Chapter 9.7, Problem 16P

To determine

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

Answer to Problem 16P

The fraction is $\frac{8}{17}$ .

Explanation of Solution

Given information:

The denominator of a fraction is 9 more than the numerator. If the numerator is decreased by 3 and the denominator is increased by 3, the value of the resulting fraction is $\frac{1}{4}$ .

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

  $=d=n+9\text{ (i)}$

Therefore, the fraction is $\frac{n}{n+9}$ .

Fraction	Numerator is decreased by 3 and the denominator is increased by 3
Numerator	$n-3$
Denominator	$\begin{array}{l}=n+9+3\\ =n+12\end{array}$

According to the question,

  $=\frac{n-3}{n+12}=\frac{1}{4}\text{ (ii)}$

Solving the fraction we get,

  $\begin{array}{l}=4(n-3)=1(n+12)\\ =4n-12=n+12\\ =4n-n=12+12\\ =3n=24\\ =n=8\text{ (iii)}\end{array}$

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

  $\begin{array}{l}=d=8+9\\ =d=17\end{array}$

Therefore, the numerator is 8 and the denominator is 17.

Hence, the fraction is $\frac{8}{17}$ .