---
title: "Equivalent and simplest ratios"
subtitle: "Number Foundations · Ratios · Lesson 21"
---

# Equivalent and simplest ratios

**Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}}

> **Key question:** A recipe for 4 uses 2 cups of flour and 3 of milk. How much milk do you need for 10 cups of flour?


:::teacher-note

**In one sentence:** Students make copies of a ratio on the board to see that multiplying both terms keeps the ratio, find missing terms, and simplify by dividing both terms by a common factor (AC9M7N08, node 2).

**Students should be able to say:** I can find an equivalent ratio by multiplying or dividing both terms by the same number. I can write a ratio in simplest form. I can scale a recipe up using equivalent ratios.

**Width:** the unmarked parts are the floor every student does (predict, run, investigate and make). Typical adds the modify practice; wider adds the real-world contexts. They add contexts, not difficulty. Students work in the online tutorial alongside this sheet.

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## Predict

What do you do to 2 and 3 to get 4 and 6?

:::keep-together

**Which ratio is equivalent to $2 : 3$?**

☐ 4 : 5

☐ 3 : 4

☐ 5 : 6

☐ 4 : 6

My prediction, and why:

{{lines-3}}

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:::teacher-note

**What happens:** 4 : 6. Both terms are doubled. 4 : 5 and 3 : 4 each add the same number to both terms, which changes the ratio.

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## Run

Try it on the board in the tutorial. Was your prediction right? What was different?

{{lines-2}}

## Multiply or divide both


:::keep-together

### Fill in the blank

**Fill in the blank: $2 : 3 = 10 : ?$**

Answer: {{field-15}}

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:::teacher-note

**Answer:** 15

2 → 10 is × 5, so 3 → 3 × 5 = 15.

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:::keep-together

### Find the mistake

**Ana scales a recipe. Click the line where her working goes wrong.**

```
 1  A recipe uses 2 cups of flour for every 3 cups of milk.
 2  For 6 cups of flour, how much milk?
 3  Flour went from 2 to 6: that is + 4.
 4  So milk goes from 3 to 7: 3 + 4.
 5  Answer: 7 cups of milk.
```

Line: {{field-15}}

Why is it wrong?

{{lines-2}}

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:::teacher-note

**Answer:** line 4

Line 4: adding 4 to the milk too is wrong. Flour went × 3 (2 to 6), so milk goes × 3: 3 × 3 = 9 cups.

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:::keep-together

### Simplify

**Simplify $12 : 18$. It is $2 : ?$**

Answer: {{field-15}}

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:::teacher-note

**Answer:** 3

The highest common factor of 12 and 18 is 6. 12 ÷ 6 = 2 and 18 ÷ 6 = 3.

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:::typical

## Scale and simplify


**Simplest form.** 

**Divide by ten.** 

**Scale the recipe.** 

**Spot the impostor.** 

Work in the tutorial. Plan first: what will you change, and how will you test it?

{{box-4}}

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## Equivalent ratios


:::keep-together

### Divide by five

**Simplify $15 : 25$. It is $3 : ?$**

Answer: {{field-15}}

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:::teacher-note

**Answer:** 5

Divide both by 5: 3 : 5.

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:::keep-together

### Multiply

**Fill in the blank: $5 : 2 = ? : 8$**

Answer: {{field-15}}

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:::teacher-note

**Answer:** 20

2 → 8 is × 4, so 5 × 4 = 20.

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:::keep-together

### Pink paint

**Pink paint is 3 parts red to 1 part white. How many parts of white go with 12 parts of red?**

Answer: {{field-15}}

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:::teacher-note

**Answer:** 4

3 → 12 is × 4, so white is 1 × 4 = 4.

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:::keep-together

### Review: part to whole

**Review: the ratio of blue to yellow is $3 : 4$. Blue to the whole is $3 : ?$**

Answer: {{field-15}}

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:::teacher-note

**Answer:** 7

3 + 4 = 7, so blue to the whole is 3 : 7.

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## Make: big copies of 3 : 5

Then explain how you know 24 : 40 is the same ratio as 3 : 5.

Work in the tutorial. Plan first: what will you change, and how will you test it?

{{box-4}}

## Exit ticket

:::keep-together

**Simplify 24 : 36, and find the missing term: 3 : 5 = 12 : ?**

{{lines-3}}

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:::teacher-note

**Answer:** 24 : 36 = 2 : 3. 3 → 12 is × 4, so 5 × 4 = 20.

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