Savvy invested \(\frac{1}{3}\)rd of her capital at 10% gain, \(\frac{1}{4}\)th at 20% gain and the remaining at 30% gain. If her income from gain is ₹50,000, then value of capital is:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → ₹2,40,000
The total gain is the sum of the gains on the three parts, each part earning its own rate. Writing each gain as a fraction of the whole capital C gives one equation in C.
The three parts and their gains:
The remaining part: \(\displaystyle 1-\frac{1}{3}-\frac{1}{4}=\frac{12-4-3}{12}=\frac{5}{12}\) of C
Gain on \(\frac{1}{3}\) at 10%: \(\displaystyle \frac{1}{3}\times\frac{10}{100}C=\frac{C}{30}\)
Gain on \(\frac{1}{4}\) at 20%: \(\displaystyle \frac{1}{4}\times\frac{20}{100}C=\frac{C}{20}\)
Gain on \(\frac{5}{12}\) at 30%: \(\displaystyle \frac{5}{12}\times\frac{30}{100}C=\frac{C}{8}\)
Finding C (the total gain is ₹50,000):
Total gain, over the common denominator 120: \(\displaystyle \frac{C}{30}+\frac{C}{20}+\frac{C}{8}=\frac{4C+6C+15C}{120}=\frac{5C}{24}\)
This equals the given gain: \(\displaystyle \frac{5C}{24}=50000\), so \(C=240000\)
The capital is ₹2,40,000.