Ratio and Proportion MCQ Question

Solution: P:Q = 8:15 and Q:R = 3:2 = 15:10

Now P:Q:R= 8:15:10

Using Trick-1:

Using Trick-2:

Solution: $\frac{\mathrm{A}}{\mathrm{B}}=\frac{6}{7}, \frac{\mathrm{B}}{\mathrm{C}}=\frac{8}{9}\,\,\mathrm{and} \frac{\mathrm{C}}{\mathrm{D}}=\frac{5}{11}$

Multiplying them

$\frac{\mathrm{A}}{\mathrm{B}}\times \frac{\mathrm{B}}{\mathrm{C}}\times \frac{\mathrm{C}}{\mathrm{D}}=\frac{6}{7}\times \frac{8}{9}\times \frac{5}{11}=\frac{80}{231}$

∴ A:D = 80:231

Using Trick:

A:D = product of antecedent : product of consequent = 240:693 = 80:251

Solution:  First we made B same for first two ratio and C same for last two ratio.

A : B = 6 : 7 = 6 × 8 × 10 : 7 × 8 × 10; B :  C = 8 : 9 = 8 × 7 × 10 : 9 ×7 ×10 ;

C :  D = 10 : 11 = 10 × 7 × 9 : 11 × 7 × 9

A : B : C : D = 6 × 8 × 10 : 7 × 8 × 10 : 9 ×7 ×10 : 11 × 7 × 9 = 480:560:630:693

Using trick:

Solution: A : B = 2: 3, B : C = 5: 7 and C : D = 3:10

∴ $\frac{\mathrm{A}}{\mathrm{D}}=\frac{\mathrm{A}}{\mathrm{B}}\times \frac{\mathrm{B}}{\mathrm{C}}\times \frac{\mathrm{C}}{\mathrm{D}}$

= $\frac{2}{3}\times \frac{5}{7}\times \frac{3}{10}=\frac{1}{7}$

∴ A:D = 1:7

Using Trick:

A:D = product of antecedent : product of consequent = 2×5×3:3×7×10 = 1:7

Solution: Let 4th proportional = x

∴ $\frac{4}{16}=\frac{7}{\mathrm{x}}\,\,$

⇒ 4x = 16×7

⇒x = 28

Solution: Let 3rd proportional = x

∴ 15:30 :: 30: x

⇒ $\frac{15}{30}=\frac{30}{\mathrm{x}}$

⇒ 15x = 900 ⇒x = 60

Solution: mean proportional = $\sqrt{9\times 64}=24$

Mean Proportion – Let x be the mean proportion between a and b, then a:x::x:b (Real condition)

i.e. $\frac{a}{x}=\frac{x}{b}\,\,or\,\,x^2=ab\,\,or\,\,x\,\,=\,\,\sqrt{ab}$

∴ Mean proportional of a and b = $\sqrt{ab}$

Solution: inverse ratio = 19 : 17

Solution: 4a = 5b ⇒ a : b = 5:4 and 7b = 9c ⇒b:c = 9:7

Solution: Let ratio constant k

$\frac{1}{\mathrm{a}}:\frac{1}{\mathrm{b}}:\frac{1}{\mathrm{c}}=2:3:5$

$\frac{1}{\mathrm{a}}=2\mathrm{k}\Rightarrow \mathrm{a}=\frac{1}{2\mathrm{k}}$

$\frac{1}{\mathrm{b}}=3\mathrm{k}\Rightarrow \mathrm{b}=\frac{1}{3\mathrm{k}}$

$\frac{1}{\mathrm{c}}=2\mathrm{k}\Rightarrow \mathrm{c}=\frac{1}{3\mathrm{k}}$

a : b : c = $\frac{1}{2\mathrm{k}}:\frac{1}{3\mathrm{k}}:\frac{1}{5\mathrm{k}}=15:10:6$ = 5:3:2