Mathematics of FinanceMCQPYQ Nov 19Question 1296 of 512
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At six months intervals A deposited $1000\displaystyle in a savings account which credit interest at $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 $10^{th}$ birthday. How much did he receive? $(1.05)^{16} = 2.1829$

Select Your Answer (Keyboard: A, B, C, D)
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Correct Answer

✅ Option b — $28755$

All Options:

  • A$25740$
  • B$28755$
  • C$27860$
  • D$29760$

Detailed Solution & Explanation

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

About This Chapter: Mathematics of Finance

Paper

Paper 3: Quantitative Aptitude

Weightage

12-16 Marks

Key Topics

Simple & Compound Interest, Annuity, Perpetuity

The most important mathematical chapter in the entire syllabus. It covers Simple Interest (SI), Compound Interest (CI), Nominal vs Effective rates, Present and Future Value, Annuities (Ordinary and Due), Sinking Funds, and Perpetuities. The concepts learned here are applied heavily in CA Intermediate and Final.

View Official ICAI Syllabus

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