Chapter 9.7, Problem 15P

To determine

To solve: the system of two equations in two variables.

Answer to Problem 15P

The fraction is $\frac{5}{13}$ .

Explanation of Solution

Given information:

The denominator of a fraction is 8 more than the numerator. If 3 is added to both the numerator and the denominator, the value of the resulting fraction is $\frac{1}{2}$ .

Formula used:

Fraction problem

Step1. The problem asks for .

Step2. Let n = the numerator of the fraction

Let d = the denominator of the fraction.

Then $\frac{n}{d}=\text{the fraction}$ .

Step3. Use the facts of the problem to write two equations.

Step4. Solve the equations.

Calculation:

Consider ,

Let n = the numerator of the fraction.

The denominator of a fraction

  $=d=n+8\text{ (i)}$

Therefore, the fraction is $\frac{n}{n+8}$ .

Fraction	Add 3
Numerator	$n+3$
Denominator	$\begin{array}{l}=n+8+3\\ =n+11\end{array}$

According to the question,

  $=\frac{n+3}{n+11}=\frac{1}{2}\text{ (ii)}$

Solving the fraction we get,

  $\begin{array}{l}=2(n+3)=1(n+11)\\ =2n+6=n+11\\ =2n-n=11-6\\ =n=5\text{ (iii)}\end{array}$

Putting equation (iii) in (i) we get,

  $\begin{array}{l}=d=5+8\\ =d=13\\ \end{array}$

Therefore, the numerator is 5 and the denominator is 13.

Hence, the fraction is $\frac{5}{13}$ .