Question

.$x=12,y=10,andz=24$Write each ratio in simplest form.

$x:(x+y):(y+z)$

Short answer

The simplest form is$x:(x+y):(y+z)=12:22:34$

Step-by-step solution

Step 1. Given information.

According to the given values,

$\begin{array}{l}x=12\\ y=10\\ z=24\end{array}$

Thus,

$\begin{array}{l}\frac{x}{x+y}=\frac{12}{12+10}\\ =\frac{12}{22}\\ \text{and}\\ \frac{x+y}{y+z}=\frac{12+10}{10+24}\\ =\frac{22}{34}\end{array}$

Step 2. Concept Used.

To write the fraction in simplest form, divide the numerator and denominator with the GCF (Greatest Common Factor).

Step 3. Now find the GCF.

Write the prime factors of numerator and denominator.

$\begin{array}{l}12=2\times 2\times 3\\ 22=2\times 11\\ 34=2\times 17\end{array}$

Here GCF is2.

Step 4. Let’s divide the numerator and denominator with the GCF.

Here we will divide the numerator and denominator by.2

$\begin{array}{l}\frac{x}{x+y}=\frac{12/2}{22/2}\\ =\frac{6}{11}\end{array}$

And

Here we will divide the numerator and denominator by2.

$\begin{array}{c}\frac{x+y}{y+z}=\frac{22/2}{34/2}\\ =\frac{11}{17}\end{array}$

Therefore, the simplest form of the given ratio is$z:(x+y):(y+z)=6:11:17$.