Uniot - 2 Compound Interest

 2.1

Question: Compound Interest Problems

---

1. Definitions

(a) Yearly Compound Interest: Compound interest calculated on a yearly basis means that the interest is added to the principal once every year. The formula for compound amount is:

$$CA = P \left( 1 + \frac{R}{100} \right)^T$$

where:

• $P$ = Principal amount,
• $R$ = Yearly rate of interest,
• $T$ = Time in years.

---

(b) Half-Yearly Compound Interest: Compound interest calculated on a half-yearly basis means that the interest is compounded twice a year (every six months). The formula is:

$$CA = P \left( 1 + \frac{R/2}{100} \right)^{2T}$$

where:

• $R/2$ = Half-yearly rate of interest,
• $2T$ = Number of half-yearly compounding periods.

---

(c) Quarterly Compound Interest: Compound interest calculated quarterly means that the interest is compounded four times a year. The formula is:

$$CA = P \left( 1 + \frac{R/4}{100} \right)^{4T}$$

where:

• $R/4$ = Quarterly rate of interest,
• $4T$ = Number of quarterly compounding periods.

---

2. Questions

(a) Formula to Compute Compound Amount for Yearly Compound Interest: If the principal is $P$, the yearly rate of interest is $R$% and the time is $T$ years, the formula for the compound amount is:

$$CA = P \left( 1 + \frac{R}{100} \right)^T$$

---

(b) Relation Between $P, T, R$, and $CI$: The compound interest $CI$ is the difference between the compound amount $CA$ and the principal $P$. The relationship is:

$$CI = CA - P = P \left( 1 + \frac{R}{100} \right)^T - P$$

---

(c) Formula for Compound Amount When Interest Rate Varies Yearly: If the interest rate varies for the first, second, and third years as $R_1\%$, $R_2\%$, and $R_3\%$, respectively, the compound amount $CA$ is calculated as:

$$CA = P \left( 1 + \frac{R_1}{100} \right) \left( 1 + \frac{R_2}{100} \right) \left( 1 + \frac{R_3}{100} \right)$$

This formula accounts for the compounding effect of varying yearly interest rates.

Question 3: Calculate Compound Interest and Amount Without Using Formula

---

Given Problems:

1. (a) Principal ($P$) = Rs. 10,000, Time ($T$) = 2 years, Rate ($R$) = 6% p.a.

Solution:

• Year 1:
  Interest = $\frac{6}{100} \times 10,000 = 600$.
  New principal for Year 2 = $10,000 + 600 = 10,600$.

• Year 2:
  Interest = $\frac{6}{100} \times 10,600 = 636$.
  Compound Amount = $10,600 + 636 = 11,236$.
  Compound Interest = $11,236 - 10,000 = 1,236$.

---

2. (b) Principal ($P$) = Rs. 64,000, Time ($T$) = 3 years, Rate ($R$) = 6% p.a.

Solution:

• Year 1:
  Interest = $\frac{6}{100} \times 64,000 = 3,840$.
  New principal for Year 2 = $64,000 + 3,840 = 67,840$.

• Year 2:
  Interest = $\frac{6}{100} \times 67,840 = 4,070.40$.
  New principal for Year 3 = $67,840 + 4,070.40 = 71,910.40$.

• Year 3:
  Interest = $\frac{6}{100} \times 71,910.40 = 4,314.62$.
  Compound Amount = $71,910.40 + 4,314.62 = 76,225.02$.
  Compound Interest = $76,225.02 - 64,000 = 12,225.02$.

---

3. (c) Principal ($P$) = Rs. 20,000, Time ($T$) = 2 years, $R_1 = 10\%$ (Year 1), $R_2 = 12\%$ (Year 2).

Solution:

• Year 1:
  Interest = $\frac{10}{100} \times 20,000 = 2,000$.
  New principal for Year 2 = $20,000 + 2,000 = 22,000$.

• Year 2:
  Interest = $\frac{12}{100} \times 22,000 = 2,640$.
  Compound Amount = $22,000 + 2,640 = 24,640$.
  Compound Interest = $24,640 - 20,000 = 4,640$.

More Solution Update soon . Keep using app.