Question Video: Using an Exponential Function to Model a System Involving Compound Interest Mathematics

Video Transcript

If 800 dollars is earning interest semiannually at two percent per annum, what is the amount after 𝑛 years?

This question involves compound interest. And we begin by recalling the compound interest formula. This states that 𝐴 is equal to 𝑃 multiplied by one plus 𝑟 over 𝑛 all raised to the power of 𝑛𝑡, where 𝑃 is the principal or original amount, 𝑟 is the interest rate written as a decimal, and 𝑛 is the number of times interest is compounded per unit of 𝑡, which is the time.

In this question, the original amount is 800 dollars. So the value of 𝑃 is 800. The interest rate of two percent written as a decimal is 0.02. And this is the value of 𝑟. As the interest is compounded semiannually, the value of 𝑛 from our general formula is two. Finally, the time 𝑡 in the general formula is equal to 𝑛, as we need to find the amount after 𝑛 years.

Substituting these values into the general formula, we have 𝐴 is equal to 800 multiplied by one plus 0.02 divided by two all raised to the power of two 𝑛. 0.02 divided by two is 0.01. And adding this to one gives us 1.01. Our expression simplifies to 800 multiplied by 1.01 to the power of two 𝑛. This is the amount of money in the account after 𝑛 years.

Whilst it is not required in this question, let’s assume we wanted to calculate the amount of money after 10 years. Substituting 𝑛 equals 10 into our expression, we have 800 multiplied by 1.01 raised to the power of two multiplied by 10. Typing this into our calculator gives us 976.1520 and so on. Rounding to two decimal places, we have 976.15. This means that after 10 years there will be 976 dollars and 15 cents in the account. This process could be repeated for any value of 𝑛, as the amount after 𝑛 years is equal to 800 multiplied by 1.01 raised to the power of two 𝑛.