their win-loss ratios, which helps determine which team performed better overall.

Continued Ratio

If three quantities be such that the ratio of the first two is a : b and the ratio of the last two is b : c then the three quantities are said to be in the continued ratio a : b : c.

Examples: (i) If a : b = 1 : 3 and b : c = 3 : 7 then a : b : c = 1 : 3 : 7.

(ii) Let m, n and p be such that m : n = 1 : 2 and n : p = 3 : 5. Then, m : n = (1 \times 3) : (2 \times 3) = 3 : 6 and n : p = (3 \times 2) : (5 \times 2) = 6 : 10.

\(\therefore\) m : n : p = 3 : 6 : 10.

Note: 3 is the antecedent in the second ratio and 2 is the consequent in the first ratio. It would be useful to remember that if the ratio of two quantities be a : b then the two quantities can be expressed as ak and bk. Also, if the continued ratio of three quantities be a : b : c then the three quantities can be expressed as ak, bk and ck.

Dividing A Quantity In A Given Ratio

If a number n is divided into two parts in the ratio a : b,

first part = \(\frac{a}{a + b} \times n\), second part = \(\frac{b}{a + b} \times n\)

Teacher's Note

Dividing inheritance or prize money among family members or beneficiaries in predetermined ratios is a common real-world application of this concept.

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