Video Transcript
Liam deposited 100 dollars in an account with an annual interest rate of 5.3 percent, where the amount of interest is added to his account at the end of each year. Given that he did not withdraw any money in three years, determine the amount of money in dollars and cents in his account at the end of each year.
We are told that Liam deposited 100 dollars into an account. This is known as the principal value or initial investment. We are told that the interest rate is 5.3 percent. As percentages are out of 100, this is equivalent to the decimal 0.053. We could calculate 5.3 percent of 100 dollars and then add this amount on to our initial investment. This would give us the amount of money after the first year. However, a quicker way to do this is using the multiplier method. This involves adding the interest rate written as a decimal to one, in this question giving us 1.053.
To calculate the amount of money in Liam’s account at the end of year one, we multiply 100 by 1.053. This is equal to 105.3 or 105 dollars and 30 cents. At the end of the first year, Liam has 105 dollars and 30 cents in his account.
As Liam does not withdraw any money from his account, we can calculate the amount of money at the end of year two by multiplying 105.30 by 1.053. This is equal to 110.8809. Rounding this to the nearest cent gives us 110 dollars and 88 cents. This is the amount of money in Liam’s account at the end of year two.
Multiplying our nonrounded value by 1.053 gives us an answer of 116.7575 and so on. Rounding this to the nearest cent, we see that Liam has 116 dollars and 76 cents in his bank account after three years. The amount of money in Liam’s account at the end of each of the first three years is 105 dollars and 30 cents, 110 dollars and 88 cents, and 116 dollars and 76 cents.
An alternative way of calculating these three values would be using our compound interest formula. This states that the new value 𝑣 is equal to the principal value 𝑃 multiplied by one plus 𝑟 over 100 all raised to the power of 𝑦. The expression in our parentheses is the multiplier, which we calculated in this question to be 1.053. The value of 𝑦 is the number of years. In this question, we would need to substitute one, two, and three into the formula.
Our year-one calculation would be exactly the same, 100 multiplied by 1.053. In year two, we multiply 100 by 1.053 squared. This gives us an answer of 110 dollars and 88 cents. In year three, we multiply 100 dollars by 1.053 cubed. This gives us an answer of 116 dollars and 76 cents. Either method will work in this question. But when we are dealing with a high number of years, it is easier to use the compound interest formula.
