A sum of money lent out at simple interest amounts to ₹600 after two years and to ₹900 after a further period of 5 years. The interest rate is:

Correct option is C

Given:
Amount after 2 years ($$A_1) = ₹600$$​
Amount after additional 5 years (i.e., total 7 years)$$(A_2) = ₹900$$​
Formula Used:

Simple Interest (SI):

A = P + SI 
SI =$$\frac{P \times R \times T}{100}$$​​
Since P remains the same, the difference in amounts gives the interest for the additional time period.
Solution:

Interest for 5 years (from year 2 to year 7):
Interest = $$A_2 - A_1 = 900 - 600 = ₹300$$​​
Rate of Interest (R):
The interest of ₹300 is earned in 5 years.
Let the principal at the end of 2 years be ₹600 (which includes the principal + interest for 2 years).
Let the original principal = P .
After 2 years:
​$$A_1 = P + \frac{P \times R \times 2}{100} = 600$$​​

​$$P \left(1 + \frac{2R}{100}\right) = 600 ...(1)$$​​
After 7 years:
​$$A_2 = P + \frac{P \times R \times 7}{100} = 900$$​​

​$$P \left(1 + \frac{7R}{100}\right) = 900 ...(2)$$​​
Divide (2) by (1):

​$$\frac{1 + \frac{7R}{100}}{1 + \frac{2R}{100}} = \frac{900}{600} = 1.5 \\ \ \\1 + \frac{7R}{100} = 1.5 + \frac{3R}{100}\\ \ \\\frac{7R}{100} - \frac{3R}{100} = 1.5 - 1\\ \ \\\frac{4R}{100} = 0.5$$​​

4R = 50 

R = 12.5%

Alternate Method:​

Interest for 5 years = ₹900 - ₹600 = ₹300

So, SI for 5 years = ₹300
SI for 1 year =$$\frac{ ₹300 } 5$$​ = ₹60
Now, amount after 2 years = ₹600
SI for 2 years = ₹60 × 2 = ₹120
Principal (P) = ₹600 - ₹120 = ₹480
R = $$\frac{120\times 100}{480\times 2}$$ = $$\frac{100}8=$$ 12.5%​

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