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Math  /  Algebra

QuestionParents wish to have $110,000\$ 110,000 available for a child's education. If the child is now 8 years old, how much money must be set aside at 7%7 \% compounded semiannually to meet their financial goal when the child is 18?18 ? Click the icon to view some finance formulas.
The amount that should be set aside is $\$ □\square (Round up to the nearest dollar.)

Studdy Solution

STEP 1

1. The future value needed for the child's education is $110,000.
2. The child is currently 8 years old and will need the funds at age 18, which is 10 years from now.
3. The interest rate is 7% compounded semiannually.
4. We need to find the present value or the amount to be set aside now.

STEP 2

1. Identify the appropriate finance formula.
2. Define the variables in the formula.
3. Substitute the known values into the formula.
4. Solve for the unknown variable.

STEP 3

Identify the appropriate finance formula. Since we are dealing with compound interest, we use the future value formula for compound interest:
A=P(1+rn)nt A = P \left(1 + \frac{r}{n}\right)^{nt}
Where: - A A is the future value of the investment/loan, including interest. - P P is the principal investment amount (the initial deposit or loan amount). - r r is the annual interest rate (decimal). - n n is the number of times that interest is compounded per year. - t t is the time the money is invested for in years.

STEP 4

Define the variables in the formula.
Given: - A=110,000 A = 110,000 - r=0.07 r = 0.07 (7% as a decimal) - n=2 n = 2 (since interest is compounded semiannually) - t=10 t = 10 (years until the child is 18)

STEP 5

Substitute the known values into the formula to solve for P P .
110,000=P(1+0.072)2×10 110,000 = P \left(1 + \frac{0.07}{2}\right)^{2 \times 10}
110,000=P(1+0.035)20 110,000 = P \left(1 + 0.035\right)^{20}
110,000=P(1.035)20 110,000 = P (1.035)^{20}

STEP 6

Solve for P P .
First, calculate (1.035)20 (1.035)^{20} .
(1.035)20≈1.9837 (1.035)^{20} \approx 1.9837
Now, solve for P P :
110,000=P×1.9837 110,000 = P \times 1.9837
P=110,0001.9837 P = \frac{110,000}{1.9837}
P≈55,462.50 P \approx 55,462.50
Since we need to round up to the nearest dollar, the amount that should be set aside is:
55,463 \boxed{55,463}

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