For each of the following cases, take time (n) = 1 year, rate of interest per year (r) = 6% and sum invested (P) = ₹ 4,000.

• When the interest is compounded every four months, then N = the number of times the interest is compounded in one year = $\dfrac{12}{4}$ = 3.

∴ A = $P\Big(1 + \dfrac{r}{100 × N}\Big)$^n×N

= ₹ 4,000 $\Big(1 + \dfrac{6}{100×3}\Big)$^1×3 = ₹ 4,244.83

• When the interest is compounded half-yearly, i.e. two times in a year ⇒ N = 2

∴ A = $P\Big(1 + \dfrac{r}{100 × N}\Big)$^n×N

= ₹ 4,000 $\Big(1 + \dfrac{6}{100×2}\Big)$^1×2 = ₹ 4,243.60

(a) Calculate the amount when the interest is compounded quarterly.

(b) What to do you observe from the two cases discussed above ?

Given,

Principal (P) = ₹ 4,000

Rate (r) = 6% per year

Time (n) = 1 year

Formula for compound interest (when compounded N times per year) :

A = $P\Big(1 + \dfrac{r}{100 × N}\Big)$^n×N

(a) Quarterly means 4 times in a year.

So, N = 4

By formula,

$\Rightarrow A = P\Big(1 + \dfrac{r}{100 × N}\Big)^{n×N} \\[1em] \Rightarrow A = 4000\Big(1 + \dfrac{6}{100 × 4}\Big)^{1×4} \\[1em] \Rightarrow A = 4000\Big(1 + \dfrac{6}{400}\Big)^4\\[1em] \Rightarrow A = 4000 (1 + 0.015)^4 \\[1em] \Rightarrow A = 4000 (1.015)^4 \\[1em] \Rightarrow A = 4000 × 1.06136 \\[1em] \Rightarrow A \approx 4245.45$

Hence, amount = ₹ 4,245.45.

(b) From question:

Compounded every 4 months = ₹ 4,244.83.

Compounded half-yearly = ₹ 4,243.60

Compounded quarterly = ₹ 4,245.45

∴ As the number of times interest is compounded increases, the amount also increases.

Hence, when the number of times the interest is compounded in one year is more, the C.I accrued is also more.