Although this question does not actually ask you about interest rates, it calls for you to use the formula for compound interest. This is a formula that is commonly used in calculations in the fields of business and economics. The formula is used when you have a given amount of something (usually money) at the beginning of a period and that amount increases by a given percentage each year. In this case, we know how much money it costs Rachael to attend school for one year right now, we know how many years she still has to go, and we know the percentage by which the cost will increase each year. With this information, we can calculate her fees during her senior year by using the formula for compound interest.
The formula for compound interest is
FV = PV × (1+r)^n
Where FV = future value, PV = present value, r = interest rate and n = number of years.
We can now plug in Rachael’s numbers, which are $12,000 for present value (the cost of a year right now), .04 for the interest rate (it has to be expressed as a decimal), and 3 for the number of years. That gives us
FV = 12000 (1+.04)^3
Because of the rules of order of operations, we first add the numbers in the parenthesis and we get
FV = 12000 (1.04)^3
Next, we use the exponent, giving us
FV = 12000 x 1.125
Finally, we multiply those numbers and get
FV = $13498.37
This means that, in Rachael’s senior year, her tuition, fees, and expenditures for books will come to $13,498.37.
I definitely agree with the answer being 12000 * (1.04)^3 = $13498.37 for her expenditures during senior year. However, I just wanted to provide some added explanation as to the intuition behind the calculation.
One potential pitfall for people is to assume that since the percentage cost rise is constant (4%), the amount of cost rise is the same. This would lead people to calculate 4% of 12000 ($480), multiply that by 3 years for the 3 year difference, then tack that on to the current value of 12000 to get a final answer of $13440. The reason why this approach is incorrect is because there are two factors in determining the amount of cost rise each year: the percentage cost rise and the amount of money upon which the percentage is based. While it is true that the percentage is staying the same, the base amount of money is increasing each year. Therefore, the net effect is that the amount of cost increase is increasing across the years as well, making this approach incorrect.
Multiplying by 1.04^3 works because the it can also be expressed as (1.04 * (1.04 * (1.04 * 12000))). Written out like this, it is easier to see that the 4 percent additional cost from the first year (in the very inside parenthesis) is being factored into the base cost for the next year.
If Rachael's tuition, fees, and expenditures for books this year total $12,000, what will they be during her senior year (3 years from now), assuming cost rise 4% annually?
Here's a different way of solving it.
Year 1: 12000*4% = 480
12000 + 480 = 12480
Year 2: 12480*4% = 499.2
12480 + 499.2 = 12979.2
Year 3: 12979.2*4% = 519.168
12979.2 + 519.168 = 13498.368 or 13498.37 Answer.