Samantha invests 26000 pound in an
account which offers compound interest. For the first two years, the rate
of interest is 2.6 percent. For the next three years, the rate
of interest is 1.7 percent. Calculate the value of Samantha’s
investment after five years. Give your answer to the nearest 100
If we are solving any problem
involving compound interest, our first step is to calculate the decimal
multiplier. In this question, for the first two
years, there was an increase of 2.6 percent. For the next three years, there was
an increase of 1.7 percent. We need to calculate the decimal
multipliers for these two percentages.
Our initial value is 100
percent. Therefore, to calculate a 2.6
percent increase, we need to add 100 percent and 2.6 percent. This is equal to 102.6 percent. The word “percent” means out of
100. Therefore, we need to divide 102.6
percent by 100 to work out the decimal multiplier. This gives us an answer of
1.026. The decimal multiplier equivalent
to a 2.6 percent increase is 1.026.
We need to repeat this process for
the 1.7 percent increase. 100 percent plus 1.7 percent is
equal to 101.7 percent. Dividing this by 100 gives us a
decimal multiplier of 1.017. A 1.7 percent increase is equal to
a decimal multiplier of 1.017.
An alternative method to calculate
the decimal multipliers would be to add the percentage divided by 100 to one. One plus 2.6 divided by 100 is
equal to 1.026 and one plus 1.7 divided by 100 is equal to 1.017.
We were asked to calculate the
value of Samantha’s investment after five years. Samantha’s original investment was
26000 pound. For the first two years, the rate
of interest was 2.6 percent. Therefore, we need to multiply by
1.026 for year one and 1.026 for year two. As the interest rate was 1.7
percent for the next three years, we need to multiply by 1.017 three times.
This calculation can be rewritten
as 26000 multiplied by 1.026 squared multiplied by 1.017 cubed. We square as it was this interest
for two years and we cube because it was this interest rate for three years. Typing this into the calculator
gives us 28789.288 and so on.
We were asked to give our answer to
the nearest 100 pounds. Therefore, the deciding number is
the eight in the tens column. As this is greater than five, we
will round up. The value of Samantha’s investment
after five years to the nearest 100 pound is 28800 pounds.
In some compound interest
questions, we are asked to calculate the amount of interest earned. In order to do this, we subtract
the original investment from the final amount. In this case, we subtract 26000
from 28800. This is equal to 2800. Therefore, Samantha has earned 2800
pound interest over the five-year period.
A longer method which is sometimes
used for compound interest problems on a noncalculator paper is to calculate the
amount of money after year one, then year two, then year three, and so on. To do this, we would work out 2.6
percent of 26000 and then add on our answer.
In the second year, we would work
out 2.6 percent of this new amount and add it on again. We would then repeat this process
with 1.7 percent for years three, four, and five. If we have a calculator though, it
is much quicker to do the calculation highlighted: 26000 multiplied by 1.026 squared
multiplied by 1.017 cubed.