Ratio & Proportion — Assertion-Reason Type Questions

Assertion (A): If $\dfrac{a}{b} = \dfrac{c}{d}$, then $\dfrac{a+c}{b+d} = \dfrac{a}{b}$.

Reason (R): If two or more than two ratios are equal, then each ratio = $\dfrac{\text{sum of antecedents}}{\text{sum of consequents}}$.

1. Both A and R are true, and R is the correct explanation of A.

2. Both A and R are true, but R is not the correct explanation of A.

3. A is true, but R is false.

4. A is false, but R is true.

Answer

We know that,

Each ratio = $\dfrac{\text{sum of antecedents}}{\text{sum of consequents}}$.

Let $\dfrac{a}{b} = \dfrac{c}{d}$ = k for some constant k.

a = kb, c = kd

Substituting value of a and c in $\dfrac{a+c}{b+d}$, we get :

$\Rightarrow \dfrac{kb + kd}{b + d} \\[1em] \Rightarrow \dfrac{k(b + d)}{(b + d)} \\[1em] \Rightarrow k \\[1em] \Rightarrow \dfrac{a}{b} \text{ or } \dfrac{c}{d}.$

∴ Both A and R are true, and R is the correct explanation of A.

Hence, option 1 is the correct option.

Assertion (A): The mean proportion between a²b and $\dfrac{1}{b}$ is $\dfrac{a}{b}$.

Reason (R): The mean proportion between x and y is given by $\sqrt{xy}$.

1. Both A and R are true, and R is the correct explanation of A.

2. Both A and R are true, but R is not the correct explanation of A.

3. A is true, but R is false.

4. A is false, but R is true.

Answer

We know that,

Mean proportion between two numbers

= $\sqrt{\text{First number} \times \text{Second number}}$

Thus,

The mean proportion between x and y is given by $\sqrt{xy}$.

∴ Reason (R) is true.

Thus,

The mean proportion between a²b and $\dfrac{1}{b}$ = $\sqrt{a^2b \times \dfrac{1}{b}}$

$= \sqrt{a^2} = a$.

∴ Assertion (A) is false.

Hence, option 4 is the correct option.

Assertion (A): If 2x = 3y and 4y = 5z, then x : y : z is equal to 15 : 10 : 8.

Reason (R): If x = y and y = z, then we cannot find the ratio x : y : z.

1. Both A and R are true, and R is the correct explanation of A.

2. Both A and R are true, but R is not the correct explanation of A.

3. A is true, but R is false.

4. A is false, but R is true.

Answer

Given,

⇒ 2x = 3y

⇒ $\dfrac{x}{y} = \dfrac{3}{2}$

⇒ x : y = 3 : 2

⇒ 4y = 5z

⇒ $\dfrac{y}{z} = \dfrac{5}{4}$

⇒ y : z = 5 : 4

Since, x : y = 3 : 2 and y : z = 5 : 4. L.C.M of 2 and 5 is 10.

⇒ x : y = 3 : 2 = (3 × 5) : (2 × 5) = 15 : 10

⇒ y : z = 5 : 4 = (5 × 2) : (4 × 2) = 10 : 8

⇒ x : y : z = 15 : 10 : 8.

∴ Assertion (A) is true.

If x = y and y = z, then by the transitive property of equality, it follows that x = y = z.

x : y : z = x : x : x = 1 : 1 : 1.

∴ Reason (R) is false.

Hence, option 3 is the correct option.