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The difference of compound interest and simple interest for 3 years and for 2 years are in ratio 23: 7 respectively. What is rate of interest per annum (in %)?
- a)200/7
- b)100/7
- c)300/7
- d)400/7
Correct answer is option 'A'. Can you explain this answer?
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Given,
Ratio of difference of compound interest and simple interest for 3 years and 2 years = 23:7
Let P be the principal amount, r% be the rate of interest per annum and n be the number of years.
The formula for simple interest is given by:
Simple Interest = (P * r * n) / 100
The formula for compound interest is given by:
Compound Interest = P * (1 + r/100)^n - P
Let the difference between compound interest and simple interest for 3 years and 2 years be D1 and D2 respectively.
D1 = P * [(1 + r/100)^3 - 1 - (3r/100)]
D2 = P * [(1 + r/100)^2 - 1 - (2r/100)]
Dividing D1 by D2, we get:
D1/D2 = [(1 + r/100)^3 - 1 - (3r/100)] / [(1 + r/100)^2 - 1 - (2r/100)]
Simplifying the above equation, we get:
D1/D2 = (1 + r/100) - (1/[(1 + r/100)^2 - 1 - (2r/100)])
Let (1 + r/100) = x. Substituting this in the above equation, we get:
D1/D2 = x - (1/(x^2 - 1 - (2x - 2)))
D1/D2 = x - (1/[(x-1)^2 - 2])
Let (D1/D2) = 23/7. Substituting this in the above equation, we get:
23/7 = x - (1/[(x-1)^2 - 2])
Solving the above equation, we get:
x = 2
Therefore, (1 + r/100) = 2
r/100 = 1
r = 100/7 %
Hence, the rate of interest per annum is 100/7 %.
Ratio of difference of compound interest and simple interest for 3 years and 2 years = 23:7
Let P be the principal amount, r% be the rate of interest per annum and n be the number of years.
The formula for simple interest is given by:
Simple Interest = (P * r * n) / 100
The formula for compound interest is given by:
Compound Interest = P * (1 + r/100)^n - P
Let the difference between compound interest and simple interest for 3 years and 2 years be D1 and D2 respectively.
D1 = P * [(1 + r/100)^3 - 1 - (3r/100)]
D2 = P * [(1 + r/100)^2 - 1 - (2r/100)]
Dividing D1 by D2, we get:
D1/D2 = [(1 + r/100)^3 - 1 - (3r/100)] / [(1 + r/100)^2 - 1 - (2r/100)]
Simplifying the above equation, we get:
D1/D2 = (1 + r/100) - (1/[(1 + r/100)^2 - 1 - (2r/100)])
Let (1 + r/100) = x. Substituting this in the above equation, we get:
D1/D2 = x - (1/(x^2 - 1 - (2x - 2)))
D1/D2 = x - (1/[(x-1)^2 - 2])
Let (D1/D2) = 23/7. Substituting this in the above equation, we get:
23/7 = x - (1/[(x-1)^2 - 2])
Solving the above equation, we get:
x = 2
Therefore, (1 + r/100) = 2
r/100 = 1
r = 100/7 %
Hence, the rate of interest per annum is 100/7 %.
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The difference of compound interest and simple interest for 3 years and for 2 years are in ratio 23: 7 respectively. What is rate of interest per annum (in %)?a)200/7b)100/7c)300/7d)400/7Correct answer is option 'A'. Can you explain this answer?
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