Veer invested Rs.10000 at simple interest for 2 years at the rate of *R*% and gets an interest of Rs.1400. He invested the total amount (Principal + Interest) in a scheme, which offered compound interest at the rate of *(R% + x%)* for two years.

What are the possible integral values of *x%* so that the obtained compound interest is less than Rs.2400?

(i) 1%

(ii) 2%

(iii) 3%

(iv) 4%

(v) 5%

Question

Veer invested Rs.10000 at simple interest for 2 years at the rate of *R*% and gets an interest of Rs.1400. He invested the total amount (Principal + Interest) in a scheme, which offered compound interest at the rate of *(R% + x%)* for two years.

What are the possible integral values of *x%* so that the obtained compound interest is less than Rs.2400?

(i) 1%

(ii) 2%

(iii) 3%

(iv) 4%

(v) 5%

Accepted Answer

**Solution:**

1. Calculate the simple interest rate *R*:
   - Given: Principal = Rs.10000, Simple Interest (SI) for 2 years = Rs.1400
   - Using SI formula: $ SI = \frac{P \times R \times T}{100} $
   - $ 1400 = \frac{10000 \times R \times 2}{100} \implies 1400 = 200R \implies R = 7\% $

2. Total amount after 2 years of SI:
   - $ A = P + SI = 10000 + 1400 = 11400 $

3. Compound interest rate is $ (R + x)\% = (7 + x)\% $.

4. Calculate compound interest (CI) for 2 years at rate $ (7 + x)\% $ on Rs.11400:
   - Compound amount after 2 years:
     $$
     A_{CI} = 11400 \times \left(1 + \frac{7 + x}{100}\right)^2
     $$
   - Compound interest:
     $$
     CI = A_{CI} - 11400
     $$

5. Calculate CI for each value of $ x $ and check if CI < Rs.2400:

- For $ x = 1 $ (i.e., 8%):
     $$
     CI = 11400 \times \left(1.08^2 - 1\right) = 11400 \times (1.1664 - 1) = 11400 \times 0.1664 = 1897\text{ (approx)}
     $$

- For $ x = 2 $ (i.e., 9%):
     $$
     CI = 11400 \times (1.09^2 - 1) = 11400 \times (1.1881 - 1) = 11400 \times 0.1881 = 2144\text{ (approx)}
     $$

- For $ x = 3 $ (i.e., 10%):
     $$
     CI = 11400 \times (1.10^2 - 1) = 11400 \times (1.21 - 1) = 11400 \times 0.21 = 2394
     $$

- For $ x = 4 $ (i.e., 11%):
     $$
     CI = 11400 \times (1.11^2 - 1) = 11400 \times (1.2321 - 1) = 11400 \times 0.2321 = 2645.94
     $$

- For $ x = 5 $ (i.e., 12%):
     $$
     CI = 11400 \times (1.12^2 - 1) = 11400 \times (1.2544 - 1) = 11400 \times 0.2544 = 2900.16
     $$

6. Compare CI values with Rs.2400:
   - For $ x = 1, 2, 3 $, CI < 2400
   - For $ x = 4, 5 $, CI > 2400

**Answer:** (c) Only (i), (ii) and (iii)

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