Chapter 9.7, Problem 17P

To determine

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

Answer to Problem 17P

The fraction is $\frac{5}{8}$ .

Explanation of Solution

Given information:

If 1 is subtracted from the numerator of a fraction , the value of the resulting fraction is $\frac{1}{2}$ . However, if 7 is added to the denominator of the original fraction , the value of the resulting fraction is $\frac{1}{3}$ .

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 d = the denominator of the fraction.

The numerator of a fraction

  $=n=n-1\text{ (i)}$

Therefore, the fraction is $\frac{n-1}{d}$ .

According to the question,

  $\begin{array}{l}=\frac{n-1}{d}=\frac{1}{2}\\ =2(n-1)=1(d)\\ =2n-2=d\text{ (ii)}\end{array}$

Fraction	denominator is increased by 7
Numerator	$n$
Denominator	$=d+7$

Therefore,

  $\begin{array}{l}=\frac{n}{d+7}=\frac{1}{3}\\ =3(n)=1(d+7)\\ =3n=d+7\\ =3n-d=7\text{ (iii)}\end{array}$

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

  $\begin{array}{l}=3n-d=7\\ =3n-(2n-2)=7\\ =3n-2n+2=7\\ =n+2=7\\ =n=7-2\\ =n=5\text{ (iv)}\end{array}$

Equation (iv) in (ii)

  $\begin{array}{l}=2(5)-2=d\\ =10-2=d\\ =8=d\end{array}$

Therefore, the numerator is 5 and the denominator is 8.

Hence, the fraction is $\frac{5}{8}$ .