Compound Interest [Applications] — Case-Study Based Questions

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 ?

Answer

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.

In the year 2024-25 period, India collected approximately 1.46 crore (14.6 million) units of blood, which was about 15% more than the previous year. This volume is roughly equal to the estimated annual national requirement, indicating progress toward meeting demand. The Indian Red Cross Society (IRCS) typically collects around 30,000 units per year as per the report in 2024-25.

Based on the above information, answer the following :

(i) If the number of blood donors increases by 15% every year, then what will be the number of units to be collected by IRCS in the year 2026-27.

(ii) If 2024-25 is taken as a base year and it is predicted that the number of units to be collected by IRCS in 3 years will be 39,930, then what will be rate of increase of donors?

Answer

(i) Given,

Units collected by IRCS in 2024-25 = 30,000

Increase rate = 15% per year

From 2024-25 to 2026-27 = 2 years

By formula, A = $P\Big(1 + \dfrac{r}{100}\Big)$ⁿ

$\Rightarrow A = 30000 \Big(1 + \dfrac{15}{100}\Big)^2 \\[1em] \Rightarrow A = 30000(1 + 0.15)^2 \\[1em] \Rightarrow A = 30000(1.15)^2 \\[1em] \Rightarrow A = 30000 × 1.3225 \\[1em] \Rightarrow A = 39,675 \text { units}$

Hence, in 2026-27 number of units collected = 39,675 units.

(ii) Given,

Initial (P) = 30,000

Final (A) = 39,930

Time (n) = 3 years

By formula, A = $P\Big(1 + \dfrac{r}{100}\Big)$ⁿ

$\Rightarrow 39930 = 30000 \Big(1 + \dfrac{r}{100}\Big)^3 \\[1em] \Rightarrow \dfrac{39930}{30000} = \Big(1 + \dfrac{r}{100}\Big)^3 \\[1em] \Rightarrow 1.331 = \Big(1 + \dfrac{r}{100}\Big)^3 \\[1em] \Rightarrow 1 + \dfrac{r}{100} = \sqrt[3]{1.331} \\[1em] \Rightarrow 1 + \dfrac{r}{100} = 1.1 \\[1em] \Rightarrow \dfrac{r}{100} = 0.1 \\[1em] \Rightarrow r = 100 × 0.1 \\[1em] \Rightarrow r = 10$

Hence, rate of increase = 10%.