Assertion (A) : The simple interest on a certain sum is $\dfrac{9}{16}$ of the principal. If the number representing the rate of interest in percent and the time in year are equal then the time for which the principle is lent out is $7\dfrac{1}{2}$ years.

Reason (R) : In simple interest,

Time = $\dfrac{\text{S.I.} \times 100}{\text{\text{Principal}} \times \text{Rate}}$.

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

2. Both A and R are correct, and R is not the correct explanation for A.

3. A is true, but R is false.

4. A is false, but R is true.

Let the principal be ₹ P.

S.I. = $\dfrac{9}{16}$ x P

Rate of interest = a%

Time = a years

By formula,

S.I. = $\dfrac{P \times R \times T}{100}$

Substituting values we get :

$\Rightarrow \dfrac{9}{16}P = \dfrac{P \times a \times a}{100}\\[1em] \Rightarrow \dfrac{9}{16} \cancel{P} = \dfrac{\cancel{P} \times a \times a}{100}\\[1em] \Rightarrow \dfrac{9}{16} = \dfrac{a \times a}{100}\\[1em] \Rightarrow \dfrac{9}{16} = \dfrac{a^2}{100}\\[1em] \Rightarrow a^2 = \dfrac{9 \times 100}{16}\\[1em] \Rightarrow a^2 = \dfrac{900}{16}\\[1em] \Rightarrow a = \sqrt{\dfrac{900}{16}}\\[1em] \Rightarrow a = \dfrac{30}{4}\\[1em] \Rightarrow a = \dfrac{15}{2}\\[1em] \Rightarrow a = 7\dfrac{1}{2}$

Time = $7\dfrac{1}{2}$ years.

So, assertion (A) is true.

By formula;

$\Rightarrow S.I. = \dfrac{P \times R \times T}{100} \\[1em] \Rightarrow T = \dfrac{\text{S.I.} \times 100}{P \times R} \\[1em] \Rightarrow \text{Time } = \dfrac{\text{Simple Interest} \times 100}{\text{Principal} \times \text{Rate}}$

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

Hence, option 1 is the correct option.