CBSE Class 8 Mathematics Ratio and Proportion Notes Set 12

Ratio

A ratio compares two quantities of the same kind by division. The ratio of a to b is written a : b and equals the fraction \( \frac{a}{b} \). Both quantities must be in the same unit before they are compared.

1. Multiplying or dividing both terms by the same non-zero number gives an equivalent ratio, so 12 : 8 = 3 : 2 = 24 : 16.
2. A ratio is in simplest form when the HCF of its terms is 1. Divide both terms by their HCF to reach it.
3. To compare two ratios, turn them into fractions with the same denominator. For 5 : 12 and 3 : 5 the common denominator is 60, giving 25/60 and 36/60, so 3 : 5 is larger.
4. Fractional terms are cleared by multiplying both terms by the LCM of the denominators, for example \( \frac{1}{6} : \frac{1}{8} : \frac{1}{12} \) becomes 4 : 3 : 2 after multiplying by 24.

Proportion

Four numbers a, b, c, d are in proportion when a : b = c : d, written a : b :: c : d. The first and last terms are the extremes and the middle two are the means.

1. Test for proportion: product of extremes equals product of means, that is a × d = b × c. For 63, 42, 33, 22 both products are 1386, so they are in proportion.
2. If three terms are known, the fourth is found by cross multiplication. If a : b :: c : x then x = (b × c)/a.
3. The unitary method finds the value of one unit first and then scales up. If 12 bowls cost Rs. 72, one bowl costs Rs. 6 and 20 bowls cost Rs. 120.

Sharing in a Ratio

To divide a quantity in the ratio a : b : c, add the parts to get a + b + c. Each person gets their part divided by this total, times the quantity.

1. Divide Rs. 2600 in the ratio 4 : 5 : 4. The total of the parts is 13 and one part is 2600/13 = Rs. 200. The shares are Rs. 800, Rs. 1000 and Rs. 800.
2. A class of 60 has boys and girls in the ratio 7 : 8. One part is 60/15 = 4, so there are 28 boys and 32 girls.
3. Always check that the shares add back to the whole quantity.