Question 1190943: Mr. Mehta deposited Rs. 10,000 in HSBC bank at his sixth marriage anniversary and
borrowed Rs. 20,000 from a money lender but he could not pay any amount in a period
of 1460 days. Accordingly, the money lender demands now Rs. 26,500 from him.
a. What is the difference between the ‘Exact Method’ and ‘Ordinary Method’?
b. At what rate percent per annum compound interest did the latter lend his
money? (Use the exact method).
c. Find the number of years and the fraction of a year in which the deposit money
will treble itself at compound interest 8 percent per annum.
d. In what time will a sum of Rs. 1,234 amount to Rs. 5,678 at 8% per annum
compound interest, payable quarterly?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! a. What is the difference between the ‘Exact Method’ and ‘Ordinary Method’?
here's a reference on exact interest versus ordinary interest.
https://www.smartcapitalmind.com/what-is-exact-interest.htm
b. At what rate percent per annum compound interest did the latter lend his
money? (Use the exact method).
you lend 20,000 and need to pay the lender in 1460 days.
with exact interest, this comes out to be 1460 / 365 = 4 years.
with ordinary interest, this could come out to be 1460 / 360 = 4.05555555.... years.
the formula is f = p * (1 + r) ^ n
f is the future value
p is the present value
r is the interest rate per time period.
n is the number of time periods.
your time periods are in years.
i believe you want to use exact interest.
therefore, you need to pay back the loan in 4 years.
f = p * (1 + r) ^ n becomes:
26500 = 20000 * (1 + r) ^ 4
divide both sides of the equation by 20000 to get:
26500 / 20000 = (1 + r) ^ 4
take the fourth root of both sides of this equation to get:
(26500 / 20000) ^ (1/4) = 1 + r
solve for 1 + r to get:
1 + r = 1.072886967.
subtract 1 from that and multiply it by 100 to get:
7.2886967% per year.
solve for f in the equation by replacing 1 + r with 1.072886967 to get:
f = 20000 * 1.072886967 ^ 4 = 26500.
c. Find the number of years and the fraction of a year in which the deposit money will treble itself at compound interest 8 percent per annum.
deposit money is 10,000
interest rate is 8% per year compounded annualy.
formula to use is f = p * (1 + r) ^ n
f = 30,000
p = 20,000
1 + r = 1.08
n is what you want to find.
formula becomes:
30,000 = 10,000 * 1.08 ^ n
divide both sides by 10,000 to get:
3 = 1.08 ^ n
take the log of both sides of the equation to get:
log(3) = log(1.08 ^ n)
by one of the properties of logs, this becomes:
log(3) = n * log(1.08)
divide both sides of this equation by log(1.08) to get:
log(3) / log(1.08) = n
solve for n to get:
n = 14.27491459.
confirm by replacing n in the original equation by that and solving for f to get:
f = 10000 * 1.08 ^ 14.27491459 = 30,000.
d. In what time will a sum of Rs. 1,234 amount to Rs. 5,678 at 8% per annum
compound interest, payable quarterly?
i believe you want the inteest rate to be compounded quarterly.
8% per year, compounded quarterly, is equal to 8/4 = 2% per quarter.
the formula used would still be f = p * (1 + r) ^ N
the formula would becomes:
5678 = 1234 * 1.02 ^ n
divide both sides of the equation by 1234 to get:
5678/1234 = 1.02 ^ n
take the log of both sides of the equation to get:
log(5678/1234) = log(1.02 ^ n)
by one of the properties of logs, this becomes:
log(5678/1234) = n * log(1.02)
divide both sides of this equation by log(1.02) to get:
log(5678/1234) / log(1.02) = n
solve for n to get:
n = 77.07755692 quarters.
divide by 4 to get:
n = 19.26938923 years.
confirm by replacing n with the number of quarters to get:
f = 1234 * 1.02 ^ 77.07755692 = 5678.
the nominal growth rate is 1.08 per year.
the effective growth rate is 1.02 ^ 4 = 1.08243216.
the equation using years instead of quarters is:
f = 1234 * 1.08243216 ^ 19.26938923 = 5678.
same answer using years as you got using months.
you had to use the effective interest rate rather than the nominal interest rate because the effective interest rate assumes the actual annual growth rate when the interest rate is compounded quarterly.
relationship between growth rate and interest rate is shown below:
annual interest rate is 8%
the decimal equivalent of this is .08.
you take the percent and divide it by 100 to get the decimal equivalent.
the nominal growth rate is 1 plus this = 1.08
when you compound the interest rate quarterly, you divide the annual interest rate by 4.
you get quarterly interest rate of 8%/4 = 2%.
the decimal equivalent of this is .02
the quarterly growth rate is 1 plus this = 1.02.
the effective annual growth rate if 1.02 ^ 4 = 1.08243216
the decimal equivalent of the effect interest rate per year is equal to this minus 1 = .08243216.
multiply that by 100 to get 8.243216% effective interest rate per year.
all of the above is what i think you are asking.
if it's not, let me know.
i'll be available to answer any questions or concerns about this that you might have.
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