# How to Write Ratios

In Math a ratio is an effective method of comparing different values of certain things. Enabling us to see how much of some quantity there is compared to another quantity of the same measurement.

This page will demonstrate how ratios work, and show how to write ratios, and how they can help compare values such as lengths, weights and areas.

The units of measure used for a comparison in a ratio do need to be the same.

### Introduction, How to Write Ratios

Say a clothes drawer contains &nbsp5&nbsp shirts.

Inside the drawer are &nbsp3&nbsp red shirts and &nbsp2&nbsp red shirts.

3 RED SHIRTS
2 BLUE SHIRTS

The ratio of red shirts to blue shirts is &nbsp3 : 2.

There are &nbsp4&nbsp times as many red balls as there are yellow balls.

### Notation:

a : b&nbsp is the common general notation for Ratios.

Though sometimes we can see that ratios can be represented as a fraction also, &nbsp{\frac{\tt{a}}{\tt{b}}}.

So with the red and blue shirt ratio above could also be written as,&nbsp \bf{\frac{3}{2}}.

Also, the ratio could if one wished, be written in a different order,
perhaps as blue shirts to red shirts instead,&nbsp 2 : 3.

### Ratio, Simplified Ratios:

The ratio &nbsp3 : 2 &nbsp doesn’t only have to represent a group/collection of &nbsp5&nbsp things though, such as in the case earlier with the shirts.

A ratio can be a what is known as a simplified ratio, representing a group that is larger than &nbsp5&nbsp in size.

If the drawer contained &nbsp6&nbsp red shirts and &nbsp2&nbsp blue shirts.

Then the ratio of red shirts to blue shirts would be &nbsp6&nbsp red shirts to &nbsp2&nbsp blue shirts,
6 : 2&nbsp which can be simplified further to &nbsp4 : 1.

Even in this drawer with more shirts, the ratio is still &nbsp4 : 1.

As there is still &nbsp3&nbsp times as many red shirts as blue shirts overall.

Like in the case with fractions,&nbsp ratios are often simplified where possible to a neater and tidier equivalent ratio.

For example &nbsp16 : 4&nbsp and &nbsp4 : 1&nbsp are equivalent ratios, because they represent the same scale.

## Writing Ratios Examples

(1.1)&nbsp

Express &nbsp8 : 40&nbsp as a ratio in its simplest form.

Solution&nbsp &nbsp

Largest common factor of both numbers is &nbsp8.

8 ÷ 8 &nbsp=&nbsp 1 &nbsp&nbsp , &nbsp&nbsp 40 ÷ 8 &nbsp=&nbsp 5

8 : 40&nbsp can be simplified to &nbsp1 : 5

(1.2)&nbsp

Express &nbsp27 : 9&nbsp as a ratio in its simplest form.

Solution&nbsp &nbsp

27 ÷ 9 &nbsp=&nbsp 3 &nbsp&nbsp , &nbsp&nbsp 9 ÷ 9 &nbsp=&nbsp 1

27 : 9&nbsp can be simplified to &nbsp3 : 1

(1.3)&nbsp

Express &nbsp1 YEAR &nbspto&nbsp 10 MONTHS&nbsp as a ratio in its simplest form.

Solution&nbsp &nbsp

1 YEAR &nbsp:&nbsp 10 MONTHS

We first look to make both measurements/units in the ratio the same.

They aren’t initially in this example, as a year and a month are different values.

Something we can do is to change &nbsp1 YEAR&nbsp into &nbsp12 MONTHS.

1 YEAR : 10 MONTHS &nbsp=&nbsp 12 MONTHS : 10 MONTHS &nbsp=&nbsp 12 : 10

Now the ratio &nbsp12 : 10&nbsp can now be simplified to &nbsp6 : 5

(1.4)&nbsp

Share &nbsp$3000&nbsp in the ratio &nbsp2 : 3. Solution&nbsp &nbsp 2 : 3&nbsp is &nbsp5&nbsp parts in total. As a first step we split &nbsp$3000&nbsp into &nbsp5&nbsp equal parts.

$3000 ÷ 5 &nbsp=&nbsp$600

Sometimes it can be helpful to draw a suitable number line to illustrate the example and understand what’s going on visually.

2 parts &nbsp=&nbsp $1200 &nbsp &nbsp , &nbsp &nbsp 3 parts &nbsp=&nbsp$1800

So for &nbsp$3000: The ratio &nbsp2 : 3 &nbsp = &nbsp$1200 : \$1800

## Ratios, another case

There can be the case that sometimes a ratio can represent a part of a smaller group, to a greater whole group.

Part of Group : Larger Whole Group

With the shirts in the drawer example, the &nbsp3&nbsp red shirts alone, could be written as the ratio &nbsp3 : 5.

Representing that there is &nbsp4&nbsp red shirts in the drawer, in relation to all shirts that in the drawer in total.

Example &nbsp &nbsp

(2.1)&nbsp

An oil barrel has a capacity of &nbsp40&nbsp gallons, but has only been filled with &nbsp10&nbsp gallons of oil.
What ratio of the barrel has been filled with oil.

Solution&nbsp &nbsp

Part of Group : Larger Whole Group &nbsp=&nbsp 10 : 40

10 : 40&nbsp can be simplified to the ratio &nbsp1 : 4

Or if we wanted to represent the ratio in fraction form.

\boldsymbol{\tt\frac{Part \space of \space Group}{Larger \space Whole \space Group}} &nbsp = &nbsp \bf{\frac{10}{40}} &nbsp = &nbsp \bf{\frac{1}{4}}

\bf{\frac{1}{4}}&nbsp of the barrel has been filled with oil.

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