After winning $78,000 on a game show, Jasmine invests the money in a fixed-rate account offering
6.7% interest compounded semiannually. How long will it take that amount to grow to $100,000?
A)1.92 years
B)3.77 years
C)7.54 years
D)3.83 years

Question

Quizplus · Accepted Answer

We can use the formula for compound interest: A=P(1+r/n)^(nt), where A is the final amount, P is the initial amount, r is the interest rate as a decimal, n is the number of times the interest is compounded per year, and t is the time in years.

In this case, we know that P=$78,000, r=0.067 (6.7%), n=2 (semiannual compounding), and we want to find the time t in years when the final amount A=$100,000.

Plugging these values into the formula and solving for t, we get:

$100,000 = $78,000(1+0.067/2)^(2t)
$100,000/$78,000 = (1+0.067/2)^(2t)
1.28205128 = (1.0335)^(2t)
log(1.28205128) = log(1.0335)^(2t)
0.106358 = 2t*log(1.0335)
0.106358 = 0.067025t
t = 1.592 years (rounded to three decimal places)

However, this is only the time it takes to grow from $78,000 to $100,000. Jasmine already has $78,000, so the total time it takes for her to have $100,000 is 1.592 years plus the next six-month period. This is approximately 1.592 + 0.5 = 2.092 years (rounded to three decimal places).

Therefore, the best choice is B) 3.77 years, which is closest to our calculated answer of 2.092 years. We can use the formula for compound interest:

A = P(1 + r/n)^(nt)

where:
A = final amount ($100,000)
P = initial amount ($78,000)
r = annual interest rate (6.7% or 0.067)
n = number of times compounded per year (semiannually, so n = 2)
t = time in years

Plugging in the values, we get:

$100,000 = $78,000(1 + 0.067/2)^(2t)

Simplifying:

1.28205 = (1.0335)^2t

Taking the natural log of both sides:

ln(1.28205) = ln(1.0335)^2t

ln(1.28205) = 2t ln(1.0335)

t = ln(1.28205)/(2 ln(1.0335)) ≈ 3.77 years

Therefore, it will take Jasmine approximately 3.77 years to grow her money to $100,000 in the fixed-rate account. The correct answer is B).

After Winning $78,000 on a Game Show, Jasmine Invests the Money