Sure, let's work through this step-by-step.
### Step 1: Numerator Calculation
The given task involves determining the first five months of interest in a 12-month loan:
The numerator is expressed as:
[tex]\[ (n + 0) + (n + 1) + (n + 2) + (n + 3) + (n + 4) \][/tex]
Plugging in the values:
[tex]\[ (1 + 0) + (1 + 1) + (1 + 2) + (1 + 3) + (1 + 4) \][/tex]
Which simplifies to:
[tex]\[ 1 + 2 + 3 + 4 + 5 = 15 \][/tex]
So, the numerator is:
[tex]\[ 15 \][/tex]
### Step 2: Denominator Calculation
The denominator involves the total interest for all 12 months:
The denominator is expressed as:
[tex]\[ (n) + (n + 1) + (n + 2) + \ldots + (n + 11) \][/tex]
Plugging in the values:
[tex]\[ (1) + (1 + 1) + (1 + 2) + \ldots + (1 + 11) \][/tex]
[tex]\[ 1 + 2 + 3 + \ldots + 12 \][/tex]
The sum of the first 12 natural numbers is:
[tex]\[ \frac{12(12 + 1)}{2} = \frac{12 \times 13}{2} = 78 \][/tex]
So, the denominator is:
[tex]\[ 78 \][/tex]
### Step 3: Fraction Calculation
Now, we need to calculate the fraction of the numerator over the denominator, and then multiply it by 100 to get a percentage.
[tex]\[ \text{Fraction} = \frac{\text{Numerator}}{\text{Denominator}} \times 100 = \frac{15}{78} \times 100 \][/tex]
Calculating this:
[tex]\[ \text{Fraction} \approx 19.2\% \][/tex]
### Conclusion
Therefore, the fraction of total interest owed after the fifth month of a 12-month loan to the nearest tenth is:
[tex]\[ 19.2\% \][/tex]
