How much money (in dollars) should be invested in an account that earns 5% interest, compounded quarterly, to yield $10,000 in 7 years?
The correct answer and explanation is:
To solve this, we use the formula for compound interest, which is: A=P(1+rn)ntA = P \left(1 + \frac{r}{n}\right)^{nt}
Where:
- AA is the amount of money accumulated after interest (in this case, $10,000),
- PP is the principal amount (the initial investment),
- rr is the annual interest rate (as a decimal, so 5% becomes 0.05),
- nn is the number of times the interest is compounded per year (quarterly means n=4n = 4),
- tt is the time the money is invested or borrowed for, in years (7 years in this case).
We are solving for PP, the initial investment. So, rearranging the formula: P=A(1+rn)ntP = \frac{A}{\left(1 + \frac{r}{n}\right)^{nt}}
Substitute the known values:
- A=10,000A = 10,000,
- r=0.05r = 0.05,
- n=4n = 4,
- t=7t = 7.
P=10,000(1+0.054)4×7P = \frac{10,000}{\left(1 + \frac{0.05}{4}\right)^{4 \times 7}}
Now calculate the expression inside the parentheses first: 1+0.054=1+0.0125=1.01251 + \frac{0.05}{4} = 1 + 0.0125 = 1.0125
Next, calculate the exponent: 4×7=284 \times 7 = 28
So now we have: P=10,000(1.0125)28P = \frac{10,000}{(1.0125)^{28}}
Now calculate (1.0125)28(1.0125)^{28}: (1.0125)28≈1.421897(1.0125)^{28} \approx 1.421897
Now divide: P=10,0001.421897≈7,031.36P = \frac{10,000}{1.421897} \approx 7,031.36
Thus, the amount that should be invested today is approximately $7,031.36.
Explanation:
When interest is compounded, the investment grows not just by a simple percentage each year, but by compounding the interest every quarter. The formula reflects this, adjusting the interest rate and number of compounding periods to account for quarterly compounding. By rearranging the formula and plugging in the known values, you can determine the principal amount that needs to be invested to reach the target of $10,000 in 7 years.