The simple interest on a sum of money is \(\frac{9}{35}\) of the sum. If the number of years is numerically \(\frac{5}{7}\) times the rate per cent per annum, what is the rate per cent per annum?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 6%
Since \(\displaystyle \mathrm{SI}=\frac{P\times R\times T}{100}\), the interest as a fraction of the sum is \(\displaystyle \frac{R\times T}{100}\). Writing the time in terms of the rate leaves one unknown.
\(\displaystyle \frac{R\times T}{100}=\frac{9}{35}\)
\(\displaystyle T=\frac{5}{7}R\), so \(\displaystyle \frac{R\times\frac{5}{7}R}{100}=\frac{9}{35}\)
\(\displaystyle \frac{5R^{2}}{700}=\frac{9}{35}\), so \(\displaystyle R^{2}=\frac{9\times700}{35\times5}=36\)
\(R=6\) (a rate cannot be negative)
Check: time \(\displaystyle =\frac{5}{7}\times6=\frac{30}{7}\) years, and \(\displaystyle \frac{6\times\frac{30}{7}}{100}=\frac{180}{700}=\frac{9}{35}\)
The rate is 6% per annum.