Chapter 6.1, Problem 8E

To determine

To arrange the given ratios from least to greatest.

Answer to Problem 8E

The ratios arranged from least to greatest are:

  $\frac{2}{8},5:18,1\text{ to 3,}7\text{ to 20, }\frac{9}{25}$

Explanation of Solution

Given information :

the ratios given in the question:

  $1\text{ to 3,}\frac{2}{8},5:18,7\text{ to 20, }\frac{9}{25}$

Calculation :

As per the given problem,

Consider the ratios:

  $1\text{ to 3,}\frac{2}{8},5:18,7\text{ to 20, }\frac{9}{25}$

Writing each ratio in the fraction form:

  $\frac{1}{3},\frac{2}{8},\frac{5}{18},\frac{7}{20},\frac{9}{25}.$

Writing the ratios in their simplest form:

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

  $\frac{2}{8}=\frac{2÷2}{8÷2}=\frac{1}{4}$ is the simplest form. [ $\text{GCF of 2 and 8 is 2}$ , divide numerator and denominator by $\text{2}$ ]

  $\frac{5}{18}$ is in its simplest form. [ $\text{GCF of 5 and 18 is 1}$ ]

  $\frac{7}{20}$ is in its simplest form. [ $\text{GCF of 7 and 20 is 1}$ ]

  $\frac{9}{25}$ is in its simplest form. [ $\text{GCF of 9 and 25 is 1}$ ]

Rewriting the ratios in simplest form:

  $\begin{array}{l}\frac{1}{3},\frac{1}{4},\frac{5}{18},\frac{7}{20},\frac{9}{25}\\ \text{The LCD of 3,4,18,20 and 25 is 1800}\text{.}\end{array}$

Making $\text{1800}$ the denominator of each ratio:

  $\begin{array}{l}\frac{\text{1×600}}{\text{3×600}}\text{=}\frac{\text{600}}{\text{1800}}\text{ [Multiply numerator and denominator by 600]}\\ \frac{\text{1×450}}{\text{4×450}}\text{=}\frac{\text{450}}{\text{1800}}\text{ [Multiply numerator and denominator by 450]}\\ \frac{\text{5×100}}{\text{18×100}}\text{=}\frac{\text{500}}{\text{1800}}\text{ [Multiply numerator and denominator by 100]}\\ \frac{\text{7×90}}{\text{20×90}}\text{=}\frac{\text{630}}{\text{1800}}\text{ [Multiply numerator and denominator by 90]}\\ \frac{\text{9×72}}{\text{25×72}}\text{=}\frac{\text{648}}{\text{1800}}\text{ [Multiply numerator and denominator by 72]}\end{array}$

Rewrite the ratios with denominator $\text{1800}$ :

  $\frac{600}{1800},\frac{450}{1800},\frac{500}{1800},\frac{630}{1800},\frac{648}{1800}$

Arrange the ratios with numerators from least to greatest:

  $\frac{450}{1800},\frac{500}{1800},\frac{600}{1800},\frac{630}{1800},\frac{648}{1800}$

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

  $\frac{1}{4},\frac{5}{18},\frac{1}{3},\frac{7}{20},\frac{9}{25}.$

Hence,

The ratios arranged from least to greatest are:

  $\frac{2}{8},5:18,1\text{ to 3,}7\text{ to 20, }\frac{9}{25}$