GCSE · Maths · AQA · Spec 8300

Four operations on integers, decimals and fractions

Multiplying makes numbers bigger; dividing makes them smaller. You've trusted that rule for years. Start at 4 and find exactly where it breaks.

Predict first

Where does 4 end up?

Every calculation below starts from 4 — and so does every slider. Don't work anything out yet: drag each one to where you think its answer lands, then check your line.

×3 — 4 × 3×0.5 — 4 × 0.5÷0.5 — 4 ÷ 0.5÷¼ — 4 ÷ ¼×−1.5 — 4 × (−1.5)−(−3) — 4 − (−3)

Place value

Which decimal claim is true?

Four students each make one claim about decimals. Only one of the claims is true.

Which claim do you think is the true one?

0.25 is bigger than 0.5, because 25 is bigger than 5.0.3 × 0.2 = 0.6, because 3 × 2 = 6.0.006 ÷ 10 = 0.0006, because every digit moves one column to the right.2.4 × 10 = 2.40, because times 10 means add a zero.

How sure are you?

I'm sureI think soI'm guessing

0.25 is bigger than 0.5, because 25 is bigger than 5.

This is a common misconception

It feels right, but read the columns, not the digits. 0.5 is 5 tenths. 0.25 is 2 tenths and 5 hundredths. Compare the tenths first: 5 tenths beats 2 tenths, so 0.5 is bigger. If it helps, write 0.5 as 0.50 — now you're comparing 50 hundredths with 25 hundredths.

0.3 × 0.2 = 0.6, because 3 × 2 = 6.

This is a common misconception

3 × 2 = 6 is the right start, but the size has gone missing. 0.3 × 0.2 means three tenths of two tenths. A tenth of a tenth is a hundredth, so the answer is 6 hundredths: 0.06. Quick sense check: 0.3 is less than 1, so the answer must be smaller than 0.2 — and 0.6 isn't.

0.006 ÷ 10 = 0.0006, because every digit moves one column to the right.

This matches the evidence

Yes. Dividing by 10 makes every digit worth ten times less, so each one moves one column to the right: 6 thousandths become 6 ten-thousandths. The same idea handles huge numbers too — 5 000 000 ÷ 1000 = 5000, because every digit moves three columns right.

2.4 × 10 = 2.40, because times 10 means add a zero.

This is a common misconception

Adding a zero works for whole numbers (24 × 10 = 240), which is why the shortcut feels safe. But 2.40 is exactly the same number as 2.4 — a zero on the end of a decimal changes nothing. Multiplying by 10 makes every digit worth ten times more, so each one moves one column left: 2 ones become 2 tens, 4 tenths become 4 ones. 2.4 × 10 = 24.

Never judge a decimal by how many digits it has. Ask what each digit is worth — which column it sits in — and the right answer follows.

Multiplying decimals

Build 4.3 × 2.6 from its parts

Estimate first: 4.3 × 2.6 is about 4 × 3 = 12. Now fill every cell — each one is its row times its column — then add them up.

Grid: each cell is its row times its column
×	2	0.6
4
0.3

Add the four cells: 4.3 × 2.6 =

Type x² as x^2 if you cannot type ². Spaces do not matter.

Spot the slip

Where does this answer go wrong?

Work out −2¼ ÷ 1½. Give your answer as a mixed number.

A student's answer — which line goes wrong?

Change both mixed numbers into improper fractions.
−2¼ = −8/4 + 1/4 = −7/4
1½ = 3/2
−7/4 ÷ 3/2 = −7/4 × 2/3 (flip the fraction you are dividing by, then multiply)
= −14/12 = −7/6 (negative ÷ positive gives a negative)
= −1⅙

Money in context

Priya's account and her old bike

Priya has £42.50 in her bank account. She pays a £65.80 phone bill by debit card. Then she sells her old bike for £62.40 and the money is credited to her account. She originally bought the bike for £85. What is her balance now, and did she make a profit or a loss on the bike?

Balance at the start: £42.50
missing step

Which line is step 2?

£42.50 + £65.80 = £108.30£42.50 − £65.80 = −£23.30£65.80 − £42.50 = £23.30

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

The same four operations work on whole numbers, decimals, fractions and negatives. Place value tells you how big each answer should be before you even start.

What you need to know

How to add, subtract, multiply and divide whole numbers, decimals, fractions and mixed numbers — positive and negative.
How place value (what each digit is worth) tells you where the decimal point goes and roughly how big an answer should be.
What everyday money words mean — profit, loss, cost price, selling price, debit, credit, balance, income tax, VAT and interest rate — so you can choose the right operation in a word problem.

The big picture

The four operations work the same way on every kind of number — whole numbers, decimals, fractions, mixed numbers and negatives. What changes is the size of the answer, and that depends on what you're working with. Multiply a positive number by something between 0 and 1 and it shrinks; divide by it and it grows. Negatives send answers across zero. Place value tells you where the decimal point belongs, and in money problems the words tell you which operation to use.

Key points

1For a positive number: multiplying by more than 1 makes it bigger, multiplying by a number between 0 and 1 makes it smaller, and dividing by a number between 0 and 1 makes it bigger.

2Multiplying or dividing two numbers: same signs give a positive answer, different signs give a negative one. Adding and subtracting negatives is movement along a number line — and subtracting a negative is the same as adding: 5 − (−2) = 7.

3Multiplying decimals: multiply without the decimal points, then give the answer the same total number of decimal places as the question. 0.3 × 0.2 = 0.06.

4Dividing by a decimal: multiply both numbers by the same power of 10 until you're dividing by a whole number. 7.2 ÷ 0.3 = 72 ÷ 3 = 24.

5Fractions: add or subtract over a common denominator; multiply tops together and bottoms together; to divide, flip the fraction you're dividing by and multiply. Always change mixed numbers into improper fractions first — never work on the whole parts and fraction parts separately — and keep the sign on the whole number: −2¼ = −9/4.

6Estimate before you calculate. 19.8 × 3.1 is about 20 × 3 = 60, so 61.38 is believable and 613.8 is not.

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Worked example

Problem

Work out 1⅔ − 3¼.

⚠ Watch out

Lining up decimals by their last digit when you add or subtract in columns. In 12.5 + 3.75, the 5 tenths must sit directly above the 7 tenths — line up the decimal points, not the right-hand ends. Fill any gap with a zero: 12.50 + 3.75 = 16.25.

🧠

Memory hook

Times by a bit — it shrinks. Divide by a bit — it grows. (A 'bit' is any number between 0 and 1.) And a minus sign sends it across zero.

✓

Check yourself

Is 9 ÷ 0.1 bigger or smaller than 9? What is 0.5 × 0.07? (Answers: bigger — 90, since ten tenths fit into every whole; 0.035, since 5 × 7 = 35, three decimal places.)

Flashcards

(10)

For a positive number, what does multiplying or dividing by a number between 0 and 1 do?

Multiplying by it makes the number smaller (you take only part of it). Dividing by it makes the number bigger (more than one of it fits into each whole).

What sign does the answer have when you multiply or divide two numbers?

Same signs → positive. Different signs → negative. For example −6 × −2 = 12, but −6 ÷ 2 = −3.

How do you know where the decimal point goes when you multiply two decimals?

Multiply as whole numbers, then count the decimal places in the question. The answer has the same total number of decimal places.

How do you divide by a decimal, like 4.5 ÷ 0.09?

Multiply both numbers by the same power of 10 until you're dividing by a whole number: 4.5 ÷ 0.09 = 450 ÷ 9 = 50.

What are the steps for dividing mixed numbers?

Change each mixed number into an improper fraction (keeping any minus sign on the whole thing), flip the fraction you're dividing by, multiply, then simplify.

What do debit, credit and balance mean on a bank account?

A debit is money going out. A credit is money coming in. The balance is how much is in the account — a negative balance means the account is overdrawn and you owe the bank.

How do cost price and selling price tell you whether there's a profit or a loss?

Cost price is what you paid; selling price is what you sold it for. Selling price higher → profit (selling − cost). Selling price lower → loss (cost − selling).

What is income tax?

A tax taken from the money people earn (their income). How much is taken usually depends on how much they earn.

What is VAT?

Value added tax: a tax added to the price of many goods and services, so the price you pay includes it.

What is an interest rate?

The percentage added to an amount of money over a period of time — paid to you on savings, or charged to you on money you borrow.

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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Where does 4 end up?

Which decimal claim is true?

Build 4.3 × 2.6 from its parts

Where does this answer go wrong?

Priya's account and her old bike

More AQA GCSE Maths topics