__Objectives__

At the end of this lesson, students should be able to:

- Understand what ratio is.
- Find equivalent ratios.
- Work with ratios that have different units.

__Understanding Ratio__

**A RATIO **is a comparison of two numbers or quantities that indicates how many times the first number contains the second. The quantities being compared are called the **terms **of the ratio and they are quantities of a comparable kind, such as objects, persons, lengths, etc.

For example, if a bowl of fruit contains eight oranges and six lemons, then the ratio of oranges to lemons is **eight** **to** **six**. Note that, in the expression “the ratio of oranges to lemons”,” oranges” comes **first**. This order is very important and must be respected. Whichever quantity comes first; its number must come first. If the expression had been “the ratio of lemons to oranges“, then it would have been “six to eight“.

__Ratio Notation__

A ratio is usually written in one of these three ways.

The ratio of 15 men to 20 women can be express as

__Example 1__

There are 16 ducks and 9 geese in a certain park. Express the ratio of ducks to geese in all three formats.

__Solution __

__Equivalent Ratios__

If the numerator and denominator of the reduced form have no common factors, the ratio is said to be in its **lowest terms**. Since there are an infinite number of fractions that are equivalent to a given ratio such as 3 out of 4, **when comparing ratios, it is convenient to write the fractions in reduced or simplified form. **

Let’s return to the 15 men and 20 women in our original group. Ratio of men to women is 15:20. This means that you can also express the ratio of men to women as** 3/4, 3:4, or “3 to 4” **

This points out something important about ratios; the numbers used in the ratio might not be the **absolute measured values**. The ratio “15 to 20” refers to the absolute number of men and women respectively in the group of thirty-five people. The simplified or reduced ratio “3 to 4” implies that, for every three men, there are four women. The simplified ratio also indicates that, in any representative set of seven people (3 + 4 = 7) from this group, three will be men.

In other words, the men comprise 3/4 of the people in the group. These relationships and reasoning are what you use to solve many word problems.

__Example 1__

__Solution__

**Method 1: Using Multiplication**

**Method 2: Using Divisors **

The fraction will be in the lowest or simplest form if the numerator and denominator of the fraction have no common factors, which is to say that they are **RELATIVELY PRIME.**

__Note__

Equivalent ratios are just like equivalent fractions. If two ratios have the same value, then they are equivalent even though they may look very different.

__Common Units __

When working with ratio involving different units, always convert them to the same units. A ratio can be simplified only when the units of the quantities are the same.

__Example 1__

Express 25 minutes: 1 hour as a ratio in its simplest form.

__Solution__

The units must be the same so change 1 hour to 60 minutes

25 minutes: 1 hour

=25 minutes: 60minutes Cancel the units

= 25:60 Divide both sides by 5

= 5:12

So 25 minutes: 1 hour simplifies to 5:12

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