Equivalent and simplest ratios

2 : 3 = 4 : 6 = 6 : 9: multiply or divide both numbers by the same number. 12 : 18 simplifies to 2 : 3 by dividing both by 6.

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

Doubling a recipe keeps the taste the same. Two ratios are equivalent when one is the other with both numbers multiplied (or both divided) by the same number:

2 : 3 = 4 : 6 = 6 : 9 \qquad (\times 2,\ \times 3)

A ratio is in simplest form when no number greater than 1 divides both terms. To simplify, divide both by their highest common factor: $12 : 18$ , divide by 6, gives $2 : 3$ . It is the same idea as equivalent fractions.

Adding the same number to both terms does not keep the ratio: $2 : 3$ is not $4 : 5$ .

Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Try: Try it on the board and compare the result with your prediction.
Investigate: Work out why it happens, one step at a time.
Modify: Change the problem on purpose and see what happens.
Practise: A few on your own, to check you have it.
Make: Make your own problem using what you have learned. Expect a few tries; that is normal.

PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.

Which is the same as 2 : 3?

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

Which ratio is equivalent to

2 : 3

?4 : 53 : 45 : 64 : 6

TryTry it on the board and compare the result with your prediction.

Copies of a ratio

The groups of blocks repeat, so the ratio stays the same.

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Make 3 copies of 2 : 3. What is the new ratio?

Use the board. This ticks when you get there.

InvestigateWork out why it happens, one step at a time.

Multiply or divide both

Find what the first term was multiplied by.

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

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

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

ModifyChange the problem on purpose and see what happens.

Scale and simplify

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The board shows 4 : 6. Change it to the simplest ratio equal to 4 : 6.

Use the board. This ticks when you get there.

Simplify $20 : 30$ . It is $2 : ?$

A recipe uses 2 cups of flour for every 3 cups of milk. How many cups of milk go with 10 cups of flour?

Which of these is NOT equivalent to

6 : 8

?24 : 323 : 412 : 169 : 11

PractiseA few on your own, to check you have it.

Equivalent ratios

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

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

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

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

MakeMake your own problem using what you have learned. Expect a few tries; that is normal.

Big copies of 3 : 5

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

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Need a hint?

Set a = 3 and b = 5 first.

Another hint

Try 7 copies.

Make a ratio equal to 3 : 5 with both terms bigger than 20.

Use the board. This ticks when you get there.

For teachers

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). About 40 minutes.

Open the teacher guide: answers, solutions, slides and a worksheet.

Students will be able to

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.

Before class

Do the Run board yourself first. Students need equivalent fractions (the same multiplying and dividing) and highest common factors.

The one idea: equivalent ratios are copies of the pattern: both terms multiplied (or divided) by the same number. The common mistake is additive thinking (Ana: 2 → 6 is +4, so 3 → 7). This lesson is the hinge for sharing in a ratio: students who cannot scale and simplify will struggle next.

Link to equivalent fractions: 2 : 3 behaves like 2/3 for scaling, but a ratio is not a fraction (part to part).

Running the lesson

Predict: students who chose 4 : 5 added 2 to both terms.
Run: ask how many blocks of each colour with 3 copies, then how the numbers are related to 2 and 3.
Investigate: ask students to draw Ana's recipe as blocks.
Modify: for the impostor, ask what the multiplier would have to be.
Make: collect the multipliers used: all give the same ratio.

The hard part (on purpose)

Multiplying both terms, not adding the same amount to both. Ask 'how many copies is that?' before any arithmetic.

When students get stuck

A student adds the same number to both terms (Ana's slip). Ask: Draw 2 flour and 3 milk as blocks. What happens to the blocks when you make 3 copies? Fix: The number of blocks is multiplied, not added to.
A student simplifies one term only. Ask: Did you divide both terms by the same number? Fix: Divide both by their highest common factor.
A student cannot find the multiplier. Ask: What do you multiply 2 by to get 10? Fix: Find the multiplier from the term you know, then use it on the other.

Stretch and support

Stretch: A ratio is 2 : 3. Find all ratios equivalent to it with both terms less than 30, and say how you know you found all of them.

Support: Keep the copy blocks on the board and count the groups.

Where it leads

Next: Sharing in a ratio, which uses equivalent ratios to split a total.

Where it leads: rates and unit rates, scale drawings, similar shapes, and proportion in Years 8 and 9.

What each step is for

which is the same as 2 : 3?: Committing shows whether they multiply both terms (4 : 6) or add the same amount to both (4 : 5).
copies of a ratio: Copies of the same pattern of blocks keep the ratio; the number of copies is the multiplier.
multiply or divide both: Equivalent ratios multiply both terms by the same factor; Ana adds. Simplifying divides both terms by the same factor.
scale and simplify: Simplifying on the board, simplifying by hand, scaling a recipe and spotting an impostor use the same rule from several angles.
equivalent ratios: Simplify, scale up, and scale by a factor found from one term; the review keeps part to whole fresh.
big copies of 3 : 5: Finding copies of 3 : 5 with both terms over 20 shows the multiplier is what keeps the ratio, and many copies work.

Now you can

Find an equivalent ratio by multiplying or dividing both terms by the same number.
Write a ratio in simplest form.
Scale a recipe up using equivalent ratios.

Practise this skillNew questions, new numbers each timeGet 3 right in a row to mark it practised.Next lesson22. Sharing in a ratioShare $60 in the ratio 2 : 3. Add the parts (5), find one part ($60 ÷ 5 = $12), then multiply: $24 and $36.

Source: content/maths/number/equivalent-ratios/lesson.md