Answer: (c)

A : B =\(\displaystyle \frac{1}{2}:\frac{3}{8}\)= 4 : 3 = 8 : 6

B : C =\(\displaystyle \frac{1}{3}:\frac{5}{9}\)= 3 : 5 = 6 : 10

C : D =\(\displaystyle \frac{5}{6}:\frac{3}{4}\)= 10 : 9

Therefore, A : B : C : D = 8 : 6 : 10 : 9

Answer: (c)

A : B : C = 2 : 3 : 4

\(\displaystyle \Rightarrow \) \(\displaystyle \frac{A}{B}=\frac{2}{3},\frac{B}{C}=\frac{3}{4},\frac{C}{A}=\frac{4}{2}=2\)

\(\displaystyle \Rightarrow \) \(\displaystyle \frac{A}{B}:\frac{B}{C}:\frac{C}{A}=\frac{2}{3}:\frac{3}{4}:\frac{2}{1}\)

= 8 : 9 : 24

Answer: (a)

a : (b+c) = 1 : 3

\(\displaystyle \Rightarrow \) \(\displaystyle \frac{{b+c}}{a}=\frac{3}{1}\) Þ \(\displaystyle \frac{{b+c}}{a}+1=\frac{3}{1}+1\)

\(\displaystyle \frac{{a+b+c}}{a}=\frac{{3+1}}{1}=\frac{4}{1}\)  ….. (i)

Similarly, \(\displaystyle \frac{{a+b}}{c}=\frac{7}{5}\)

\(\displaystyle \Rightarrow \) \(\displaystyle \frac{{a+b+c}}{c}=\frac{{12}}{5}\) …..(ii)

On dividing (i) by (ii),

\(\displaystyle \frac{c}{a}=\frac{{4\times 5}}{{12}}=\frac{5}{3}=k\) ….. (iii)

From equation (i), b = 4k

\(\displaystyle \frac{b}{{a+c}}=\frac{{4k}}{{3k+5k}}=1:2\)

The Ratio and Proportion questions answer is (b)

\(\displaystyle \frac{p}{1}=\frac{q}{2}=\frac{r}{4}=k\)

\(\displaystyle \Rightarrow \) p = k, q = 2k, r = 4k

\(\displaystyle \sqrt{{5{{p}^{2}}+{{q}^{2}}+{{r}^{2}}}}=\sqrt{{5{{k}^{2}}+4{{k}^{2}}+16{{k}^{2}}}}=\sqrt{{25{{k}^{2}}}}\)

= 5k = 5p

Answer for this MCQ on Ratio and Proportion is (b)

Mean proportional = \(\displaystyle \sqrt{{(3+\sqrt{2})(12-\sqrt{{32}})}}\)

= \(\displaystyle \sqrt{{(3+\sqrt{2})4(3-\sqrt{2})}}\)

= \(\displaystyle 2\sqrt{{9-2}}=2\sqrt{7}\)