Chapter 6.1, Problem 7E

To determine

To arrange the given ratios from least to greatest.

Answer to Problem 7E

The ratios arranged from least to greatest are:

  $\frac{1}{7},\frac{2}{9},\frac{7}{28},\frac{3}{10},\frac{2}{6}.$

Explanation of Solution

Given information :

The ratios given in the question:

  $\text{1:7,}2\text{ to 9, , }\frac{7}{28},\frac{3}{10},2\text{ to 6}.$

Calculation :

As per the given problem,

Consider the ratios:

  $2\text{ to 9, 1:7, }\frac{7}{28},2\text{ to 6,}\frac{3}{10}.$

Writing each ratio in the fraction form:

  $\frac{2}{9},\frac{1}{7},\frac{7}{28},\frac{2}{6},\frac{3}{10}$

Writing the ratios in their simplest form:

  $\frac{2}{9}$ is in its simplest form. [ $\text{GCF of 2 and 9 is 1}$ ]

  $\frac{1}{7}$ is in its simplest form. [ $\text{GCF of 1 and 7 is 1}$ ]

  $\frac{7}{28}=\frac{7÷7}{28÷7}=\frac{1}{4}$ is the simplest form. [ $\text{GCF of 7 and 28 is 7}$ , divide numerator and denominator by $7$ ]

  $\frac{2}{6}=\frac{2÷2}{6÷2}=\frac{1}{3}$ is the simplest form. [ $\text{GCF of 2 and 6 is 2}$ , divide numerator and denominator by $2$ ]

  $\frac{3}{10}$ is in its simplest form. [ $\text{GCF of 3 and 10 is 1}$ ]

Rewriting the ratios in simplest form:

  $\begin{array}{l}\frac{2}{9},\frac{1}{7},\frac{1}{4},\frac{1}{3},\frac{3}{10}\\ \text{The LCD of 9,7,4,3 and 10 is 1260}\end{array}$

Making $\text{1260}$ the denominator of each ratio:

  $\begin{array}{l}\frac{2\times 140}{9\times 140}=\frac{280}{1260}\text{ [Multiply numerator and denominator by 140]}\\ \frac{1\times 180}{7\times 180}=\frac{180}{1260}\text{ [Multiply numerator and denominator by 180]}\\ \frac{\text{1×315}}{\text{4×315}}\text{=}\frac{\text{315}}{\text{1260}}\text{ [Multiply numerator and denominator by 315]}\\ \frac{\text{1×420}}{\text{3×420}}\text{=}\frac{\text{420}}{\text{1260}}\text{ [Multiply numerator and denominator by 420]}\\ \frac{\text{3×126}}{\text{10×126}}\text{=}\frac{\text{378}}{\text{1260}}\text{ [Multiply numerator and denominator by 126]}\end{array}$

Rewrite the ratios with denominator $\text{1260}$ :

  $\frac{280}{1260},\frac{180}{1260},\frac{315}{1260},\frac{420}{1260},\frac{378}{1260}$

Arrange the ratios with numerators from least to greatest:

  $\frac{180}{1260},\frac{280}{1260},\frac{315}{1260},\frac{378}{1260},\frac{420}{1260}$

Write the ratios in fraction form as given in the question after arranging in ascending order:

  $\frac{1}{7},\frac{2}{9},\frac{7}{28},\frac{3}{10},\frac{2}{6}.$

Hence,

The ratios arranged from least to greatest are:

  $\text{1:7,}2\text{ to 9, , }\frac{7}{28},\frac{3}{10},2\text{ to 6}.$