Ratio and Proportion — Assertion-Reason Type Questions

Assertion (A): 1.7 : 0.34 and 3.4 : 0.68 are two equivalent ratios.

Reason (R): If each term of a ratio is multiplied or divided by the same non-zero number, the ratio remains the same.

1. A is true, R is false.

2. A is false, R is true.

3. Both A and R are true.

4. Both A and R are false.

Answer

Solving,

$1.7 : 0.34 = \dfrac{1.7}{0.34} = \dfrac{170}{34} = 5$ and,

$3.4 : 0.68 = \dfrac{3.4}{0.68} = 5$.

Both ratios equal 5 : 1, so they are equivalent. Assertion (A) is true.

We know that,

Multiplying or dividing both terms of a ratio by the same non-zero number gives an equivalent ratio. Reason (R) is true.

Both A and R are true.

Hence, option 3 is the correct option.

Assertion (A): $\dfrac{3}{20} \text{ is the fourth proportional to } \dfrac{1}{3}, \dfrac{1}{4} \text{ and } \dfrac{1}{5}$.

Reason (R): If a, b, c and d are in proportion, then d is called the fourth proportional to a, b and c.

1. A is true, R is false.

2. A is false, R is true.

3. Both A and R are true.

4. Both A and R are false.

Answer

Let $a = \dfrac{1}{3}$, $b = \dfrac{1}{4}$ and $c = \dfrac{1}{5}$.

Let the fourth proportional to a, b and c be x, then a : b :: c : x,

$\Rightarrow \dfrac{a}{b} = \dfrac{c}{x} \\[1em] \Rightarrow x = \dfrac{b \times c}{a} \\[1em] \Rightarrow x = \dfrac{\dfrac{1}{4} \times \dfrac{1}{5}}{\dfrac{1}{3}} \\[1em] \Rightarrow x = \dfrac{1}{20} \times 3 \\[1em] \Rightarrow x = \dfrac{3}{20}$

So, $\dfrac{3}{20}$ is the fourth proportional to $\dfrac{1}{3}, \dfrac{1}{4}$ and $\dfrac{1}{5}$. Assertion (A) is true.

We know that,

Four quantities a, b, c and d are said to be in proportion if the ratio of the first two is equal to the ratio of the last two, i.e. if

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

Here a and d are the extremes and b and c are the means, so a × d = b × c.

The fourth term d of such a proportion is called the fourth proportional to a, b and c. Reason (R) is true.

Hence, option 3 is the correct option.

Assertion (A): If a : b = 4 : 5 and b : c = 6 : 7 then a : b : c = 24 : 30 : 35

Reason (R): Compounded ratio of a : b and c : d is ac : bd.

1. A is true, R is false.

2. A is false, R is true.

3. Both A and R are true.

4. Both A and R are false.

Answer

Solving,

a : b = 4 : 5 and b : c = 6 : 7

Making the value of b the same in both the ratios (L.C.M. of 5 and 6 is 30),

a : b = 4 : 5 = (4 × 6) : (5 × 6) = 24 : 30

b : c = 6 : 7 = (6 × 5) : (7 × 5) = 30 : 35

∴ a : b : c = 24 : 30 : 35. Assertion (A) is true.

We know that,

The compounded ratio of a : b and c : d is (a × c) : (b × d) = ac : bd. Reason (R) is true.

Both A and R are true.

Hence, option 3 is the correct option.