Question

Order the ratios from least to greatest.

$2to9,1:7,\frac{7}{28},2to6,\frac{3}{10}$

Short answer

The order of the ratios from least to greatest is:$1:7,2to9,\frac{7}{28},\frac{3}{10},2to6.$

Step-by-step solution

. Given

The given ratios are:$1:7,2to9,\frac{7}{28},\frac{3}{10},2to6.$

. To determine

We have to order the given ratios from least to greatest.

. Calculation

First, we willwrite each ratio in the fraction form:

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

Then we write the ratios in their simplest forms:

$\frac{2}{9}$is in the simplest form because GCF of 2 and 9 = 1.

$\frac{1}{7}$is in simplest form because GCF of 1 and 7 = 1.

$\frac{7}{28}$is not in the simplest form because GCF of 7 and 28 = 7

So we reduce it by 7,$\frac{7}{28}=\frac{7÷7}{28÷7}=\frac{1}{4}$ .

$\frac{2}{6}$is not in the simplest form because GCF of 2 and 6 = 2

So we reduce it by 2,$\frac{2}{6}=\frac{2÷2}{6÷2}=\frac{1}{3}$ .

$\frac{3}{10}$is in the simplest form because GCF of 3 and 10 = 1.

Then we will rewrite the ratios in their simplest forms:

$\begin{array}{l}\frac{2}{9},\frac{1}{7},\frac{1}{4},\frac{1}{3},\frac{3}{10}\\ \text{TheLCDof9,7,4,3and10is1260}\end{array}$

We try to get common denominator of 1260 for each ratio:

$\begin{array}{l}\frac{2\times 140}{9\times 140}=\frac{280}{1260}\text{[Multiplynumeratoranddenominatorby140]}\\ \frac{1\times 180}{7\times 180}=\frac{180}{1260}\text{[Multiplynumeratoranddenominatorby180]}\\ \frac{\text{1×315}}{\text{4×315}}\text{=}\frac{\text{315}}{\text{1260}}\text{[Multiplynumeratoranddenominatorby315]}\\ \frac{\text{1×420}}{\text{3×420}}\text{=}\frac{\text{420}}{\text{1260}}\text{[Multiplynumeratoranddenominatorby420]}\\ \frac{\text{3×126}}{\text{10×126}}\text{=}\frac{\text{378}}{\text{1260}}\text{[Multiplynumeratoranddenominatorby126]}\end{array}$

Then we rewrite the ratios with denominator = 1260:

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

After that, order the ratios with numerators from least to greatest:

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

Then we write the ratios in their given form in the question keeping the ascending order.

$1:7,2to9,\frac{7}{28},\frac{3}{10},2to6.$

So, the order of the ratios from least to greatest is:$1:7,2to9,\frac{7}{28},\frac{3}{10},2to6.$