Question

The ratio of a to b is $4:3$. The sum of a and b is $14$. What is the ratio of $a-b$ to $a+b$?

Short answer

The ratio is $\frac{1}{7}$.

Step-by-step solution

Step 1. Given information.

Consider the given question,

$$\begin{gathered} \frac{a}{b}=\frac{4}{3}......\left(i\right) \\ a+b=14......\left(ii\right) \end{gathered}$$

Solve equation (ii) for a,

$a=14-b$

Substitute the value of a in equation (i),

$$\begin{gathered} \frac{14-b}{b}=\frac{4}{3} \\ 42-3b=4b \\ 42=7b \\ b=6 \end{gathered}$$

Step 2. Find the value of a.

Substitute the value of bin equation (ii),

$$\begin{gathered} a+6=14 \\ a=8 \end{gathered}$$

Then,

$$\begin{gathered} a-b=8-6 \\ a-b=2 \end{gathered}$$

Therefore, the required ration is given below,

$$\begin{gathered} \frac{a-b}{a+b}=\frac{2}{14} \\ \frac{a-b}{a+b}=\frac{1}{7} \end{gathered}$$