\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)

Ratios

What is a Ratio?

Definition Ratio
A ratio is a comparison of two quantities. The ratio of \(\textcolor{colordef}{2}\) to \(\textcolor{colorprop}{3}\) can be expressed as the fraction \(\dfrac{\textcolor{colordef}{2}}{\textcolor{colorprop}{3}}\).

Part-to-Part Ratios

Definition Part-Part Ratio
A part-part ratio compares two distinct parts of a whole.$$\textcolor{colordef}{\text{Part 1}}:\textcolor{colorprop}{\text{Part 2}}$$
Example
A fruit bowl contains 3 cherries and 2 apples. What is the ratio of cherries to apples?
The ratio of cherries to apples is \(\textcolor{colordef}{3}:\textcolor{olive}{2}\). This compares the two parts of the fruit collection to each other.

The ratio of cherries to apples is \(\textcolor{colordef}{3}:\textcolor{olive}{2}\). This compares the two parts of the fruit collection to each other.

Part-Whole Ratios

Definition Part-Whole Ratio
A part-whole ratio compares one part of a whole to the whole.$$\textcolor{colordef}{\text{Part 1}}:\textcolor{olive}{\text{Whole}}\text{ or }\textcolor{colorprop}{\text{Part 2}}:\textcolor{olive}{\text{Whole}}$$
Example
A juice is made with 1 lemon and 2 oranges. What is the ratio of oranges to the total number of fruits?

  • First, determine the total number of fruits. The total is \(1 + 2 = 3\) fruits.
  • The ratio of oranges (the part) to the total number of fruits (the whole) is \(\textcolor{orange}{2}:3\).
  • This part-to-whole ratio can also be expressed as the fraction \(\dfrac{\textcolor{orange}{2}}{3}\).

Equivalent Ratios

Definition Equivalent Ratios
Two ratios are equivalent if they represent the same relationship. You can find equivalent ratios by multiplying or dividing both parts of the ratio by the same number.
Example
The ratio of red apples to all apples is \(\dfrac{2}{4}\), which is equivalent to \(\dfrac{1}{2}\) (half of the apples are red).
Method Using Fractions
To check if two ratios are equivalent, we can compare their fractions. If the fractions are equivalent, then the ratios are equivalent.
Example
Since , the ratios are equivalent: \(\textcolor{colordef}{1}:\textcolor{colorprop}{2}=\textcolor{colordef}{2}:\textcolor{colorprop}{4}\)

Proportion


  • For one glass of juice, you need \(\textcolor{colordef}{1}\) lemon and \(\textcolor{colorprop}{2}\) oranges. The ratio of lemons to oranges is \(\textcolor{colordef}{1}:\textcolor{colorprop}{2}\).
  • To make two glasses of juice, you need to double the ingredients: \(\textcolor{colordef}{2}\) lemons and \(\textcolor{colorprop}{4}\) oranges. The new ratio is \(\textcolor{colordef}{2}:\textcolor{colorprop}{4}\).
  • The amount of fruit is proportional because the ratios are equivalent: \(\dfrac{\textcolor{colordef}{1}}{\textcolor{colorprop}{2}} = \dfrac{\textcolor{colordef}{2}}{\textcolor{colorprop}{4}}\).

Definition Proportion
A proportion is an equation stating that two ratios are equivalent.
Example
To make \(\textcolor{colorprop}{1}\) chocolate cake, you need \(\textcolor{colordef}{4}\) eggs. How many eggs are needed for \(\textcolor{colorprop}{2}\) cakes?

For \(\textcolor{colorprop}{1}\) cake, you need \(\textcolor{colordef}{4}\) eggs. To find the number of eggs for \(\textcolor{colorprop}{2}\) cakes, we set up a proportion:
By multiplying both parts of the ratio by 2, we find you need \(\textcolor{colordef}{8}\) eggs.

Part in Whole-Part Relationships

Method Finding a Part in Whole-Part Relationships
To find the number of apples corresponding to \(\dfrac{1}{2}\) of 4 apples, we start with the whole:
  • Divide the whole into 2 equal parts (the denominator) and select 1 part (the numerator):
  • Count the apples in the selected part: there are 2 apples.