Find the payment necessary to amortize a loan of compounded quarterly, with 16 quarterly payments.
The payment size is .
(Round to the nearest cent.)

Ask by Harper Bolton. in the United States

Mar 21,2025

Question

Find the payment necessary to amortize a $$ 4 \% $$ loan of $$ \$ 1600 $$ compounded quarterly, with 16 quarterly payments. The payment size is $$ \$ \square $$. (Round to the nearest cent.)

UpStudy AI · Accepted Answer

We start with the formula for an amortizing loan payment:

$$
\text{Payment} = P\frac{r(1+r)^n}{(1+r)^n - 1}
$$

where  
$$ P $$ is the principal,  
$$ r $$ is the interest rate per period, and  
$$ n $$ is the total number of payments.

**Step 1. Identify the variables.**

- The principal is $$ P = 1600 $$.
- The annual interest rate is $$ 4\% $$, or $$ 0.04 $$ in decimal. Since the loan is compounded quarterly, the interest rate per quarter is

$$
r = \frac{0.04}{4} = 0.01.
$$

- The number of payments is $$ n = 16 $$ (16 quarterly payments).

**Step 2. Substitute the values into the formula.**

$$
\text{Payment} = 1600 \cdot \frac{0.01 \left(1+0.01\right)^{16}}{\left(1+0.01\right)^{16} - 1}
$$

This simplifies to:

$$
\text{Payment} = 1600 \cdot \frac{0.01 \left(1.01\right)^{16}}{\left(1.01\right)^{16} - 1}
$$

**Step 3. Compute $$\left(1.01\right)^{16}$$.**

$$
\left(1.01\right)^{16} \approx 1.172578644
$$

**Step 4. Substitute and calculate the fraction.**

The numerator becomes:

$$
0.01 \cdot 1.172578644 \approx 0.01172578644
$$

The denominator is:

$$
1.172578644 - 1 = 0.172578644
$$

Thus, the fraction is:

$$
\frac{0.01172578644}{0.172578644} \approx 0.067909
$$

**Step 5. Multiply by the principal.**

$$
\text{Payment} \approx 1600 \times 0.067909 \approx 108.6544
$$

**Step 6. Round to the nearest cent.**

$$
\text{Payment} \approx \$108.65
$$

So, the payment size is $$ \$108.65 $$.
The payment size is \$108.65.

Find the payment necessary to amortize a loan of compounded quarterly, with 16 quarterly payments.
The payment size is .
(Round to the nearest cent.)

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Find the payment necessary to amortize a loan of compounded quarterly, with 16 quarterly payments. The payment size is . (Round to the nearest cent.)