To determine the highest interest rate at which Harry would be better off paying Sally $10,000 today rather than $12,000 two years from today, we can compare the present value of the $12,000 payment to the $10,000 payment.
The formula for present value (PV) is:
\[ PV = \frac{FV}{(1 + r)^n} \]
Where:
- \( FV \) = future value ($12,000 in this case)
- \( r \) = interest rate
- \( n \) = number of years (2 years here)
We want to find the maximum interest rate \( r \) where it is still true that:
\[ 10,000 \geq \frac{12,000}{(1 + r)^2} \]
Rearranging gives us:
\[ (1 + r)^2 \geq \frac{12,000}{10,000} \]
\[ (1 + r)^2 \geq 1.2 \]
Taking the square root of both sides:
\[ 1 + r \geq \sqrt{1.2} \]
\[ r \geq \sqrt{1.2} - 1 \]
Now, we calculate \( \sqrt{1.2} \):
\[ \sqrt{1.2} \approx 1.0954 \]
Thus,
\[ r \geq 1.0954 - 1 \approx 0.0954 \]
So,
\[ r \approx 0.0954 \text{ or } 9.54% \]
Harry would be better off paying today as long as the interest rate is less than about 9.54%.
Therefore, of the given options:
- a. 9 percent
- b. 6 percent
- c. 4 percent
- d. 11 percent
The highest rate at which Harry would still prefer paying today (9.54%) is 9 percent (option a).
Thus, the answer is a. 9 percent.
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