Ratio & Proportion — Statement I-II Type Questions

Statement 1 : Two numbers $A$ and $B$ are in the ratio 3 : 5, then $A = \dfrac{3}{5} × B$.

Statement 2 : $\dfrac{A}{3} = \dfrac{B}{5}$

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

Given,

$A$ and $B$ are in the ratio 3 : 5.

∴ $\dfrac{A}{B} = \dfrac{3}{5}$

$\Rightarrow A = \dfrac{3}{5} × B$.

So, Statement 1 is true.

From $A : B$ = 3 : 5, we get $\dfrac{A}{B} = \dfrac{3}{5}$, which on cross multiplication gives $5A = 3B$, i.e. $\dfrac{A}{3} = \dfrac{B}{5}$.

So, Statement 2 is true.

Hence, option 1 is the correct option.

Statement 1 : If $m : n$ = 5 : 4, then $5m : 4n$ = 25 : 16

Statement 2 : $5m : 4n = \dfrac{5m}{4n} = \dfrac{5}{4} \times \dfrac{5}{4} = 25 : 16$

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

Given,

$m : n$ = 5 : 4, i.e. $\dfrac{m}{n} = \dfrac{5}{4}$.

Solving,

$\Rightarrow 5m : 4n = \dfrac{5m}{4n} \\[1em] = \dfrac{5}{4} \times \dfrac{m}{n} \\[1em] = \dfrac{5}{4} \times \dfrac{5}{4} \\[1em] = \dfrac{25}{16} \\[1em] = 25 : 16$

So, Statement 2 is true.

From the working above, $5m : 4n$ = 25 : 16.

So, Statement 1 is true.

Hence, option 1 is the correct option.