Let's break down the problem step-by-step to determine the fraction of total interest paid after the seventh month of a 12-month loan.
Given:
- The numerator is the sum of the interest paid from the 5th month to the 11th month.
- The denominator is the sum of the interest paid from the 0th month to the 11th month.
1. Calculate the Numerator:
The numerator is given by the sum of interest payments from the 5th month to the 11th month:
[tex]\[
\{(n+11) + (n+10) + (n+9) + (n+8) + (n+7) + (n+6) + (n+5)\}
\][/tex]
We have already determined that the numerator is:
[tex]\[
63
\][/tex]
2. Calculate the Denominator:
The denominator is the sum of interest payments from the 0th month to the 11th month:
[tex]\[
(n) + (n+1) + (n+2) + \ldots + (n+11)
\][/tex]
We have already determined that the denominator is:
[tex]\[
78
\][/tex]
3. Determine the Fraction:
We need to find the fraction of the total interest paid, which is the ratio of the numerator to the denominator, and then express this fraction as a percentage to the nearest tenth:
[tex]\[
\text{Fraction} = \left( \frac{\text{Numerator}}{\text{Denominator}} \right) \times 100
\][/tex]
Plugging in the values we have:
[tex]\[
\text{Fraction} = \left( \frac{63}{78} \right) \times 100 = 80.8\%
\][/tex]
Therefore, the fraction of total interest paid after the seventh month of the 12-month loan, rounded to the nearest tenth, is [tex]\(80.8\%\)[/tex].
