If the compound interest (CI) and simple interest (SI) of a certain sum of money for 2 years is $410 and $400 respectively, find the rate of compound interest and principle.
Understand the Problem
The question is asking to find the rate of compound interest and the principle, given that the compound interest (CI) and simple interest (SI) for a certain sum of money over 2 years are $410 and $400 respectively.
Answer
Rate of compound interest = $5\%$ Principal = $4000$
Answer for screen readers
Rate of compound interest = $5%$ Principal = $4000$
Steps to Solve
- Calculate simple interest for 1 year
Since the simple interest (SI) for 2 years is $400, the simple interest for 1 year is: $SI_{1} = \frac{400}{2} = 200$
- Calculate the difference between CI and SI for 2 years
The difference between compound interest (CI) and simple interest (SI) for 2 years is given as $410 - 400 = 10$. This difference is the interest on the first year's simple interest.
- Find the rate of interest
The interest on the first year's simple interest is $10. The rate of interest (r) can be calculated as: $r = \frac{Interest \ on \ first \ year's \ SI}{SI \ for \ 1 \ year} \times 100 = \frac{10}{200} \times 100 = 5%$
- Calculate the principal
We know that the simple interest for 1 year is $200 and the rate of interest is 5%. We can use the formula for simple interest to find the principal (P): $SI = \frac{P \times r \times t}{100}$
Where: $SI = 200$ $r = 5$ $t = 1$
So, $200 = \frac{P \times 5 \times 1}{100}$. Solving for P:
$P = \frac{200 \times 100}{5} = 4000$
Rate of compound interest = $5%$ Principal = $4000$
More Information
The rate of compound interest is the same as the simple interest in this case, since they are calculated over the same period and the difference between the CI and SI is used to derive the rate. The principal represents the initial sum of money on which the interest is calculated.
Tips
A common mistake is to confuse the compound interest and simple interest formulas. Another mistake might involve incorrectly calculating the interest earned in the first year, which stems from misunderstanding the fundamental theory of the problem.
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