If Rs. 2100 and Rs. 1500 are given at x% per annum for 5 years and 8 years respectively, and the total interest received is Rs. 1350, find the rate of interest.

Steps to Solve

1. Understand the interest formula

The formula for simple interest is given by:

$$ I = \frac{P \times r \times t}{100} $$

where (I) is the interest, (P) is the principal amount, (r) is the rate of interest (in percent), and (t) is the time in years.

2. Calculate the interest from the first investment

For the first investment of Rs. 2100 for 5 years at (x%), the interest can be calculated as:

$$ I_1 = \frac{2100 \times x \times 5}{100} $$

3. Calculate the interest from the second investment

For the second investment of Rs. 1500 for 8 years at (x%), the interest can be calculated as:

$$ I_2 = \frac{1500 \times x \times 8}{100} $$

4. Set up the equation for total interest

According to the problem, the total interest from both investments is Rs. 1350. Therefore, we can write the equation:

$$ I_1 + I_2 = 1350 $$

Substituting the equations from steps 2 and 3:

$$ \frac{2100 \times x \times 5}{100} + \frac{1500 \times x \times 8}{100} = 1350 $$

5. Simplify the equation

First, we can multiply the entire equation by 100 to eliminate the denominators:

$$ 2100 \times x \times 5 + 1500 \times x \times 8 = 135000 $$

Simplifying further:

$$ 10500x + 12000x = 135000 $$

6. Combine like terms and solve for (x)

Combine the terms with (x):

$$ 22500x = 135000$$

Now, divide both sides by 22500:

$$ x = \frac{135000}{22500} = 6 $$