If the simple interest on a certain sum for 1 year 3 months at 8 ½ % per annum exceeds the simple interest on the same sum for 8 months at 7 ½ % per annum by Rs. 45, then the sum is:
Rs. 800
To solve this problem, we need to calculate the simple interest for two different periods and rates, and find the principal sum where the difference between the two interests is Rs. 45.
Let's denote the principal sum as \(P\).
The formula for simple interest is given by:
\(SI = \frac{P \times R \times T}{100}\)
- First, calculate the simple interest for 1 year 3 months at 8 ½ % per annum:
- Convert 1 year 3 months to years: \(T = 1 + \frac{3}{12} = \frac{5}{4}\) years.
- The rate of interest is \(R = 8 \frac{1}{2} \% = \frac{17}{2} \%\).
- Using the formula, we have: \(SI_1 = \frac{P \times \frac{17}{2} \times \frac{5}{4}}{100} = \frac{P \times 17 \times 5}{2 \times 4 \times 100} = \frac{85P}{800}\)
- Next, calculate the simple interest for 8 months at 7 ½ % per annum:
- Convert 8 months to years: \(T = \frac{8}{12} = \frac{2}{3}\) years.
- The rate of interest is \(R = 7 \frac{1}{2} \% = \frac{15}{2} \%\).
- Using the formula, we have: \(SI_2 = \frac{P \times \frac{15}{2} \times \frac{2}{3}}{100} = \frac{P \times 15 \times 2}{2 \times 3 \times 100} = \frac{30P}{600}\)
- The difference between the two interests is Rs. 45:
- \(SI_1 - SI_2 = 45\)
- Substitute the expressions for \(SI_1\) and \(SI_2\): \(\frac{85P}{800} - \frac{30P}{600} = 45\)
- To simplify the equations, find a common denominator: \(\frac{85P}{800} = \frac{255P}{2400}, \quad \frac{30P}{600} = \frac{120P}{2400}\)
- So we have: \(\frac{255P}{2400} - \frac{120P}{2400} = 45\)
- Simplify: \(\frac{135P}{2400} = 45\)
- Therefore, \(135P = 45 \times 2400 = 108000\)
- Solving for \(P\), we get: \(P = \frac{108000}{135} = 800\)
Thus, the principal sum is Rs. 800, making the correct answer Rs. 800.
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