Ratio and Proportion — Assertion-Reason Type Questions

Assertion (A): Mean proportion between $\dfrac{2}{3} \text{ and } \dfrac{15}{2}$ is 5.

Reason (R): Mean proportion between two numbers is the positive square root of their product.

1. Assertion (A) is true, but Reason (R) is false.

2. Assertion (A) is false, but Reason (R) is true.

3. Both Assertion (A) and Reason (R) are correct, and Reason (R) is the correct reason for Assertion (A).

4. Both Assertion (A) and Reason (R) are correct, and Reason (R) is incorrect reason for Assertion (A).

Answer

The mean proportion between two numbers a and b is defined as $\sqrt{ab}$

So, reason (R) is true.

When two numbers are $\dfrac{2}{3} \text{ and } \dfrac{15}{2}$

$\text{Mean proportion} = \sqrt{\dfrac{2}{3} \times \dfrac{15}{2}}\\[1em] = \sqrt{\dfrac{30}{6}}\\[1em] = \sqrt{5}$

So, assertion (A) is false.

Thus, Assertion (A) is false, but Reason (R) is true.

Hence, option 2 is the correct option.

a, b and c are in continued proportion.

Assertion (A): a is first proportion.

Reason (R): The first proportion is always the smallest of the three numbers.

1. Assertion (A) is true, but Reason (R) is false.

2. Assertion (A) is false, but Reason (R) is true.

3. Both Assertion (A) and Reason (R) are correct, and Reason (R) is the correct reason for Assertion (A).

4. Both Assertion (A) and Reason (R) are correct, and Reason (R) is incorrect reason for Assertion (A).

Answer

Three numbers a, b, c are in continued proportion if:

$\dfrac{a}{b} = \dfrac{b}{c}$

⇒ b² = ac

Here, a is called the first proportion, b is the mean (or geometric mean), c is the third proportion.

So, assertion (A) is true.

Lets take an example, a = 9, b = 6 and c = 4.

When three numbers a, b, c are in continued proportion if:

$\Rightarrow \dfrac{a}{b} = \dfrac{b}{c}\\[1em] \Rightarrow \dfrac{9}{6} = \dfrac{6}{4}\\[1em] \Rightarrow\dfrac{3}{2} = \dfrac{3}{2}$

Here, a = 9 is first proportion but it is not the smallest number among three.

So, reason (R) is false.

Thus, Assertion (A) is true, but Reason (R) is false.

Hence, option 1 is the correct option.

a, b, c and d are in proportion.

Assertion (A): a + b, a - b, c + d and c - d are in proportion.

Reason (R): $\dfrac{a}{b} = \dfrac{c}{d}$.

1. Assertion (A) is true, but Reason (R) is false.

2. Assertion (A) is false, but Reason (R) is true.

3. Both Assertion (A) and Reason (R) are correct, and Reason (R) is the correct reason for Assertion (A).

4. Both Assertion (A) and Reason (R) are correct, and Reason (R) is incorrect reason for Assertion (A).

Answer

Given,

a, b, c and d are in proportion.

$\therefore \dfrac{a}{b} = \dfrac{c}{d}$

So, reason (R) is true.

Applying componendo and dividendo, we get

$\Rightarrow \dfrac{a + b}{a - b} = \dfrac{c + d}{c - d}$

Therefore, a + b, a - b, c + d and c - d are in proportion.

So, assertion (A) is true and reason (R) correctly explains assertion (A).

Thus, both Assertion (A) and Reason (R) are correct, and Reason (R) is the correct reason for Assertion (A).

Hence, option 3 is the correct option.

a, b, c and d are in proportion.

Assertion (A): a - c, c, b - d, d are in proportion.

Reason (R): a, c, b and d are in proportion.

1. Assertion (A) is true, but Reason (R) is false.

2. Assertion (A) is false, but Reason (R) is true.

3. Both Assertion (A) and Reason (R) are correct, and Reason (R) is the correct reason for Assertion (A).

4. Both Assertion (A) and Reason (R) are correct, and Reason (R) is incorrect reason for Assertion (A).

Answer

Given,

a, b, c and d are in proportion.

$\therefore \dfrac{a}{b} = \dfrac{c}{d} \\[1em] \Rightarrow \dfrac{a}{c} = \dfrac{b}{d}.$

∴ a, c, b and d are in proportion.

So, reason (R) is true.

$\Rightarrow \dfrac{a}{c} = \dfrac{b}{d}\\[1em] \Rightarrow \dfrac{a}{c} - 1 = \dfrac{b}{d} - 1\\[1em] \Rightarrow \dfrac{a - c}{c} = \dfrac{b - d}{d}$

We can say that a - c, c, b - d, d are in proportion.

So, assertion (A) is true and reason (R) correctly explains assertion (A).

Thus, both Assertion (A) and Reason (R) are correct, and Reason (R) is the correct reason for Assertion (A).

Hence, option 3 is the correct option.