Select the correct answer.
Aaron borrows $150 from his friend Austin. He promises to pay back the money in 4 monthly installments. Each month he wants to pay half the amount he paid the previous month. Assuming Austin does not charge any interest, how much should Aaron pay the first month to repay the money as scheduled?
A.
$60
B.
$70
C.
$80
D.
$90
E.
$100
Each month, Aaron will pay half of what he paid the previous month. This is a geometric sequence with first term $x$ (the first payment), common ratio $\frac{1}{2}$ (since each payment is half of the previous one), and four terms. Therefore, the sum of the sequence is given by: $$S = \frac{x\left(1-\frac{1}{2^4}\right)}{1-\frac{1}{2}} = 2x\left(1-\frac{1}{16}\right) = \frac{31}{16}x$$ Since $S$ is equal to the amount Aaron borrowed ($150), we have $\frac{31}{16}x = 150$. Solving for $x$, we get $x=\frac{480}{31} \approx \boxed{\textbf{(B)}\ $70}$.
To determine how much Aaron should pay in the first month, we need to calculate the monthly installment amount.
Since Aaron wants to pay half the amount he paid the previous month, we can set up an equation to represent the total amount he needs to pay back over the four months:
Let x be the amount Aaron needs to pay in the first month.
Then, the installment amounts for the next three months would be: x/2, (x/2)/2, and ((x/2)/2)/2.
The total amount Aaron needs to pay over the four months would be:
x + x/2 + (x/2)/2 + ((x/2)/2)/2 = $150
To simplify the equation, we can multiply both sides by 2:
2x + x + (x/2) + (x/2)/2 = 2 * $150
2x + x + x/2 + x/4 = $300
8x/4 + 4x/4 + 2x/4 + x/4 = $300
(8x + 4x + 2x + x)/4 = $300
15x/4 = $300
Now, we can solve for x:
15x = 4 * $300
15x = $1200
x = $1200 / 15
x = $80
Therefore, Aaron should pay $80 in the first month to repay the money as scheduled.
The correct answer is C. $80.