Arjun invests Rs. 25,000 for 4 years in a scheme offering 4% interest per annum, compounded half-yearly. Separately, Varun invests a certain amount in the same scheme for 3 years. At the end of 3 years, he reinvests the entire amount received for one more year at 8% simple interest. If the amounts received by both at the end of 4 years are equal, then the initial investment, in rupees rounded to two decimal places, made by Varun is:
Answer and solution
Correct answer: B
Solution
Answer: B — 24,083.33
1. Arjun invests Rs. 25,000 at 4% per annum, compounded half-yearly.
2. The rate for each six-month period is 2%. In 4 years, there are 8 such periods.
3. Arjun's final amount = 25,000 × (1.02)⁸.
4. Let Varun's initial investment be P.
5. Varun invests for 3 years at the same half-yearly compound rate. There are 6 half-year periods, so his amount becomes P × (1.02)⁶.
6. He then reinvests this amount for one year at 8% simple interest. The final amount becomes:
7. P × (1.02)⁶ × 1.08
8. Both final amounts are equal:
9. 25,000 × (1.02)⁸ = P × (1.02)⁶ × 1.08
10. Cancel (1.02)⁶ from both sides:
11. P = 25,000 × (1.02)² ÷ 1.08
12. P = Rs. 24,083.33 approximately.
13. Answer: Rs. 24,083.33.
14. Rounding check: 25000×1.0404/1.08 = 72250/3 = 24083.333… . Option B is the value rounded to two decimal places, not an exact equality.
Why this method works:
A 4% annual nominal rate compounded half-yearly means 2% per half-year. Both investments share the first six compound periods, so those factors cancel from the equality. Varun’s last year is one simple-interest factor 1.08.
Independent check and common trap:
The remaining growth factor for Arjun is 1.02²=1.0404, less than 1.08. Varun therefore needs less than₹25,000 initially, consistent with₹24,083.33. Keep the exact fraction 72250/3 until the final rounding.
Apply: Cancel the common first 3 years. P=25000×1.02²/1.08=72250/3=24083.333…; nearest paisa 24083.33.
Check: The remaining growth factor for Arjun is 1.02²=1.0404, less than 1.08. Varun therefore needs less than₹25,000 initially, consistent with₹24,083.33. Keep the exact fraction 72250/3 until the final rounding.
1. Arjun invests Rs. 25,000 at 4% per annum, compounded half-yearly.
2. The rate for each six-month period is 2%. In 4 years, there are 8 such periods.
3. Arjun's final amount = 25,000 × (1.02)⁸.
4. Let Varun's initial investment be P.
5. Varun invests for 3 years at the same half-yearly compound rate. There are 6 half-year periods, so his amount becomes P × (1.02)⁶.
6. He then reinvests this amount for one year at 8% simple interest. The final amount becomes:
7. P × (1.02)⁶ × 1.08
8. Both final amounts are equal:
9. 25,000 × (1.02)⁸ = P × (1.02)⁶ × 1.08
10. Cancel (1.02)⁶ from both sides:
11. P = 25,000 × (1.02)² ÷ 1.08
12. P = Rs. 24,083.33 approximately.
13. Answer: Rs. 24,083.33.
14. Rounding check: 25000×1.0404/1.08 = 72250/3 = 24083.333… . Option B is the value rounded to two decimal places, not an exact equality.
Why this method works:
A 4% annual nominal rate compounded half-yearly means 2% per half-year. Both investments share the first six compound periods, so those factors cancel from the equality. Varun’s last year is one simple-interest factor 1.08.
Independent check and common trap:
The remaining growth factor for Arjun is 1.02²=1.0404, less than 1.08. Varun therefore needs less than₹25,000 initially, consistent with₹24,083.33. Keep the exact fraction 72250/3 until the final rounding.
G Strategy shortcut
Spot: Arithmetic – Compound Interest.Apply: Cancel the common first 3 years. P=25000×1.02²/1.08=72250/3=24083.333…; nearest paisa 24083.33.
Check: The remaining growth factor for Arjun is 1.02²=1.0404, less than 1.08. Varun therefore needs less than₹25,000 initially, consistent with₹24,083.33. Keep the exact fraction 72250/3 until the final rounding.
G Strategy: Spot: Arithmetic – Compound Interest. Apply: Cancel the common first 3 years. P=25000×1.02²/1.08=72250/3=24083.333…; nearest paisa 24083.33. Check: The remaining growth factor for Arjun is 1.02²=1.0404, less than 1.08. Varun therefore needs less than₹25,000 initially, consistent with₹24,083.33. Keep the exact fraction 72250/3 until the final rounding.