Consider the given question and decide which of the following statements is sufficient to answer the question. X took a loan from Y on compound interest. Find the rate per annum? Statements: 1. After 3 years, X paid Rs. 500 as interest. 2. After 3 years, X paid Rs. 1,500 to clear his loan with Y.
Together, both 1 and 2 are sufficient to answer the question.
Solution:
Step 1: Recall the compound interest formula:
A = P(1 + r)^n
Where:
- A = Total amount after n years
- P = Principal (loan amount)
- r = Annual rate (in decimal)
- n = Number of years
Step 2: Analyze statement 1:
Interest after 3 years = 500
CI = P((1+r)^3 - 1) = 500
Here, P is unknown, so we cannot find r. Statement 1 alone is NOT sufficient.
Step 3: Analyze statement 2:
Total amount paid after 3 years = 1500
A = P(1+r)^3 = 1500
Here, P is unknown, so we cannot find r. Statement 2 alone is NOT sufficient.
Step 4: Combine statements 1 and 2:
From statement 2, A = 1500. From statement 1, CI = 500 ⇒ P = 1500 - 500 = 1000
Now we can solve for r:
1500 = 1000(1+r)^3
(1+r)^3 = 1.5
r = (1.5)^(1/3) - 1
Both statements together are sufficient to find the rate.
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