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06 Aug 2026, 11:19
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An investor deposits $62,500 in a one-year savings certificate that earns \(p\) percent annual interest, compounded quarterly. What is the value of \(p\)?
(1) Over the one-year term, the certificate earns $152.01 more interest than it would have earned if the same annual interest rate had been applied as simple interest.
(2) The interest credited at the end of the first quarter is $1,250.
(1) Over the one-year term, the certificate earns $152.01 more interest than it would have earned if the same annual interest rate had been applied as simple interest.
(2) The interest credited at the end of the first quarter is $1,250.
06 Aug 2026, 11:19
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Official Solution:
An investor deposits $62,500 in a one-year savings certificate that earns \(p\) percent annual interest, compounded quarterly. What is the value of \(p\)?
(1) Over the one-year term, the certificate earns $152.01 more interest than it would have earned if the same annual interest rate had been applied as simple interest.
Let \(r = \frac{p}{100}\) be the annual interest rate in decimal form.
The interest earned with quarterly compounding is:
\(62,500\left[\left(1 + \frac{r}{4}\right)^4 - 1\right]\)
The simple interest earned is:
\(62,500r\)
So the difference is:
\(62,500\left[\left(1 + \frac{r}{4}\right)^4 - 1 - r\right]\)
For a positive interest rate, this difference increases as \(r\) increases, so an exact difference determines a unique value of \(r\).
In fact, when \(r = 0.08\):
\(62,500\left[(1.02)^4 - 1 - 0.08\right] = 152.01\)
Thus, \(p = 8\).
Sufficient.
(2) The interest credited at the end of the first quarter is $1,250.
Since the first-quarter interest is $1,250 on a deposit of $62,500, the quarterly interest rate is:
\(\frac{1,250}{62,500} = 0.02 = 2\%\)
So the annual interest rate is:
\(4 * 2\% = 8\%\)
Thus, \(p = 8\).
Sufficient.
Answer: D
An investor deposits $62,500 in a one-year savings certificate that earns \(p\) percent annual interest, compounded quarterly. What is the value of \(p\)?
(1) Over the one-year term, the certificate earns $152.01 more interest than it would have earned if the same annual interest rate had been applied as simple interest.
Let \(r = \frac{p}{100}\) be the annual interest rate in decimal form.
The interest earned with quarterly compounding is:
\(62,500\left[\left(1 + \frac{r}{4}\right)^4 - 1\right]\)
The simple interest earned is:
\(62,500r\)
So the difference is:
\(62,500\left[\left(1 + \frac{r}{4}\right)^4 - 1 - r\right]\)
For a positive interest rate, this difference increases as \(r\) increases, so an exact difference determines a unique value of \(r\).
In fact, when \(r = 0.08\):
\(62,500\left[(1.02)^4 - 1 - 0.08\right] = 152.01\)
Thus, \(p = 8\).
Sufficient.
(2) The interest credited at the end of the first quarter is $1,250.
Since the first-quarter interest is $1,250 on a deposit of $62,500, the quarterly interest rate is:
\(\frac{1,250}{62,500} = 0.02 = 2\%\)
So the annual interest rate is:
\(4 * 2\% = 8\%\)
Thus, \(p = 8\).
Sufficient.
Answer: D
