To find the fraction, we'll let the numerator be denoted as [tex]\( n \)[/tex] and the denominator be denoted as [tex]\( d \)[/tex]. Given the information:
1. The numerator [tex]\( n \)[/tex] is five less than the denominator [tex]\( d \)[/tex]. This can be expressed as:
[tex]\[ n = d - 5 \][/tex] (Equation 1)
2. Four times the numerator is one more than the denominator. This relationship can be written as:
[tex]\[ 4n = d + 1 \][/tex] (Equation 2)
We want to find the values of [tex]\( n \)[/tex] and [tex]\( d \)[/tex] that satisfy both equations. We'll start with Equation 1 and use it to express [tex]\( n \)[/tex] in terms of [tex]\( d \)[/tex].
Substitute [tex]\( n \)[/tex] into Equation 2:
[tex]\[ 4(d - 5) = d + 1 \][/tex]
Expanding and simplifying:
[tex]\[ 4d - 20 = d + 1 \][/tex]
Combine like terms to isolate [tex]\( d \)[/tex]:
[tex]\[ 4d - d = 1 + 20 \][/tex]
[tex]\[ 3d = 21 \][/tex]
Divide both sides by 3 to solve for [tex]\( d \)[/tex]:
[tex]\[ d = \frac{21}{3} \][/tex]
[tex]\[ d = 7 \][/tex]
Now we have the value of the denominator. Using Equation 1, we find the numerator:
[tex]\[ n = d - 5 \][/tex]
Substitute the value of [tex]\( d \)[/tex]:
[tex]\[ n = 7 - 5 \][/tex]
[tex]\[ n = 2 \][/tex]
So, we found out that the numerator [tex]\( n \)[/tex] is 2 and the denominator [tex]\( d \)[/tex] is 7. Therefore, the fraction is [tex]\( \frac{2}{7} \)[/tex], which corresponds to option (C) [tex]\( \frac{2}{7} \)[/tex].
