Question : In $t$ years, the simple interest earned on a certain amount at the rate of 10% per annum is $\frac{5}{8}$th of the principal amount. If the rate of interest is made three-fourths of 10% and the simple interest remains the same, which of the following statements is true about the changed value of time?
Option 1: It remains the same as $t$.
Option 2: It becomes three-fourths of the time $t$.
Option 3: It increases to $8 \frac{1}{3}$ years.
Option 4: It decreases to $5 \frac{1}{2}$ years.
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Correct Answer: It increases to $8 \frac{1}{3}$ years.
Solution :
Let the principal amount be $P$.
Simple interest after $t$ years at the rate of 10% per annum = $\frac{5}{8}P$
⇒ $\frac{P(10)t}{100}=\frac{5}{8}P$---(1)
⇒ $t=\frac{25}{4}$ years
New rate of interest $=\frac{3}{4}\times 10 = 7.5\%$
Let the new time taken for $P$ to become $\frac{5}{8}P$ at new rate of interest be $x$ years
⇒ $\frac{P(7.5)x}{100}=\frac{5}{8}P$---(2)
Dividing (1) and (2), we get,
$\frac{10t}{7.5x} = 1$
⇒ $x = \frac{4}{3}t$
⇒ $x=\frac{25}{3}$ years
⇒ $x=8\frac{1}{3}$ years
Hence, the correct answer is $8\frac{1}{3}$ years.
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