KS3 · Maths

Multiplicative relationships as ratios and fractions

Your recipe feeds 4 people and 12 turn up. Do you add 8 spoonfuls of everything? Of course not — so what do you do instead? That's today's lesson.

Flour and water · one multiplier

Drag the flour. Watch the water keep up.

Flour (tbsp): 4/8. Flour (tbsp) 4. Water if × 60 240 ml. ml per tbsp (× 60) 60 ml. Water if + same 240 ml. ml per tbsp (+ same) 60 ml

Flour (tbsp)4Water if × 60240 mlml per tbsp (× 60)60 mlWater if + same240 mlml per tbsp (+ same)60 ml

The mix starts at 4 tablespoons (tbsp) of flour to 240 ml of water, so flour × 60 = water. Drag the flour bar, then compare the two water chips. “+ same” means whatever you add to the flour, you add to the water too.

Watch out: Adding the same amount to each part is not scaling. Only multiplying every part by the same number keeps the ratio.

Judge it from the multiplier

Lemonade: bitter, sweet or just right?

Sort each mix: too bitter, just right or too sweet?

Still to sort

Too bitter (0)

Not enough sugar for the lemons and water.

Where the line is: Same lemons-and-water multiplier, but the sugar falls short of it.

Just right (0)

Sugar uses the same multiplier as the lemons and water.

Where the line is: All three parts are multiplied by the same number.

Too sweet (0)

Too much sugar for the lemons and water.

Where the line is: Same lemons-and-water multiplier, but the sugar overshoots it.

6 of 6 still to sort.

The recipe is 1 lemon : 500 ml water : 2 tablespoons sugar. Don't recalculate every part. Find the multiplier on the lemons, then ask what it does to the sugar.

What does a multiplier mean, and what is it the other way round?

Problem

Three pairs of quantities, each with a multiplier. What does each multiplier say, and what takes you back the other way?

Your turn to build the multiplier

One multiplier, from any number to any other

Connect numbers using a single multiplier: first 6 to 8, then 8 to 15, then 6 to 14.

Any two numbers can be linked by an add-on and by a multiplier. 10 to 30 is + 20, and it is also × 3. For ratios, the multiplier is the link that matters.Start here.
A multiplier can come in steps. To get from 6 to 8: ÷ 3 gives 2, then × 4 gives 8, so 6 ÷ 3 × 4 = 8.
missing step

Which line is step 3?

6 ÷ 3 × 4 = 6 × 1/3 × 46 ÷ 3 × 4 = 6 × 3 × 46 ÷ 3 × 4 = 6 − 3 × 4

Ratio or fraction?

Ratio (part to part)vsFraction (part to whole)

Same mix, two ways of writing it. Start with the one everyone mixes up.

Focus

2 cups of flour to 3 cups of water

Ratio (part to part)

2 : 3 compares flour with water

Fraction (part to whole)

Flour is 2/5 of the batter, not 2/3

The insight

2 : 3 is not 2/3. For the fraction, compare the flour with the whole batter: 2 out of 2 + 3 = 5 parts.

What goes on the bottom?

Ratio (part to part)

Nothing. You just write the parts: 2 and 3.

Fraction (part to whole)

The total number of parts: 5

The water

Ratio (part to part)

3 parts of water to every 2 of flour

Fraction (part to whole)

Water is 3/5 of the batter

Linking flour and water

Ratio (part to part)

flour × 3/2 = water, and water × 2/3 = flour

Fraction (part to whole)

Flour is 2/5 of the whole and water is 3/5 of the whole

1 cat for every 2 dogs, 33 pets in all

Ratio (part to part)

Total 3 parts. The multiplier from 3 to 33 is 11.

Fraction (part to whole)

Cats are 1/3 of 33 = 11. Dogs are 2/3 of 33 = 22.

Which one's bigger?

Bigger number, bigger proportion?

You're comparing proportions: pay rises, percentages, allowance increases. Which of these ideas is closest to yours?

Pick the idea closest to what you think right now.

A bigger percentage always means a bigger amount. 300% of £2,000 has to beat 2% of £1,000,000.The bigger rise is the bigger proportion. A £500 pay rise on £4,000 has to beat a £400 rise on £400.Two £6 allowance increases are the same proportion, because £6 is £6.Turn each one into a fraction of its own whole, then compare the fractions.

How sure are you?

I'm sureI think soI'm guessing

A bigger percentage always means a bigger amount. 300% of £2,000 has to beat 2% of £1,000,000.

This is a common misconception

Check the wholes. 2% of £1,000,000 is £20,000, but 300% of £2,000 is only £6,000. A smaller percentage of a much bigger number can be the greater amount.

The bigger rise is the bigger proportion. A £500 pay rise on £4,000 has to beat a £400 rise on £400.

This is a common misconception

£500 on £4,000 is 500/4000 = 1/8. £400 on £400 is 400/400 = 1, a 100% rise. The smaller amount is the bigger proportion, because its whole is so much smaller.

Two £6 allowance increases are the same proportion, because £6 is £6.

This is a common misconception

£6 on £15 is 2/5, but £6 on £30 is 1/5. Same rise, different wholes, so £6 on £15 is the bigger proportion.

Turn each one into a fraction of its own whole, then compare the fractions.

This matches the evidence

Exactly. 12 out of 20 is 3/5 and 40 out of 50 is 4/5, so 40 out of 50 is the higher proportion.

Before you compare, ask: out of what? A proportion is always measured against its own whole.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

What you need to know

A ratio shows how big one part is compared with another part of the same whole.
The link between the parts is multiplicative, not additive. Two variables are in proportion when that multiplier stays constant.
Multiply every part by the same number and the ratio stays the same. 2 : 3 : 4 becomes 10 : 15 : 20 for five sandwiches.
Have a goSame sandwich: 2 bread, 3 cheese, 4 tomato. You've got 9 slices of cheese. How many slices of bread and tomato go with them?
6 bread and 12 tomato.
9 cheese is 3 × 3, so multiply the bread and the tomato by 3 as well: 2 × 3 = 6 and 4 × 3 = 12.
Parts can be multiplied by decimals or fractions too. If 2 smoothies need 300 ml of yoghurt, 1 smoothie needs 150 ml.
Add the same amount to every part, or take the same amount away, and the ratio changes.
Have a goZainab scales 2 red : 5 green by adding 2 to each part, and gets 4 : 7. How much green should go with 4 red?
10 green.
2 red to 4 red is × 2, so the green has to be × 2 as well: 5 × 2 = 10. Adding 2 to each part isn't scaling.
A multiplier tells you how much of one quantity there is for every one of the other. Think of it as a scaling factor.
Going the other way uses the reciprocal. If one way is × 3/2, the way back is × 2/3.
Have a goOne way, the multiplier is × 4/5. What multiplier takes you back the other way?
× 5/4.
It's the reciprocal: 4/5 × 5/4 = 1, so the two steps cancel and you land back where you started.
Several multiplier steps become one: multiply the multipliers. ÷ 2 then × 5 is a single step of × 5/2.
Using 1 as the middle step is called the unitary method: reach 1 first, then multiply up to your target.
Proportion can be shown as a fraction, decimal, percentage or ratio. 1/4, 0.25, 25% and ‘for every 1 shaded, 3 unshaded’ all say the same thing.
A ratio can be written as fractions of the whole. Add up the parts, and that total is the denominator.
To write one quantity as a fraction of another, simplify. £4 from £24 is 4/24 = 1/6, because six lots of £4 make £24.
When you compare proportions, look at the whole. A smaller percentage of a much bigger whole can be the greater amount.
Have a goWhich is the bigger amount: 10% of £500, or 50% of £60? Work out both before you decide.
10% of £500, which is £50 (50% of £60 is £30).
A bigger percentage of a small whole can lose to a smaller percentage of a big one. Compare the amounts, not the percentages.
A straight-line graph shows a multiplicative relationship: every point on the line is an equivalent ratio. That is how ratio links to linear functions.
Read a graph carefully: check the scale and which quantity is on which axis. On a map graph of 2 cm to 15 km, 12 cm is 15 × 6 = 90 km.

The big picture

Two quantities in a ratio are tied together by one multiplier. Multiply every part by it and the ratio holds; add the same amount to every part and it breaks. The multiplier has a reciprocal for the way back, it turns a ratio into fractions of a whole, and it shows up as a straight line on a graph.

Key points

1A ratio's parts are linked by one multiplier: multiply every part by the same number and the ratio stays the same.

2Adding the same amount to every part, or taking it away, changes the ratio.

3The multiplier one way and the multiplier back are reciprocals: they multiply to 1, and ÷ by a number is × by its reciprocal.

4Several multiplier steps combine into one single multiplier by multiplying them; going via 1 is the unitary method.

5In a ratio, the total number of parts is the denominator of each part's fraction, and those fractions add to one whole.

6To compare proportions, work with the whole, not just the size of the amount or the percentage.

7A straight-line graph shows a multiplicative relationship: every point on the line is an equivalent ratio.

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Worked example

Problem

Are 4 : 12 and 5 : 20 equivalent ratios? What about 4 : 12 and 5 : 15?

⚠ Watch out

Keeping a ratio by adding the same amount to each part, such as turning 1 : 3 into 3 : 5. The relationship is multiplicative, so you multiply every part by the same number: 1 : 3 becomes 3 : 9.

🧠

Memory hook

Multiply to match, add to break. Same number on every part, and the reciprocal to go back.

✓

Check yourself

A fruit drink uses 5 apples to 3 cups of juice. You use 20 apples. What do you multiply the juice by, and how much juice is that? Why wouldn't adding 15 work?

Flashcards

(14)

What does ‘multiplicative relationship’ mean for the parts of a ratio?

One multiplier links the parts, and it is the same for everything in that ratio.

How do you keep a ratio the same when you scale it?

Multiply every part by the same number. Adding the same amount to each part changes the ratio.

What must be true of a double number line?

Both lines start from zero, with the zeros aligned and a consistent scale.

Can you multiply a part of a ratio by a decimal or a fraction?

Yes. Any part can be multiplied by a decimal or a fraction, for example halving to find the amount for 1.

What does a multiplier tell you?

How much of one quantity there is for every one of the other. It acts as a scaling factor.

What is a reciprocal?

The multiplicative inverse of a non-zero number. A number times its reciprocal is 1, for example 3 × 1/3 = 1.

Dividing by a number is the same as…

Multiplying by its reciprocal. For example 6 ÷ 3 = 6 × 1/3 = 2.

How do you turn several multiplier steps into one?

Multiply them together. ÷ 3 then × 4 is 1/3 × 4 = 4/3.

What is the unitary method?

Using 1 as the middle step: multiply by the reciprocal to reach 1, then multiply to reach your target.

Write ‘for every 1 shaded there are 3 unshaded’ as a fraction, a decimal and a percentage.

1/4, 0.25 and 25%. All three show the same proportion as the ratio.

For a ratio, what is the denominator of each part's fraction?

The total number of parts. The fractions of all the parts add up to one whole.

What does a straight-line graph show about a ratio?

The line represents all the points where the ratio is equivalent.

What must you attend to when you compare proportions?

The whole. A smaller percentage of one number can be greater than a larger percentage of a different number.

How do you write one quantity as a fraction of another?

Write it as a fraction and simplify. £4 from £24 is 4/24 = 1/6.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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Lemonade: bitter, sweet or just right?

One multiplier, from any number to any other

Bigger number, bigger proportion?

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