Ratios and Rates

Lesson

We like to express ratios as whole numbers, using the simplest numbers possible. We simplify ratios in a similar way to how we simplify fractions. If we multiply or divide one side by a number, we do the same thing to the other side.

**Evaluate: **Express $15:5$15:5 as a ratio in its simplest form.

**Think:** $5$5 is a factor of both $15$15 and $5$5, so we'll divide both sides by $5$5.

**Do: **$3:1$3:1

Ok, let's look at an example of a ratio that is not made up entirely of whole numbers.

**Evaluate:** Simplify the ratio $1\frac{1}{2}:6$112:6

**Think: **the easiest way to make $1\frac{1}{2}$112 a whole number is to multiply it by $2$2 (which would make $3$3). Remember we also have to multiply $6$6 by $2$2, which would be $12$12. So our ratio would be $3:12$3:12. However, $3$3 is a factor of both these numbers, so we can simplify it further.

**Do:**

$1\frac{1}{2}:6$112:6 | $=$= | $\frac{3}{2}:6$32:6 | convert to improper fraction |

$=$= | $3:12$3:12 | multiply both parts of ratio by $2$2 | |

$=$= | $1:4$1:4 | simplify by reducing by common factor $3$3 |

Some ratios have more than $2$2 parts but we still simplify them in the same way.

Simplify the ratio $3:2.4:0.45$3:2.4:0.45

Ratios are useful because they describe the relationship between two values. However, to make sure that everyone can understand the ratios to describe, it is more useful if we use the same units of measurement on both sides of our ratio.

Let's say a recipe said I needed $150$150g of butter for every kilogram of flour I was using. Then I said to you, "This recipe is easy! The ratio of butter to flour is $150:1$150:1."

All of a sudden, you're thinking, "What! You need $150$150g of butter for every gram of flour?!"

My instructions weren't clear because I was using different units of measurement in my ratio.

To overcome this, we would make sure we have the same unit of measurement on both sides by convert one of them. I am going to change $1$1kg to $1000$1000g, so I can now write the ratio as $150:1000$150:1000 which can be simplified to $3:20$3:20.

**Convert ** $50c:\$2.10$50`c`:$2.10 to the same units, then simplify.

Equivalent means equal. For example, $2+2$2+2 is equivalent to $1+3$1+3 because both sums have the same answer which is $4$4. We can tell when ratios are equivalent when we can simplify them to give the same answer.

**Evaluate:** Is $10:20$10:20 equivalent to $4:8$4:8?

**Think:** If we simplify $10:20$10:20, we would get an answer of $1:2$1:2. If we simplify $4:8$4:8, we also get an answer of $1:2$1:2

**Do:** Yes, $10:20$10:20 is equivalent to $4:8$4:8.

**Evaluate:** Is $7:3$7:3 equivalent to $14:8$14:8?

**Think:** $7:3$7:3 is already simplified. If we simplify $14:8$14:8 it would become $7:4$7:4

**Do:** No, $7:3$7:3 is not equivalent to $14:8$14:8.

Let's try and think of some equivalent fractions ourselves. Remember to multiply both numbers in the ratio by the same number.

Consider the ratio $5:28$5:28.

a) Complete the pattern of these equivalent ratios.

$5$5 | $:$: | $28$28 |

$10$10 | $:$: | $56$56 |

$15$15 | $:$: | |

$20$20 | $:$: | |

$25$25 | $:$: | $140$140 |

b) Later in the pattern, the following ratio will appear. Fill in the missing value.

$?:168$?:168

c) Later in the pattern, the following ratio will appear. Fill in the missing value.

$80:?$80:?