KS3 · Maths

Expressing one number as a percentage of another

Laura misses the first day of the school year, then never misses another. Does she finish the year on 100% attendance?

Maths · Percentages

Slide the part. Watch the percentage keep pace.

the part of 50: 25/50. percentage 50%. Part of 50 25

Fraction25/50Percentage50%Part of 5025

The whole bar is 50, and the whole bar is 100%. Drag its edge to choose the part. The fraction shows the part over the whole, and the percentage is that same fraction of 100%.

Watch out: It's multiplying, not taking away. 35 is 15 less than 50, but 35 is not 15% less than 100%. 35 is 35/50 of 50, so it is 35/50 of 100%, which is 70%.

Route 1 · Make the bottom 100

Problem

Write (a) 16 out of 25 and (b) 17 out of 200 as percentages.

Quick check · Percentages and decimals

What is 3% as a decimal?

Before the next route, we need percentages and decimals to talk to each other. Here's a small one that catches lots of people out.

Which is closest to what you think right now?

0.3 — you just move the 3 one place to the right of the point.0.03 — 3% means 3 out of 100, so divide 3 by 100.3 — the % sign is just a label, so 3% as a decimal is still 3.

How sure are you?

I'm sureI think soI'm guessing

0.3 — you just move the 3 one place to the right of the point.

This is a common misconception

That's dividing by 10, and per cent means out of 100, not out of 10. 0.3 is 3 tenths, which is 30 out of 100 — that's 30%, ten times too big. To turn a percentage into a decimal, divide by 100.

0.03 — 3% means 3 out of 100, so divide 3 by 100.

This matches the evidence

Spot on. 3% = 3/100 = 0.03. Think of the hundred grid with the whole grid worth 1: each square is 0.01, and 3 squares is 0.03.

3 — the % sign is just a label, so 3% as a decimal is still 3.

This is a common misconception

The % sign is doing real work: it means 'out of 100'. A decimal 3 would be 3 whole ones — that's 300%. 3% is a small part of one whole: 3/100 = 0.03.

Percentage to decimal: divide by 100. Decimal to percentage: multiply by 100. 3% = 0.03, and 0.3 = 30%.

Route 2 · Go through a decimal

When the bottom won't become 100

What percentage of 270 is 90? Give your answer to one decimal place.

Write the part over the whole: 90 out of 270 is 90/270.
missing step

Which line is step 2?

Simplify: the highest common factor of 90 and 270 is 90, so 90/270 = 1/3.Flip it so the bigger number is on top: 270/90 = 3.Take 90 away from 270 to see what's left: 270 − 90 = 180.

Your turn · Choose the route

Which route would you take?

For each question, pick the route that gets you there fastest.

Still to sort

Make the bottom 100 (0)

The denominator — after simplifying, if that helps — goes into 100 or is a multiple of 100.

Where the line is: Simplify before you decide. 18/24 looks awkward, but it simplifies to 3/4, and 4 goes into 100.

Go through a decimal (0)

No whole number turns the denominator into 100, even after simplifying.

Where the line is: This route always works. It's the one to reach for when making the bottom 100 isn't possible.

7 of 7 still to sort.

Both routes give the same answer. The skill is spotting which one is quicker for the numbers in front of you.

Back to Laura

Can Laura still get 100%?

Laura misses her first day of school, then never misses another. After day 1 her attendance is 0/1 = 0%. After day 2 it's 1/2 = 50%. After day 5 it's 4/5 = 80%, and after day 10 it's 9/10 = 90%. The school year has 190 days. What is the highest attendance she can finish the year on?

Your estimate

90%

80%100%

Watch out: Missing one day changes the top of the fraction for the rest of the year — but never the fact that the top is smaller than the bottom.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

What you need to know

Per cent means 'by the hundred' (from the Latin per centum): 1% is 1 out of 100, or 1/100.
To write one number as a percentage of another, start with the fraction part/whole.
Route 1: if the denominator goes into 100 (or is a multiple of 100), use an equivalent fraction with 100 on the bottom.
Route 2: if it doesn't, divide to get a decimal, then multiply by 100. This route works for any fraction.
Percentage to decimal is ÷ 100, so 3% = 0.03 — not 0.3.

The big picture

To write one number as a percentage of another, write it as a fraction of the whole (part over whole) and then write that fraction out of 100 — by making the denominator 100 with an equivalent fraction, or by converting to a decimal and multiplying by 100.

Key points

1A percentage is a fraction out of 100, so the first job is always the fraction part/whole — part on top, whole underneath.

2On a double number line the whole sits under 100%; whatever you multiply the whole by to get the part, multiply 100% by the same amount.

3Equivalent fractions keep the value: 16/25 × 4/4 = 64/100 = 64%, because 4/4 is just 1.

4Simplify first if it helps: a fraction like 18/24 becomes 3/4, which is easy to write out of 100.

5Through a decimal: 1/3 = 0.333… so 33.3% to one decimal place; 1/12 = 0.08333… so 8.3%.

6Decimal → percentage: × 100. Percentage → decimal: ÷ 100.

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Worked example

Problem

A school sets a target of raising £250 for charity. So far it has raised £150. What percentage of the target has it raised?

⚠ Watch out

Putting the fraction upside down. 'What percentage of 50 is 35?' is 35/50 = 70% — part on top, whole underneath. Writing 50/35 gives about 142.9%: more than 100% for a part smaller than the whole. That's your cue to flip it.

🧠

Memory hook

Part on top, whole underneath — then make it out of 100.

✓

Check yourself

Without a calculator: what percentage of 40 is 10? Then explain in one sentence how you know your answer is sensible.

Flashcards

(12)

What does 'per cent' mean?

By the hundred (from the Latin per centum). So 1% means 1 out of every 100, or 1/100.

On a hundred grid, what does one small square represent?

1% of the grid — or 0.01 if the whole grid is worth 1.

What's the first step in writing one number as a percentage of another?

Write it as a fraction of the whole: the part on top, the whole underneath.

Why doesn't multiplying top and bottom by 4 change a fraction's value?

Because 4/4 = 1, and multiplying by 1 changes how a number is written, not its value.

When is the 'make the bottom 100' route quick?

When the denominator (after simplifying) goes into 100, like 20, 25 or 50 — or is a multiple of 100, like 200.

How do you switch between decimals and percentages?

Decimal to percentage: × 100 (2/5 = 0.4 = 40%). Percentage to decimal: ÷ 100 (7% = 0.07).

What percentage is the decimal 0.3?

30%. (0.3 × 100 = 30.)

What is 1/8 as a percentage?

1 ÷ 8 = 0.125, so 12.5%.

What is 1/11 as a percentage, to one decimal place?

1 ÷ 11 = 0.0909… (0.09 recurring), and × 100 gives 9.0909…%, so 9.1%.

On a double number line, what goes above the whole amount?

100%. Then the part sits under its percentage, scaled by the same multiplier.

18 is what percentage of 24, using the double number line?

24 × 18/24 = 18, so 100% × 18/24 = 75%. 18 is 75% of 24.

Why can't someone who has missed a day ever have 100% attendance?

The days attended will always be less than the days so far, so the fraction is always less than 1 whole — less than 100%.

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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What is 3% as a decimal?

When the bottom won't become 100

Which route would you take?

Can Laura still get 100%?

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