Answer :
To find both the exact and ordinary interest on [tex]$30,700 at 11 \frac{1}{4} \%$[/tex] for 193 days and determine the amount by which the ordinary interest is larger, let's break it down step-by-step.
1. Identify the components:
- Principal amount ([tex]\(P\)[/tex]): [tex]$30,700
- Annual Interest Rate (\(R\)): 11.25% or 0.1125 in decimal form
- Time period (\(T\)): 193 days
2. Calculate the exact interest:
The exact interest is calculated based on a 365-day year.
The formula for exact interest is:
\[
\text{Exact Interest} = P \times R \times \frac{T}{365}
\]
Plug in the values:
\[
\text{Exact Interest} = 30700 \times 0.1125 \times \frac{193}{365}
\]
\[
\text{Exact Interest} \approx \$[/tex]1826.23
\]
3. Calculate the ordinary interest:
The ordinary interest is calculated based on a 360-day year.
The formula for ordinary interest is:
[tex]\[
\text{Ordinary Interest} = P \times R \times \frac{T}{360}
\][/tex]
Plug in the values:
[tex]\[
\text{Ordinary Interest} = 30700 \times 0.1125 \times \frac{193}{360}
\][/tex]
[tex]\[
\text{Ordinary Interest} \approx \$1851.59
\][/tex]
4. Find the difference:
To find how much larger the ordinary interest is compared to the exact interest:
[tex]\[
\text{Difference} = \text{Ordinary Interest} - \text{Exact Interest}
\][/tex]
[tex]\[
\text{Difference} = 1851.59 - 1826.23
\][/tex]
[tex]\[
\text{Difference} \approx \$25.36
\][/tex]
So, the exact interest is approximately \[tex]$1826.23, the ordinary interest is approximately \$[/tex]1851.59, and the ordinary interest is larger by about \$25.36.
1. Identify the components:
- Principal amount ([tex]\(P\)[/tex]): [tex]$30,700
- Annual Interest Rate (\(R\)): 11.25% or 0.1125 in decimal form
- Time period (\(T\)): 193 days
2. Calculate the exact interest:
The exact interest is calculated based on a 365-day year.
The formula for exact interest is:
\[
\text{Exact Interest} = P \times R \times \frac{T}{365}
\]
Plug in the values:
\[
\text{Exact Interest} = 30700 \times 0.1125 \times \frac{193}{365}
\]
\[
\text{Exact Interest} \approx \$[/tex]1826.23
\]
3. Calculate the ordinary interest:
The ordinary interest is calculated based on a 360-day year.
The formula for ordinary interest is:
[tex]\[
\text{Ordinary Interest} = P \times R \times \frac{T}{360}
\][/tex]
Plug in the values:
[tex]\[
\text{Ordinary Interest} = 30700 \times 0.1125 \times \frac{193}{360}
\][/tex]
[tex]\[
\text{Ordinary Interest} \approx \$1851.59
\][/tex]
4. Find the difference:
To find how much larger the ordinary interest is compared to the exact interest:
[tex]\[
\text{Difference} = \text{Ordinary Interest} - \text{Exact Interest}
\][/tex]
[tex]\[
\text{Difference} = 1851.59 - 1826.23
\][/tex]
[tex]\[
\text{Difference} \approx \$25.36
\][/tex]
So, the exact interest is approximately \[tex]$1826.23, the ordinary interest is approximately \$[/tex]1851.59, and the ordinary interest is larger by about \$25.36.
