Video Transcript
Benjamin deposits 100 dollars in a savings account that gives him a 15-percent interest on his savings each month. Michael has 350 dollars in a cashing account that he withdraws five dollars from each month. After how many months do the two have approximately the same bank balance?
In this question, we’re told to consider two different people. Benjamin has 100 dollars, and Michael has 350 dollars. Notice that Benjamin has 100 dollars, but his money is in a savings account, so his bank balance will increase. Michael, on the other hand, has 350 dollars, a larger amount, but he’s withdrawing five dollars every month, so his bank balance is going to decrease. We need to work out after how many months will Benjamin and Michael have approximately the same bank balance.
Let’s start by looking at how Michael’s bank balance will change over time. We’re told that Michael withdrawals five dollars each month. So after one month, he’s going to have 350 dollars subtract five dollars, which is 345 dollars. After two months, he’ll have another five dollars less, which is 340 dollars. And we can continue listing the values for several months. Next, let’s have a look at what happens with Benjamin’s account. We’re told that he has 15 percent interest added to his savings each month. So after one month, he’ll have his starting amount of 100 dollars plus 15 percent of 100 dollars. And we know that 15 percent of 100 dollars is 15, so in total he’d have 115 dollars.
We can continue working out the value of his account each month, but we need to be careful as the starting amount in each month is the total in his account from the previous months. So after two months, he’ll have 115 dollars plus 15 percent of 115 dollars. And that’s equal to 132 dollars and 25 cents. You might realize at this point that we’re not very close to the values of Michael’s account. And there is in fact an easier way to work these values out. We can use this formula for exponential functions involving interest. 𝑉 is equal to 𝑃 times one plus 𝑟 over 100 to the power 𝑛, where the value 𝑉 represents the value of the investment, 𝑃 is the principal or starting amount, 𝑟 is the interest rate, and 𝑛 is the number of time periods.
Although we usually see this formula associated with annual interest rate, we can use it here as we’re considering an interest rate per month but we’re also considering the number of months before a certain value is reached. So let’s see how we can apply this formula in this situation. We’re trying to work out the value of 𝑉 each month. The starting amount is 100 dollars. The interest rate is 15 percent. And the value of 𝑛 will change depending on which month we’re trying to calculate.
Before we use this formula, notice that we can simplify this value of one plus 15 over 100 to the decimal 1.15. So if we wanted to find the value after one month, the value of 𝑛 would be one and the answer would be 115 dollars, which is what we got previously. For the second month, the value of 𝑛 would be equal to two and we’d get the value of 132 dollars and 25 cents. Now that we have this calculation in the calculator, it might be easier to use the replay function to go back and quickly change the value of 𝑛 for each month. After three months, Benjamin will have 152 dollars and nine cents rounded to two decimal places. We can see by using this formula how the amount of money in Benjamin’s account changes through the months.
We now need to work out after how many months do Michael and Benjamin have approximately the same in each of their bank accounts. Let’s compare the values in their accounts after eight months and after nine months. At eight months, the difference in their accounts would be 310 dollars subtract 305 dollars and 90 cents. That’s four dollars and 10 cents. After nine months, the difference between their bank balances would be 351 dollars and 79 cents subtract 305 dollars. And that’s 46 dollars and 79 cents. Therefore, we can give the answer that it’s after eight months that Benjamin and Michael have approximately the same bank balance.
