Mathematics for FinancePYQ Nov 19Question 1296 of 512
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At six months intervals A deposited 1000\displaystyle 1000 in a savings account which credit interest at 10%\displaystyle 10\% p.a., compounded semi-annually. The first deposit was made when A's son was 6\displaystyle 6 months old and last deposit was made when his son turns 8\displaystyle 8 years old. The money remained in the account and was given to the son on his 10th\displaystyle 10^{th} birthday. How much did he receive? (1.05)16=2.1829\displaystyle (1.05)^{16} = 2.1829

Options

A25740\displaystyle 25740
B28755\displaystyle 28755
C27860\displaystyle 27860
D29760\displaystyle 29760
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Correct Answer

✅ Option b — 28755\displaystyle 28755

All Options:

  • A25740\displaystyle 25740
  • B28755\displaystyle 28755
  • C27860\displaystyle 27860
  • D29760\displaystyle 29760

Detailed Solution & Explanation

**Derivation of Accumulated Son's Gift Value** Given: - Semi-annual deposit (R\displaystyle R) = Rs. 1,000\displaystyle \text{Rs. }1,000 - Nominal rate (r\displaystyle r) = 10%\displaystyle 10\% p.a. compounded semi-annually - First deposit at 0.5\displaystyle 0.5 years old, last deposit at 8\displaystyle 8 years old - Maturity age = 10\displaystyle 10 years old - Given: (1.05)16=2.1829\displaystyle (1.05)^{16} = 2.1829 **Step 1: Calculate periodic rate (i\displaystyle i) and number of deposits (n\displaystyle n)** - Periodic rate i=r2=10%2=5%=0.05\displaystyle i = \frac{r}{2} = \frac{10\%}{2} = 5\% = 0.05 per half-year. - The deposits are made at ages: 0.5,1.0,1.5,…,8.0\displaystyle 0.5, 1.0, 1.5, \dots, 8.0 years. - Total deposit events n=8.0−0.50.5+1=15+1=16\displaystyle n = \frac{8.0 - 0.5}{0.5} + 1 = 15 + 1 = 16 deposits. **Step 2: Calculate Future Value of the annuity at age 8** Since deposits are at the end of each half-year starting from 0.5\displaystyle 0.5 years, the value at 8.0\displaystyle 8.0 years is: FV8=R×(1+i)n−1iFV_8 = R \times \frac{(1+i)^n - 1}{i} FV8=1000×(1.05)16−10.05FV_8 = 1000 \times \frac{(1.05)^{16} - 1}{0.05} FV8=1000×2.1829−10.05=1000×23.658=Rs. 23,658FV_8 = 1000 \times \frac{2.1829 - 1}{0.05} = 1000 \times 23.658 = \text{Rs. }23,658 **Step 3: Accumulate the value from age 8 to 10** For 2\displaystyle 2 years (which is 4\displaystyle 4 half-years), the money remains untouched and grows: A=FV8×(1+i)4A = FV_8 \times (1 + i)^4 A=23658×(1.05)4A = 23658 \times (1.05)^4 A=23658×1.215506≈Rs. 28,755A = 23658 \times 1.215506 \approx \text{Rs. }28,755 Hence, **Option B** is the correct answer.

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