Chapter 8.1, Problem 5GP

To determine

The value of $\frac{b}{a}$ .

Answer to Problem 5GP

  $\frac{b}{a}=\frac{1}{2}$

Explanation of Solution

Given:

The proportions, $\frac{a+1}{b-1}=\frac{5}{1}\text{ and}\frac{a-1}{b+1}=\frac{1}{1}$ .

Concept Used:

• To get rid of a number in addition from one side, subtract the same number from both sides of equal sign.
• To get rid of a number in subtraction from one side, add the same number both sides of equal sign.
• To get rid of a number in multiplication from one side, divide the same number from both sides of equal sign.
• To get rid of a number in division from one side, multiply the same number both sides of equal sign.

Rules of Addition/ Subtraction:

• Two numbers with similar sign always get added and the resulting number will carry the similar sign.
• Two numbers with opposite signs always get subtracted and the resulting number will carry the sign of larger number.

Rules of Multiplication/ Division:

• The product/quotient of two similar sign numbers is always positive.
• The product/quotient of two numbers with opposite signs is always negative.

Calculation:

In order to find the value of the ratio $\frac{b}{a}$ , first simplify both proportions and get two equations as shown below

  $\begin{array}{c}\frac{a+1}{b-1}=\frac{5}{1}\\ \frac{a+1}{b-1}\times \left(b-1\right)=\frac{5}{1}\times \left(b-1\right)\\ a+1=5b-5\\ a+1-5b=5b-5-5b\\ a+1-5b=5\\ a+1-5b-1=-5-1\\ a-5b=-6\mathrm{......}\left(i\right)\end{array}$

On solving the second proportion, it gives

  $\begin{array}{c}\frac{a-1}{b+1}=\frac{1}{1}\\ \frac{a-1}{b+1}\times b+1=\frac{1}{1}\times b+1\\ a-1=b+1\\ a-1-b=b+1-b\\ a-b-1=1\\ a-b-1+1=1+1\\ a-b=2\mathrm{......}\left(ii\right)\end{array}$

Now solve equation (ii) for a and substitute that in equation (i) and simplify further as shown below

  $\begin{array}{c}a-b=2\\ a-b+b=2+b\\ a=2+b\\ \text{Put the value of }a\text{ in equation1}\\ a-5b=-6\\ 2+b-5b=-6\\ 2-4b=-6\\ 2-4b-2=-6-2\\ -4b=-8\\ \frac{-4b}{4}=\frac{-8}{4}\\ -b=-2\\ b=2\\ \Rightarrow a=b+2\\ a=2+2\\ a=4\end{array}$

Thus, the solution of the proportion is $\begin{array}{c}\frac{b}{a}=\frac{2}{4}=\frac{1}{2}\end{array}$ .