KS3 · Maths

Finding the whole or a part from a ratio

Red and yellow flowers are in the ratio 3 : 5, and 9 are red. How many flowers in the bunch? The tempting answer is 9. It isn't.

Maths · Ratio

Where does the 9 go?

boxes filled: 1/8. Flowers in the filled boxes 3. Flowers in the empty boxes 21. Flowers in the whole bunch 24

Filled boxes : Empty boxes1 : 7Flowers in the filled boxes3Flowers in the empty boxes21Flowers in the whole bunch24

Red to yellow flowers are in the ratio 3 : 5, and there are 9 red flowers. Drag the handle along the bar to fill the 3 red boxes. The scale under the bar shows how many flowers you've covered.

Watch out: The 9 isn't the whole bunch. It only fills the red boxes, so the other 5 boxes are still waiting.

Worked example: the whole from a part, using a bar model

Problem

Flour, butter and sugar go into a cookie in the ratio 3 : 2 : 1. The cookie has 45 g of flour. What is the mass of the whole cookie?

Same job, different tool: a ratio table

Problem

Orange juice and lemonade are mixed in the ratio 3 : 7. There is 750 ml of orange juice. How much drink is there altogether?

Your turn to fill the gaps

One part from another part, with a ratio table

A car park has red, blue and black cars in the ratio 8 : 5 : 12. There are 48 black cars. How many red cars and how many blue cars are there?

Set up a ratio table with a column each for red, blue and black. The ratio row reads 8, 5 and 12.
missing step

Which line is step 2?

Put the 48 in the black column, because it is the number of black cars.Put the 48 in the total column.Put the 48 in the red column, since red comes first.

Which idea is yours?

The trap in ratio questions

Lucas and Aisha share sweets in the ratio 7 : 5. Aisha gets 20 sweets. You want to know how many sweets there are altogether.

Which of these is closest to what you'd do first?

20 is the number in the question, so it's the total. I'll share 20 out in the ratio 7 : 5.The ratio starts with 7, so the 20 goes on the 7 boxes. I’ll do 20 ÷ 7.Aisha is second, so she has 5 boxes. The 20 fills those 5 boxes, so each box is 20 ÷ 5 = 4.

How sure are you?

I'm sureI think soI'm guessing

20 is the number in the question, so it's the total. I'll share 20 out in the ratio 7 : 5.

This is a common misconception

Easy trap: the number in a question often looks like the total. But read it again. Aisha gets 20, so that is one person's share and it only fills her boxes. Draw the bar first and ask which boxes the 20 fills.

The ratio starts with 7, so the 20 goes on the 7 boxes. I’ll do 20 ÷ 7.

This is a common misconception

Careful: the order of the names in the question is the order of the numbers in the ratio. Lucas and Aisha is 7 : 5, so Lucas has 7 parts and Aisha has 5. Aisha’s 20 goes on her 5 boxes.

Aisha is second, so she has 5 boxes. The 20 fills those 5 boxes, so each box is 20 ÷ 5 = 4.

This matches the evidence

Yes. One box is 4 sweets. Lucas has 7 × 4 = 28, and the whole is 12 boxes × 4 = 48 sweets. Everything came from asking which boxes the 20 fills.

Before you calculate anything, ask: which boxes does my number fill?

Bar model or ratio table?

Bar modelvsRatio table

Both expose the structure. They just have different limitations.

Focus

When to switch

Bar model

Drawing it can take a lot of time. With lots of parts it gets quite inefficient.

Ratio table

May be the more efficient way when the ratio has lots of parts.

The insight

If you catch yourself drawing box after box, that's your cue to think ratio table.

Where the given value goes

Bar model

On its own part's boxes, divided by that part's number of boxes.

Ratio table

In the correct column, then you find the multiplier.

What it shows best

Bar model

Equal boxes, so you can see which boxes your number fills.

Ratio table

The multiplicative relationship between the rows.

Finding the whole

Bar model

Count all the boxes, then multiply by the value of one box.

Ratio table

Multiply the total by the same multiplier.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

The number you're given is one part, not the total

What you need to know

A ratio shows the relative sizes of values, so you can compare one part with another.
Variables are in proportion if they have a constant multiplicative relationship.
The order in the question matches the order in the ratio: cats then dogs, 2 : 5, gives cats 2 parts.
A bar model splits the whole into equal boxes, so every box holds the same amount.
The number you're given is usually one part's value. Don't assume it is the total.
Put it on its own part, then divide by that part's number of boxes. The answer goes in every box.
Have a goTom's 12 sweets fill the 4 boxes that make up his part of a bar model. How many sweets go in each box?
3 sweets in each box.
Divide by the boxes of the part your number belongs to (12 ÷ 4). Dividing by every box in the bar would spread the 12 over boxes it doesn't fill.
To find the whole, add the parts to count every box, then multiply by the value of one box.
Have a goEach box in a 2 : 3 : 1 bar holds 5 sweets. How many sweets are in the whole bar?
30 sweets.
The whole is every box: 2 + 3 + 1 = 6 boxes, and 6 × 5 = 30. One part on its own is not the whole.
To find another part you don't need the total: multiply that part's boxes by one box's value.
Have a goRed to yellow flowers are in the ratio 3 : 5 and there are 6 red. How many yellow, without working out the total?
10 yellow flowers.
6 ÷ 3 = 2 in each box, and yellow has 5 boxes, so 5 × 2 = 10. The total never comes into it.
In a ratio table, put the given value in its correct column, find the multiplier, and multiply the total by it.
Drawing a long bar can take ages, so with lots of parts a ratio table may be more efficient.

The big picture

A ratio question usually hands you one part's value, not the total. Put it on its own part of a bar model (or in the right column of a ratio table), find the value of one box, and every other part and the whole follow.

Key points

1The number you're given describes one part. Put it on that part's boxes, not on the whole bar.

2Value of one box = the given value ÷ the number of boxes in its part.

3Whole = value of one box × every box (add the parts to count them).

4Another part = its number of boxes × the value of one box. You don't need the total.

5A ratio table does the same job: given value in its column, find the multiplier, apply it across.

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Worked example

Problem

Black and white beads are in the ratio 4 : 1 on every necklace. One necklace has 4 white beads. How many beads are on the necklace?

⚠ Watch out

Treating the number in the question as the total. With red to blue cars in the ratio 5 : 3 and 120 red cars, 120 isn't all the cars. It's the red ones. Label the bar first, then decide which part your number belongs to.

🧠

Memory hook

Find the box first. The number you are given fills its own boxes, and one box unlocks every other part and the whole.

✓

Check yourself

Red, blue and white beads are in the ratio 2 : 7 : 5, with 21 blue. Which column does the 21 go in, what is the multiplier, and how many beads in total?

Flashcards

(12)

What does a ratio like 3 : 5 tell you?

The relative sizes of two or more values, so you can compare a part with another part.

When are two variables in proportion?

When they have a constant multiplicative relationship.

Lola and Ravi share in the ratio 6 : 1. Who has 6 parts?

Lola. The order of the names in the question is the order of the numbers in the ratio.

What is true of every box in a bar model?

Each box is an equal part, so every box holds the same amount.

Is the number in a ratio question always the total?

No. Always using it as the total is a common mistake. Assign it to the part it belongs to.

You know one part’s value. What do you do first?

Divide it by that part's number of boxes. The answer goes in every box.

How do you get from one box to the whole?

Multiply by the total number of boxes (add up all the parts to count them).

How many parts are there in a 4 : 5 : 2 ratio altogether?

11, because 4 + 5 + 2 = 11.

Do you need the total to find one part from another part?

No. Find the value of one box, then multiply by the other part's number of boxes.

In a ratio table, where does the given value go?

In the correct column: the one for the part it describes, not automatically the total.

How do you find the multiplier in a ratio table?

Divide the given value by the ratio number in its column, then multiply every other column by it.

When might a ratio table beat a bar model?

When the ratio has lots of parts, because drawing a long bar takes a lot of time.

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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One part from another part, with a ratio table

The trap in ratio questions

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