Ratios and Rates

UK Secondary (7-11)

Ratios with Fractions and Percents

Lesson

We have already learnt about ratios that compare the relationship between two values. We also looked at simplifying ratios, where, just like when we simplify fractions, we reduce each value by the highest common factor.

For example, if the ratio of flour to sugar in a recipe is $3:1$3:1, it means that for every $3$3 cups of flour we add, we need to add $1$1 cup of sugar.

Now let's look at what happens with ratios that include fractions.

Let's say our recipe said the ratio of flour to sugar was $1:\frac{1}{3}$1:13. That means, that for every cup of flour we add, we need to add $\frac{1}{3}$13 of a cup of sugar.

To express a ratio in its simplest form, we want to express the whole thing in terms of whole numbers (ie. simplify our ratio so it doesn't have any fractions).

To simplify the ratio $1:\frac{1}{3}$1:13 from the example above, we could multiply both sides of the ratio by $3$3 (after all, we need to keep it equivalent). This means our simplified ratio is $3:1$3:1. So it's basically a different way of writing the ratio from our first example.

Complete the ratio table below for the $6:\frac{1}{5}$6:15.

$6$6 | $12$12 | $18$18 | $\editable{}$ |

$\frac{1}{5}$15 | $\editable{}$ | $\editable{}$ | $\frac{4}{5}$45 |

Think: Every time the first variable increases by $6$6, the second variable increases by $\frac{1}{5}$15.

Do:

$6$6 | $12$12 | $18$18 | $24$24 |

$\frac{1}{5}$15 | $\frac{2}{5}$25 | $\frac{3}{5}$35 | $\frac{4}{5}$45 |

Simplify the ratio $\frac{6}{5}:\frac{7}{10}$65:710

To make the perfect shade of green for her painting, Xanthe knows she needs to mix blue to yellow in the ratio $\frac{4}{7}:\frac{5}{8}$47:58.

How much yellow paint will she need if she wants to use $4$4 pots of blue paint?

As we have seen already, a percentage is a special fraction where the denominator is always 100.

a) Complete the ratio table below for the ratio $2:\frac{2}{5}$2:25.

$2$2 | $4$4 | $6$6 | $8$8 | $\editable{}$ |

$\frac{2}{5}$25 | $\frac{4}{5}$45 | $\frac{6}{5}$65 | $\editable{}$ | $2$2 |

Think: What patterns can you see?

Do:

$2$2 | $4$4 | $6$6 | $8$8 | $10$10 |

$\frac{2}{5}$25 | $\frac{4}{5}$45 | $\frac{6}{5}$65 | $\frac{8}{5}$85 | $2$2 |

b) Express $2:\frac{2}{5}$2:25 as a unit ratio.

Think: We need to simplify both sides of the ratio so that the second term is equal to one.

Do:

$2:\frac{2}{5}$2:25 | $=$= | $10:2$10:2 | multiplying both sides by $5$5 |

$=$= | $5:1$5:1 | dividing both sides by $2$2 |

Ellie is making muffins. Her recipe states that she needs $\frac{2}{3}$23 cup of sugar, and $\frac{5}{8}$58 cup of butter.

She accidentally adds $1$1 cup of butter. How much sugar will she now need to add? You can leave your answer as an improper fraction if necessary.

Think: What is the ratio of butter to sugar in this recipe? How would we express this as a unit ratio?

Do:

The ratio of butter to sugar is $\frac{2}{3}:\frac{5}{8}$23:58. Let's simplify this.

$\frac{2}{3}:\frac{5}{8}$23:58 | $=$= | $\frac{16}{3}:5$163:5 | multiplying both sides by $8$8 |

$=$= | $16:15$16:15 | multiplying both sides by $3$3 | |

$=$= | $\frac{16}{15}:1$1615:1 | dividing both sides by $15$15 |

She will need to add $\frac{16}{15}$1615 or $1\frac{1}{15}$1115 of a cup of sugar to her mixture.

What is the unit rate for the ratio $12$12 : $\frac{4}{7}$47 ?

Express $\frac{18}{25}$1825 kg to $230$230 g as a simplified ratio.