Question 126474: On January 1, Adam borrowed $20 from his brother, Gerry, and promised to repay the loan on February 18th. If Gerry charged 2% interest, compounded daily, from January 1 to February 18th inclusive, how much did Adam pay him back?
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! At the end of the first day Adam will owe Gerry the $20 plus 2% (or .02) of $20. So at the
end of the first day Adam owed Gerry:
.
20*(1 + .02) = 20*(1.02)
.
At the end of the second day Adam owed what he owed on the previous day [20*(1.02)] times
(1.02) again or Adam owed Gerry:
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(20*(1.02))*(1.02) = 20*(1.02)^2
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At the end of the third day Adam owed what he owed on the second day times 1.02 again. So
Adam owed:
.
(20*(1.02)^2)*(1.02) = 20*(1.02)^3
.
By now you might see the pattern. The amount owed on each day is the original 20 times 1.02
raised to the power equal to the number of days the money has been on loan. For example,
at the end of January, the money has been on loan for the 31 days in January so the amount
owed on January 31st is:
.
20*(1.02)^31
.
and a little calculator work will tell you that this amount is $36.95 rounded to the nearest
cent.
.
On February 18th the money will have been out on loan for 49 days (31 days in January and
18 days in February is a total of 49 days). This means that the amount of money Adam will
own is:
.
20*(1.02)^49. A calculator will tell you that 1.02^49 = 2.638811793 and when you multiply that
by $20 you get an answer of $52.78 rounded to the nearest cent.
.
Do you realize that if this continued for 365 days (one year) Adam would have to pay Gerry
back $27,548.17 ... this comes from 20*(1.02)^365.
.
In real life this is called "loan-sharking". At first it doesn't sound too bad. If you borrow
$20 for a single day, you would have to pay back $20.40. But when you get figuring it out
you can see that the longer you go without paying it back just how expensive it can get to be.
What a rip-off this is ... (bad news for Adam but very profitable for Gerry.)
.
In summary, the answer to this problem is $52.78. Hope this helps you to see how the problem
can be worked.
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