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The simple and compound interest that can be earned in two years at the same rate is Rs 4000 and Rs 4180 respectively. What is the rate (percent per annum) of interest?
- a)18
- b)4.5
- c)9
- d)12
Correct answer is option 'C'. Can you explain this answer?
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Given data: Simple interest = Rs 4000, Compound interest = Rs 4180, Time = 2 years
Let the principal amount be P and the rate of interest be R% per annum.
Calculation:
1. Simple interest formula: SI = (P * R * T) / 100
=> 4000 = (P * R * 2) / 100
=> 20P = 100R …(1)
2. Compound interest formula: CI = P[(1 + R/100)^T - 1]
=> 4180 = P[(1 + R/100)^2 - 1]
=> (1 + R/100)^2 = 1 + (4180/P)
=> (1 + R/100)^2 = 1 + (2090/P) …(2)
3. Divide equation (2) by equation (1):
=> [(1 + R/100)^2] / 20P = (1 + R/100) / 100
=> (1 + R/100) / 20P = 1 / 100
=> 100(1 + R/100) = 20P …(3)
4. Substitute equation (1) into equation (3):
=> 100 + 100R/100 = 20P
=> 100 + R = 20P
=> R = (20P - 100) …(4)
5. Substitute equation (4) into equation (1):
=> 20P = 100(20P - 100)/100
=> 20P = 2000 - 1000
=> P = 50
6. Substitute P = 50 into equation (4):
=> R = (20 * 50 - 100)
=> R = 900 / 10
=> R = 9%
Therefore, the rate of interest per annum is 9%.
Let the principal amount be P and the rate of interest be R% per annum.
Calculation:
1. Simple interest formula: SI = (P * R * T) / 100
=> 4000 = (P * R * 2) / 100
=> 20P = 100R …(1)
2. Compound interest formula: CI = P[(1 + R/100)^T - 1]
=> 4180 = P[(1 + R/100)^2 - 1]
=> (1 + R/100)^2 = 1 + (4180/P)
=> (1 + R/100)^2 = 1 + (2090/P) …(2)
3. Divide equation (2) by equation (1):
=> [(1 + R/100)^2] / 20P = (1 + R/100) / 100
=> (1 + R/100) / 20P = 1 / 100
=> 100(1 + R/100) = 20P …(3)
4. Substitute equation (1) into equation (3):
=> 100 + 100R/100 = 20P
=> 100 + R = 20P
=> R = (20P - 100) …(4)
5. Substitute equation (4) into equation (1):
=> 20P = 100(20P - 100)/100
=> 20P = 2000 - 1000
=> P = 50
6. Substitute P = 50 into equation (4):
=> R = (20 * 50 - 100)
=> R = 900 / 10
=> R = 9%
Therefore, the rate of interest per annum is 9%.
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