Garima started a business with a personal investment of INR 3000. After 3
months Ridhima joined the same business and invested INR 2000. After
some more time Badima joined the business with INR 1500. At the end of the
year, the profit was 16000 and the share of Badima was INR 1600. How
many months did Badima invest in the business?
3
5
4
7
Garima invested INR 3000 for 12 months in the business. Ridhima joined after 3 months with INR 2000, investing for 9 months. Badima joined later with INR 1500, and with total profit of INR 16000, Badima received INR 1600, indicating Badima invested for exactly 4 months [web:1][web:2].
Profit Sharing Calculation
Profit shares are proportional to investment amounts times duration in months. Let Badima invest for \( x \) months. Total “investment-months” = \( 3000 \times 12 + 2000 \times 9 + 1500 \times x \).
Badima’s share: \( \frac{1500x}{3000 \times 12 + 2000 \times 9 + 1500x} \times 16000 = 1600 \).
Simplify: \( 1500x = \frac{1600}{16000} \times (36000 + 18000 + 1500x) \), so \( 1500x = 0.1 \times (54000 + 1500x) \).
\( 1500x = 5400 + 150x \), \( 1350x = 5400 \), \( x = 4 \). [web:1][web:2]
Why 4 Months is Correct
Plugging \( x = 4 \): Total investment-months = \( 36000 + 18000 + 6000 = 60000 \). Badima’s share = \( \frac{6000}{60000} \times 16000 = 1600 \), matching exactly. Garima gets \( \frac{36000}{60000} \times 16000 = 9600 \); Ridhima \( \frac{18000}{60000} \times 16000 = 4800 \). [web:1]
Explanation of All Options
- 3 months: Total investment-months = 57000; Badima share = \( \frac{4500}{57000} \times 16000 \approx 1263 \) (too low).
- 5 months: Total = 63000; share = \( \frac{7500}{63000} \times 16000 = 1905 \) (too high).
- 4 months: Exactly 1600, as calculated.
- 7 months: Total = 69000; share = \( \frac{10500}{69000} \times 16000 \approx 2435 \) (way too high). [web:1][web:2]
