Interest added to an updated balance is exponential growth: the balance is multiplied by the same factor each compounding period. Find the starting amount and the factor for one period, then make the exponent count compounding periods, even if time is given in different units. Adding the monthly percentages instead would miss the interest earned on earlier interest.
Hints
- Hint 1
A growth factor includes the whole balance plus the interest it earns. Convert the monthly percent to a decimal, then add it to . What number multiplies the balance each month?
- Hint 2
Compounding means each month's interest is figured on the updated balance, so the growth factor is used repeatedly. Since counts years, how many monthly periods have passed after years?
Step-by-step
Build the monthly growth model
Step 1Convert the monthly rate
No Desmos needed. The given rate and time units determine the model directly. A percent means “out of .” Divide the monthly rate by to write it as a decimal:
- Step 2
Find the monthly growth factor
Because the interest is compounded, each month's interest is added to the current balance, not only to the original deposit. Keep the whole balance, represented by , and add the interest to get the growth factor, the number multiplied each month:
- Step 3
Count the monthly periods
There are months in a year. Count the months in years: . An exponent tells how many times to use a factor. It counts months, even though counts years.
- Step 4
Write the balance function
Start with the $500 deposit and apply the monthly factor once for each of the months: . This gives the balance in dollars after years. Choice C.