Annie opens a savings account and invests 3000 pound. The savings account pays compound interest at an annual rate of three percent. Calculate the value of Annie’s investment two years after she opened the account.
We will look at two ways of solving this problem. Firstly, by looking at the amount that Annie has after year one and then year two. And secondly, by using the compound interest formula which is far more efficient when looking at a multiple-year problem. We’re told that Annie invests 3000 pounds and the savings account pays interest at an annual rate of three percent. This means that the interest she accrues in the first year is three percent of 3000.
As percentages are out of 100, three percent is equal to three out of 100 or three hundredths. Converting this to a decimal by dividing three by 100 gives us 0.03. The word “of” in mathematics means multiply. Therefore, we need to multiply 0.03 by 3000. Typing this into our calculator gives us an answer of 90. The amount of interest that the savings account pays in the first year is 90 pounds.
Annie initially had 3000 pound. Adding 90 to this gives us 3090 pounds. At the end of the first year, Annie has 3090 pounds in her bank account.
As the interest is compound Annie will get interest on the interest in her second year. She will not just get another 90 pound. In fact, the amount of interest she will gain will be three percent of 3090 pound, the amount of money that is now in the account. Once again, we can calculate this by multiplying 0.03 by 3090. 0.03 multiplied by 3090 is equal to 92.7. This means that Annie receives 92 pounds and 70 pence interest in the second year. We can now add this to 3090. At the end of two years, Annie has 3182 pounds and 70 pence in her account. The value of Annie’s investment of 3000 pound at a compound interest of three percent for two years is 3182 pounds and 70 pence.
Now let’s look at the second method using the compound interest formula. The compound interest formula states that the new amount will be equal to the original amount multiplied by a multiplier to the power of the number of payments. The original amount in this case is the amount that Annie invested, 3000 pounds. The interest rate that the account was paying was three percent. This means it has increased by three percent. We need to add three percent to 100 percent. This is equal to 103 percent. 103 percent written as a decimal is 1.03. Therefore, our decimal multiplier is 1.03.
We were told that the interest was paid annually. And we need to calculate the value of Annie’s investment after two years. Therefore, there will be two payments, at the end of the first year and at the end of the second year. In order to calculate the value of Annie’s investment two years after she opened the account, we need to multiply 3000 by 1.03 squared. Typing this into the calculator also gives us an answer of 3182 pounds and 70 pence.
Both of our methods show that the value of Annie’s investment two years after she opened the account is 3182 pounds and 70 pence. When there are a large number of payments, in this case a high number of years, it is far more efficient to use the second method. If we wanted the value of the investment after five years, using the first method, we would have to work out year one, year two, year three, year four, and year five. Whereas using the second method, we would just need one calculation. And the power would be five.