Ratio & Proportion — Mental Maths

Fill in the blanks:

(i) Simplest form of 78 : 91 is ...............

(ii) Is it possible to find the ratio of ₹12 and 24 cm? ...............

(iii) Are 3, 6, 9, 12 in proportion? ...............

(iv) If x : 36 :: 33 : 108, then x = ...............

(v) If 4 : 9 :: 9 : x = ...............

Answer

(i) Simplest form of 78 : 91

H.C.F. of 78 and 91 is 13.

$\dfrac{78 \div 13}{91 \div 13} = \dfrac{6}{7} = 6 : 7$

Hence, the simplest form of 78 : 91 is 6 : 7.

(ii) A ratio exists only between two quantities of the same kind. Since ₹12 (money) and 24 cm (length) are quantities of different kinds, their ratio cannot be found.

Hence, no, it is not possible.

(iii) For 3, 6, 9, 12 to be in proportion, product of extremes = product of means.

Product of extremes = 3 × 12 = 36

Product of means = 6 × 9 = 54

Since 36 ≠ 54, they are not in proportion.

Hence, no, 3, 6, 9, 12 are not in proportion.

(iv) x : 36 :: 33 : 108

Product of extremes = product of means

$\Rightarrow x \times 108 = 36 \times 33\\[1em] \Rightarrow x = \dfrac{36 \times 33}{108} \\[1em] = \dfrac{1188}{108} \\[1em] = 11$

Hence, x = 11.

(v) 4 : 9 :: 9 : x

Product of extremes = product of means

$\Rightarrow 4 \times x = 9 \times 9\\[1em] \Rightarrow x = \dfrac{81}{4}$

Hence, $x = \dfrac{81}{4}$.

Write (T) for true and (F) for false statement:

(i) The simplest form of $2\dfrac{1}{5} : 1\dfrac{2}{5}$ is 11 : 7. ...............

(ii) The ratio 3 : 4 is the same as 51 : 68. ...............

(iii) The ratio 70 g : 1 kg is not defined. ...............

(iv) 70 P : ₹7 = 1 : 10. ...............

(v) If ₹120 is divided in the ratio 5 : 7, the larger part is ₹$\left(120 \times \dfrac{7}{5}\right)$. ...............

Answer

(i) The simplest form of $2\dfrac{1}{5} : 1\dfrac{2}{5}$ is 11 : 7.

Convert the mixed numbers into improper fractions: $2\dfrac{1}{5} = \dfrac{11}{5}$ and $1\dfrac{2}{5} = \dfrac{7}{5}$.

$\dfrac{11}{5} : \dfrac{7}{5} = 11 : 7$

Hence, the statement is True.

(ii) The ratio 3 : 4 is the same as 51 : 68.

H.C.F. of 51 and 68 is 17.

$\dfrac{51 \div 17}{68 \div 17} = \dfrac{3}{4} = 3 : 4$

Hence, the statement is True.

(iii) The ratio 70 g : 1 kg is not defined.

Convert kg into g: 1 kg = 1000 g. Both quantities are of the same kind (weight), so the ratio is defined: 70 : 1000 = 7 : 100.

Hence, the statement is False.

(iv) 70 P : ₹7 = 1 : 10.

Convert ₹ into paise: ₹7 = 700 paise.

$70 : 700 = \dfrac{70}{700} = \dfrac{1}{10} = 1 : 10$

Hence, the statement is True.

(v) If ₹120 is divided in the ratio 5 : 7, the larger part is ₹$\left(120 \times \dfrac{7}{5}\right)$.

The larger part must be found by multiplying the total by the larger ratio term divided by the sum of the ratio terms, i.e. ₹$\left(120 \times \dfrac{7}{12}\right)$ = ₹70, not ₹$\left(120 \times \dfrac{7}{5}\right)$.

Hence, the statement is False.