Ratio and Proportion — Statement I-II Type Questions

Statement 1: The third proportional of $\dfrac{3}{4} \text{ and } 1\dfrac{1}{8} \text{ is } \dfrac{27}{16}$.

Statement 2: If a, c and b are three quantities in continued proportion, then c is called the third proportional for a and b.

Which of the following options is correct?

1. Both the statements are true.

2. Both the statements are false.

3. Statement 1 is true, and statement 2 is false.

4. Statement 1 is false, and statement 2 is true.

Answer

Let the third proportional to a and b be x, then a, b and x are in continued proportion, i.e. a : b :: b : x,

Let $a = \dfrac{3}{4} \text{ and }b = 1\dfrac{1}{8} = \dfrac{9}{8}$.

$\Rightarrow \dfrac{a}{b} = \dfrac{b}{x} \\[1em] \Rightarrow x = \dfrac{b \times b}{a} \\[1em] \Rightarrow x = \dfrac{b^2}{a} \\[1em] = \dfrac{\left(\dfrac{9}{8}\right)^2}{\dfrac{3}{4}} \\[1em] = \dfrac{81}{64} \div \dfrac{3}{4} \\[1em] = \dfrac{81}{64} \times \dfrac{4}{3} \\[1em] = \dfrac{27}{16}$

So, Statement 1 is true.

We know that,

If a, c and b are in continued proportion, then a : c :: c : b. Here c is the mean proportional between a and b, while the third proportional to a and c is b. So, Statement 2 is false.

Hence, option 3 is the correct option.

Statement 1: In a proportion, the first two terms are of same kind with the same unit, whereas the last two terms must also be of the same kind with the same unit.

Statement 2: The ratios 1 kg : 250 g and 140 cm : 35 cm form a proportion.

Which of the following options is correct?

1. Both the statements are true.

2. Both the statements are false.

3. Statement 1 is true, and statement 2 is false.

4. Statement 1 is false, and statement 2 is true.

Answer

In a proportion, the two terms of each ratio must be of the same kind and in the same unit (though the two ratios may be of different kinds). So, Statement 1 is true.

Solving,

1 kg : 250 g = 1000 g : 250 g = 4 : 1,

140 cm : 35 cm = 4 : 1.

Since both ratios are equal, they form a proportion. So, Statement 2 is true.

Hence, option 1 is the correct option.