At age 21 Mia begins depositing 700 dollars each quarter into an account earning 7.2% APR compounded quarterly. After 20 years, how much did she deposit and how much is interest?

Step-by-step solution

1. 1

   Convert the APR to a quarterly rate and count the periods. Quarterly compounding means four periods a year:

   $i = \frac{0.072}{4} = 0.018, \qquad n = 4 \times 20 = 80$

   The rate is exact here — no repeating decimal — which removes one source of rounding error.

2. 2

   Find what Mia personally contributes. This part needs no interest at all: it is simply payment times count.

   $700 \times 80 = \$56{,}000$

3. 3

   Compute the accumulation factor.

   $(1.018)^{80} = 4.16698791$

   Using $4.158880$ instead — an error in the third decimal — understates the final balance by over \300$.

4. 4

   Evaluate the future value.

   $FV = 700 \cdot \frac{(1.018)^{80} - 1}{0.018} = 700 \times \frac{3.16698791}{0.018} = 700 \times 175.943773 = \$123{,}160.64$

5. 5

   Subtract to isolate the interest.

   $123160.64 - 56000 = \$67{,}160.64$

   Interest is $54.5\%$ of the final balance — over a $20$-year horizon at $7.2\%$, compounding contributes more than the saver does.

6. 6

   Cross-check with a rough bound. If every dollar had been invested for the full $20$ years the balance would be 56000 \times 4.16699 = \233{,}351$; if none earned interest it would be$$56{,}000$. The computed$$123{,}160.64$ sits between, consistent with an average investment horizon of roughly half the term.