Sure! Let's solve this step by step as a math teacher:
To find the fraction of the total interest owed after the fifth month, we'll start by determining the numerator and the denominator using the given values.
Step 1: Calculating the Numerator
The numerator is:
[tex]\[ (n+1) + (n+1) \times 2 + (n+1) + (n+2) \times 2 + (n+1) \times 3 \][/tex]
We can fill in the numbers as follows:
[tex]\[ (n+1) + (n+1) \times 2 + (n+1) + (n+2) \times 2 + (n+1) \times 3 \][/tex]
Here, the given values in the problem are [tex]\(n = 5\)[/tex].
So, substituting the value of [tex]\(n\)[/tex]:
[tex]\[ (5+1) + (5+1) \times 2 + (5+1) + (5+2) \times 2 + (5+1) \times 3 \][/tex]
[tex]\[ = 6 + 6 \times 2 + 6 + 7 \times 2 + 6 \times 3 \][/tex]
[tex]\[ = 6 + 12 + 6 + 14 + 18 \][/tex]
[tex]\[ = 56 \][/tex]
Thus, the numerator is:
[tex]\[ 56 \][/tex]
Step 2: Calculating the Denominator
The denominator is:
[tex]\[ 2 \times (n+1+n+2) + n + 6 \times (n+3) + 4 \times (n+4) \][/tex]
Substituting the value of [tex]\(n\)[/tex]:
[tex]\[ 2 \times (5+1 + 5+2) + 5 + 6 \times (5+3) + 4 \times (5+4) \][/tex]
[tex]\[ = 2 \times (6 + 7) + 5 + 6 \times 8 + 4 \times 9 \][/tex]
[tex]\[ = 2 \times 13 + 5 + 48 + 36 \][/tex]
[tex]\[ = 26 + 5 + 48 + 36 \][/tex]
[tex]\[ = 115 \][/tex]
Thus, the denominator is:
[tex]\[ 115 \][/tex]
Step 3: Calculating the Fraction
Finally, the fraction of the total interest owed is given by:
[tex]\[ \frac{\text{numerator}}{\text{denominator}} \][/tex]
So:
[tex]\[ \frac{56}{115} = 0.487 \][/tex]
Converting this into a percentage and rounding to the nearest tenth:
[tex]\[ 0.487 \times 100 \approx 48.7 \][/tex]
Thus, the fraction of the total interest owed after the fifth month is:
[tex]\[ 48.7\% \][/tex]
Let's fill in the blanks with the given context:
Given the problem:
- The numerator is: [tex]\((n+1)+(n+1) \times 2+(n+1)+(n+2) \times 2+(n+1) \times 3\)[/tex]
- The denominator is: [tex]\(2 \times (n+1+n+2)+n+6 \times (n+3)+4 \times (n+4)\)[/tex]
- Therefore, the fraction is [tex]\(\frac{\text{numerator}}{\text{denominator}}\)[/tex] (to the nearest tenth) = 48.7%.
