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The time by which a sum of money would treble it self at 8% p.a. C.I is
- a)14.28 yrs.
- b)14 yrs.
- c)12yrs.
- d)None of these
Correct answer is option 'A'. Can you explain this answer?
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The time by which a sum of money would treble it self at 8% p.a. C.I i...
To find the time by which a sum of money would treble itself at 8% p.a. compound interest, we need to use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = Final amount
P = Principal amount (initial sum of money)
r = Annual interest rate (in decimal form)
n = Number of times interest is compounded per year
t = Number of years
Trebling the initial sum means the final amount will be three times the principal amount, so A = 3P.
Now let's solve for t:
3P = P(1 + r/n)^(nt)
Dividing both sides by P, we get:
3 = (1 + r/n)^(nt)
Taking the natural logarithm of both sides:
ln(3) = nt ln(1 + r/n)
Now we can solve for t:
t = ln(3) / (n ln(1 + r/n))
Given that the interest rate is 8% per annum and compounded annually, we have:
r = 8% = 0.08
n = 1 (compounded annually)
Substituting these values into the equation, we have:
t = ln(3) / (1 ln(1 + 0.08/1))
t = ln(3) / (ln(1.08))
Using a calculator, we find that ln(3) ≈ 1.099 and ln(1.08) ≈ 0.076. Substituting these values into the equation, we get:
t ≈ 1.099 / 0.076
Calculating this value, we find that t ≈ 14.47.
Therefore, the time by which a sum of money would treble itself at 8% p.a. compound interest is approximately 14.47 years.
Since we are looking for the nearest whole number, the correct answer is option 'A': 14.28 years.
A = P(1 + r/n)^(nt)
Where:
A = Final amount
P = Principal amount (initial sum of money)
r = Annual interest rate (in decimal form)
n = Number of times interest is compounded per year
t = Number of years
Trebling the initial sum means the final amount will be three times the principal amount, so A = 3P.
Now let's solve for t:
3P = P(1 + r/n)^(nt)
Dividing both sides by P, we get:
3 = (1 + r/n)^(nt)
Taking the natural logarithm of both sides:
ln(3) = nt ln(1 + r/n)
Now we can solve for t:
t = ln(3) / (n ln(1 + r/n))
Given that the interest rate is 8% per annum and compounded annually, we have:
r = 8% = 0.08
n = 1 (compounded annually)
Substituting these values into the equation, we have:
t = ln(3) / (1 ln(1 + 0.08/1))
t = ln(3) / (ln(1.08))
Using a calculator, we find that ln(3) ≈ 1.099 and ln(1.08) ≈ 0.076. Substituting these values into the equation, we get:
t ≈ 1.099 / 0.076
Calculating this value, we find that t ≈ 14.47.
Therefore, the time by which a sum of money would treble itself at 8% p.a. compound interest is approximately 14.47 years.
Since we are looking for the nearest whole number, the correct answer is option 'A': 14.28 years.
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