5. Ratios & Proportional Relationships

Consider a cake recipe that uses 1 cup of milk and 4 cups of flour. What is the ratio of milk to flour used in the cake?

We can express the information given as the ratio 1:4.

What if we want to make two cakes? We will need 2 cups of milk and 8 cups of flour which is double the amount that we used. Now the ratio of milk to flour is 2:8.

We say that 1:4 and 2:8 are equivalent ratios.

**Equivalent ratios** are useful for when we want to change the value of one quantity but also keep it in the same proportion to another quantity. After calculating how much the value of the first quantity has increased, we can increase the value of the second quantity by the same multiple to preserve the ratio.

We saw in the cake example that increasing both the amount of milk and the amount of flour by the same multiple preserved the ratio.

Two cakes require twice the ingredients of one cake, but in the same proportion.

And since this is an equivalence relation, we can also say the same for the reverse:

One cake requires half the ingredients of two cakes, but in the same proportion.

Two ratios are equivalent if one of the ratios can be increased or decreased by some multiple to be equal to the other ratio.

A special type of equivalent ratio is a **simplified ratio **.

The two integers in the ratio have a greatest common factor of 1. Therefore, all ratios equivalent to the simplified ratio have to be multiples of it.

To simplify a ratio, we can identify the **greatest common factor** (or the largest integer that can evenly divide both numbers in the ratio) and divide both parts of the ratio by it.

Let's say we have a recipe for 5 cakes using 25 cups of flour. The ratio of the number of cakes to the number of cups of flour is 5:25.

To simplify the ratio we can see that both 5 and 25 can be evenly divided by 5.

\dfrac{5}{5}: \dfrac{25}{5}= 1:5

The simplified ratio tells us that one cake requires 5 cups of flour. This is very useful information for planning ingredients for lots of cakes.

The ratio of tables to chairs is 1:2. If there are 14 chairs, how many tables are there?

Worked Solution

The ratio of players to teams is 60:10. If there are only 12 students present, how many teams can be made?

Worked Solution

Simplify the ratio 10:24.

Worked Solution

Write 54 cents to \$3.00 as a fully simplified ratio.

Worked Solution

Idea summary

Two ratios are equivalent if one of the ratios can be increased or decreased by some multiple to be equal to the other ratio.

A ratio is a simplified ratio if there is no equivalent ratio with smaller integer values.

We can use a **ratio table** to represent a series of equivalent ratios.

For example, if a pie recipe calls for 1 tablespoon of brown sugar for every 2 cups of flour, we could write this as a ratio: 2:1.

In a ratio table we have:

Sugar | 2 | 4 | 6 | 8 |
---|---|---|---|---|

Flour | 1 | 2 | 3 | 4 |

We can also use a ratio table to help us determine unknown values. For example, if we wanted to find out how much flour is needed when we use 12 tablespoons of brown sugar, we have the following:

Sugar | 2 | 4 | 6 | 8 | 12 |
---|---|---|---|---|---|

Flour | 1 | 2 | 3 | 4 | ⬚ |

We can determine the corresponding amount of flour to 12 tablespoons of brown sugar by finding equivalent ratios.

\displaystyle 6:3 | \displaystyle = | \displaystyle 12 : ⬚ | Equivalent ratios |

\displaystyle 6:3 | \displaystyle = | \displaystyle 6\cdot 2 : 3 \cdot 2 | Multiply both parts of the ratio by 2 |

\displaystyle = | \displaystyle 12:6 |

Therefore, for every 12 tablespoons of brown sugar, we can use 6 cups of flour.

You may have noticed that there was another way to find this using the table.

Sugar | 2 | 4 | 6 | 8 | 12 |
---|---|---|---|---|---|

Flour | 1 | 2 | 3 | 4 | ⬚ |

We can see in the table that for 4 cups of flour we need 2 tablespoons of brown sugar, and for 8 cups of flour we need 4 tablespoons of brown sugar.

We know that 4 + 8 = 12 so we could have added 2 + 4 to get the 6 tablespoons of brown sugar.

The table shows the ratio of dogs to cats:

Dogs | to | Cats |
---|---|---|

9 | : | 5 |

18 | : | 10 |

27 | : | |

45 | : | |

: | 50 |

a

Complete the table of equivalent ratios.

Worked Solution

b

If there are 270 dogs, how many cats are there expected to be?

A

150

B

30

C

270

D

266

Worked Solution

Kate and Laura are selling cakes at a bake sale. For every 6 cakes that Kate sells, she will make \$15. For every 24 cakes that Laura sells, she will make \$53. Whose cakes are more expensive?

a

Fill in the missing gaps in the table for Kate.

\text{Cakes sold} | 6 | 18 | 30 | ||
---|---|---|---|---|---|

\text{Earning } (\$) | 30 | 60 | 75 |

Worked Solution

b

Fill in the missing gaps in the table for Laura.

\text{Cakes sold} | 48 | 72 | 96 | 120 | |
---|---|---|---|---|---|

\text{Earning } \left(\$\right) | 53 | 159 | 212 | 265 |

Worked Solution

c

Whose cakes are more expensive?

Worked Solution

Idea summary

We can use ratio tables to determine unknown values by multiplying or dividing. All of the values in the table will be equivalent ratios.