Question

Order the ratios from least to greatest.

$1\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}to\hspace{0.17em}\hspace{0.17em}}3,\frac{2}{8},5:18,7\text{\hspace{0.17em}\hspace{0.17em}to\hspace{0.17em}\hspace{0.17em}}20,\frac{9}{25}$

Short answer

The order of the ratios from least to greatest is:$\frac{2}{8},5:18,1\text{to3,}7\text{to20,}\frac{9}{25}$ .

Step-by-step solution

. Given

The given ratios are:$1\text{to3,}\frac{2}{8},5:18,7\text{to20,}\frac{9}{25}$ .

. To determine

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

. Calculation

First, we willwrite each ratio in the fraction form:

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

Then we write the ratios in their simplest forms:

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

$\frac{2}{8}$is not in the simplest form because GCF of 2 and 8 = 2

So, we reduce it by 2$\frac{2}{8}=\frac{2÷2}{8÷2}=\frac{1}{4}$, .

$\frac{5}{18}$is in the simplest form because of GCF of 5 and 13 = 1.

$\frac{7}{20}$is in the simplest form because of GCF of 7 and 20 = 1.

$\frac{9}{25}$is in the simplest form because of GCF of 9 and 25 = 1.

Then we will rewrite the ratios in their simplest forms:

$\begin{array}{l}\frac{1}{3},\frac{1}{4},\frac{5}{18},\frac{7}{20},\frac{9}{25}\\ \text{TheLCDof3,4,18,20and25is1800.}\end{array}$

We try to get a common denominator of 1800 for each ratio:

$\begin{array}{l}\frac{\text{1×600}}{\text{3×600}}\text{=}\frac{\text{600}}{\text{1800}}\text{[Multiplynumeratoranddenominatorby600]}\\ \frac{\text{1×450}}{\text{4×450}}\text{=}\frac{\text{450}}{\text{1800}}\text{[Multiplynumeratoranddenominatorby450]}\\ \frac{\text{5×100}}{\text{18×100}}\text{=}\frac{\text{500}}{\text{1800}}\text{[Multiplynumeratoranddenominatorby100]}\\ \frac{\text{7×90}}{\text{20×90}}\text{=}\frac{\text{630}}{\text{1800}}\text{[Multiplynumeratoranddenominatorby90]}\\ \frac{\text{9×72}}{\text{25×72}}\text{=}\frac{\text{648}}{\text{1800}}\text{[Multiplynumeratoranddenominatorby72]}\end{array}$

Then we rewrite the ratios with denominator = 1800:

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

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

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

Then we write the ratios in their given form in the question keeping the ascending order.$\frac{1}{4},\frac{5}{18},\frac{1}{3},\frac{7}{20},\frac{9}{25}$

$\frac{2}{8},5:18,1\text{to3,}7\text{to20,}\frac{9}{25}$

So, the order of the ratios from least to greatest is:$\frac{2}{8},5:18,1\text{to3,}7\text{to20,}\frac{9}{25}$ .