Question

Fill in the blanks.
(i) $\frac{14}{21}=\frac{\boxed{}}{3}=\frac{6}{\boxed{}}$
(ii) 90 cm : 1.5 m = ...... .
(iii) If 36 : 81 : : x : 63, then x = ...... .
(iv) If 25, 35, x are in proportion, then x = ...... .
(v) If 9, x, x, 49 are in proportion, then x = ...... .

Answer 1

(i)
$$\begin{gathered} \mathrm{Let}\frac{14}{21}=\frac{x}{3} \\ \mathrm{Thus},\mathrm{we}\mathrm{have}: \\ 21x=14\times 3\Rightarrow x=\frac{14\times 3}{21}=2 \\ \therefore \frac{14}{21}=\frac{2}{3} \\ \mathrm{Again},\mathrm{let}\frac{2}{3}=\frac{6}{y} \\ \mathrm{Thus},\mathrm{we}\mathrm{have}: \\ 2y=6\times 3\Rightarrow y=\frac{6\times 3}{2}=9 \\ \therefore \frac{2}{3}=\frac{6}{9} \\ \therefore \frac{14}{21}=\frac{2}{3}=\frac{6}{9} \end{gathered}$$

(ii) 90 cm:1.5 m (or) 90 cm:150 cm (1 m = 100 cm)
90 = 9 = 9 ÷ 3 = 3 (H.C.F. of 9 and 15 is 3.)
150 15 15 ​​÷ 3 5

(iii) If 36:81::x:63
Product of means = Product of extremes
81x = 36 × 63
x = 2268
81
x = 28

(iv) Given:
25, 35, x are in proportion.
25:35::35:x
Now, we know:
Product of extremes = Product of means
25 × x = 35 ​× 35
25x = 1225
x = 1225 = 49
25

(v) Given:
9, x, x, 49 are in proportion.
9:x::x:49
Now, we know:
Product of means = Product of extremes
x ​× x = 9 ​× 49
x² = 441
x² = 21²
x = 21