Calculate with decimals

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Keep equal place values together

Estimate first. Is the piece left more or less than 4 metres?

A ribbon is 6.2 m long. A 1.48 m piece is cut off. Is the length left between 4 m and 5 m? Estimate first. Then find the exact length and check it.

Length remaining

Unit: m

Length restored by joining the cut piece

Unit: m

Why can you write 6.2 as 6.20 before subtracting 1.48?

A final zero keeps the value and shows hundredthsA final zero changes tenths into whole unitsA final zero makes the amount ten times larger

Show workingHide answer

Removing about 1.5 m from 6.2 m suggests about 4.7 m. Write 6.20 so hundredths line up.

Exchange one tenth for ten hundredths, then one whole for ten tenths. There are now 5 ones, 11 tenths and 10 hundredths. Removing 1 one, 4 tenths and 8 hundredths leaves 4.72 m.

Check 4.72 + 1.48 = 6.20 m.

For a closer look at these exchanges, revisit Add and subtract decimals.

Multiply a decimal by a whole number

Each of these four ribbons is 2.35 m long. Find their total length.

Four equal lengths

The four cards refer to four separate ribbons, each with the stated length.

Four separate ribbons. Each card states the length of one ribbon.

Figure data

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Values shown on the cards
Set	Values
Four ribbons	2.35 metres; 2.35 metres; 2.35 metres; 2.35 metres

One ribbon is 235 hundredths of a metre. Keep that unit while multiplying.

Hundredths of a metre in all four ribbons

Total length in metres

Unit: m

Total length of zero such ribbons

Unit: m

Show workingHide answer

235 × 4 = 940 hundredths, so 2.35 × 4 = 9.40 m. An estimate of 2.4 × 4 = 9.6 m confirms the scale. The hundredths justify the decimal place; the point is not moved by guesswork.

One record is wrong. Choose its row and write the correction.

Decimal records
Row	Task	Written
A	2.14 × 2, exact decimal answer	4.28
B	1.65 × 3, exact decimal answer	0.495

Row to fix

Correct answer

Show the product repairHide answer

B needs 4.95. Three groups of 165 hundredths make 495 hundredths. The written 0.495 names 495 thousandths, a smaller amount.

Keep the total through exchanges

The two rows show the same total in different place-value units.

Three groups of 1.65, before and after exchanging equal units.

Figure data

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Values shown on the cards
Set	Values
Combined parts	3 ones; 18 tenths; 15 hundredths
After exchange	4 ones; 9 tenths; 5 hundredths

A also checks: 214 hundredths × 2 = 428 hundredths, or 4.28.

Divide into equal shares

For 7.68 ÷ 3, count 768 hundredths and share them equally.

Hundredths in each share

7.68 ÷ 3

Show workingHide answer

768 ÷ 3 = 256 hundredths = 2.56. Check 2.56 × 3 = 7.68. Each share is smaller than the original total because it is one of three equal parts.

A 6.48 m ribbon is cut into three equal pieces. Find the length of one piece in metres.

Length in metres

Unit: m

Show answerHide answer

6.48 ÷ 3 = 2.16 m. Check 2.16 × 3 = 6.48 m.

One record is wrong. Choose its row and write the correction.

Decimal records
Row	Task	Written
A	6.75 ÷ 3, exact decimal answer	22.5
B	8.64 ÷ 4, exact decimal answer	2.16

Row to fix

Correct answer

Show the share repairHide answer

A needs 2.25: 675 hundredths shared three ways gives 225 hundredths each.

Join the shares to check

Each row is one share. Together the three rows restore the total.

Three equal shares together make 6.75.

Figure data

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Values shown on the cards
Set	Values
Share 1	2 ones; 2 tenths; 5 hundredths
Share 2	2 ones; 2 tenths; 5 hundredths
Share 3	2 ones; 2 tenths; 5 hundredths

2.25 × 3 = 6.75; the written 22.5 × 3 = 67.5 is too large. B checks: 2.16 × 4 = 8.64.

Do you know one share or the whole amount?

Find one equal group

Four equal groups, one total

All four unknown amounts are equal. Their combined total is 2.80.

Four equal amounts together make 2.80.

Figure data

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Values shown on the cards
Set	Values
Total: 2.80	?; ?; ?; ?

The missing amount is one group, so share the total four ways.

Find the missing amount: square × 4 = 2.80.

Missing amount

Show the equal sharesHide answer

280 ÷ 4 = 70 hundredths, so one group is 0.70. Check: 0.70 × 4 = 2.80. Multiplying the total by four would find four totals, rather than one group.

Find the whole total

Three known shares, one missing total

Each of the three equal shares is 0.36. The missing amount is their total.

Rejoin three equal shares to recover the total.

Figure data

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Values shown on the cards
Set	Values
Total: ?	0.36; 0.36; 0.36

Here the box is the amount before sharing. Decide how to recover it.

Find the missing amount: square ÷ 3 = 0.36.

Missing amount

Show the restored totalHide answer

Three groups of 36 hundredths make 108 hundredths, or 1.08. Check: 1.08 ÷ 3 = 0.36. Dividing 0.36 again would split one share into smaller pieces.

What can you do with the units left over?

Find 7 ÷ 4. Each share first gets 1 whole, leaving 3 wholes. Exchange those three wholes for 30 tenths and continue sharing.

Tenths in each share

Tenths still left

Hundredths in each share after exchanging

7 ÷ 4, written as a decimal

Show workingHide answer

Four groups use 28 tenths, leaving 2 tenths. Those become 20 hundredths, or 5 hundredths per share. Each share has 1 whole, 7 tenths and 5 hundredths: 1.75. Check 1.75 × 4 = 7.

Write the answer as a decimal

Now find 8 ÷ 4 and write the result to one decimal place.

8 ÷ 4, to one decimal place

Show answerHide answer

2.0. The value is two; the zero records the form asked for.

One record is wrong. Choose its row and write the correction.

Decimal records
Row	Task	Written
A	6 ÷ 4, exact decimal answer	1.5
B	9 ÷ 4, exact decimal answer	2.1

Row to fix

Correct answer

Show the remainder repairHide answer

B needs 2.25. Each group gets 2 wholes, leaving 1 whole. Share its 100 hundredths four ways: each group gets 25 hundredths more. The remainder is a whole still to share, rather than a digit to append. Check 2.25 × 4 = 9. A’s 1.5 × 4 = 6 also checks.

Will rounding change the total when you join the shares?

Check a rounded answer

For 5 ÷ 3, sharing into smaller units keeps producing sixes: 1.666…. There is no last six to write in a finite decimal.

Five metres of cord are shared equally among three people. Give an exact division statement and the length to two decimal places.

Length of one share to two decimal places

Total length of three portions of 1.67 m

Unit: m

Can all three 1.67 m portions be cut from exactly 5 m?

NoYes

Why is 1.67 m only an approximate share of 5 m among 3 people?

Three such shares total 5.01 mThree such shares total exactly 5 mAny answer with two decimal places is exact

Show answerHide answer

The exact share is 5 ÷ 3 m, or 1.666… m. To two decimal places it is 1.67 m, so write 5 ÷ 3 ≈ 1.67.

Three rounded portions total 1.67 × 3 = 5.01 m. Rounding helps you describe the length. But you cannot cut three pieces exactly that long.

Find 2.01 ÷ 2 exactly, then give it to two decimal places. Use the unrounded quotient to decide the rounding.

Exact quotient

Quotient to two decimal places

Total of two portions of 1.01

What should you use to round the quotient 2.01 ÷ 2?

The starting amount rounded to 2The already rounded value 1.00The exact quotient 1.005

Show answerHide answer

The exact quotient is 1.005: 1.005 × 2 = 2.01. It is halfway between 1.00 and 1.01, so it rounds to 1.01.

Two rounded shares total 2.02, which differs from the exact total 2.01. Keep the exact quotient for later work unless the question asks you to use the rounded value.

One record is wrong. Choose its row and write the correction.

Decimal records
Row	Task	Written
A	9.79 ÷ 4, rounded to 1 decimal place	2.5
B	7.38 ÷ 3, rounded to 1 decimal place	2.5

Row to fix

Correct answer

Show the final-rounding repairHide answer

A needs 2.4. The exact quotient is 2.4475: its gap to 2.4 is 0.0475, less than the 0.0525 gap to 2.5.

Rounding first to 2.45 and then to 2.5 changes the decision. Round the exact quotient directly to the place asked for. B checks: 7.38 ÷ 3 = 2.46, which rounds to 2.5.

Exchange the leftovers, then round

For 9 ÷ 4, share nine ones among four groups: 2 ones each, with 1 one left. Exchange that one for 10 tenths. Share again, then exchange the remaining tenths for hundredths.

Keep the value when exchanging units

Ones: 9 total; 2 each; 1 left. 1 one → 10 tenths: 10 total; 2 each; 2 left. 2 tenths → 20 hundredths: 20 total; 5 each; 0 left. Read, then round: 2.25 exact; 2.3 1 place.

Share the available units, then exchange what is left for smaller units.

Figure data

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Values shown on the cards
Set	Values
Ones	9 total; 2 each; 1 left
1 one → 10 tenths	10 total; 2 each; 2 left
2 tenths → 20 hundredths	20 total; 5 each; 0 left
Read, then round	2.25 exact; 2.3 1 place

Read the quotient digits by place: 2 ones, 2 tenths, 5 hundredths = 2.25. Nothing is left. The exact quotient rounds to 2.3 to one decimal place.

Calculate 11 ÷ 4 by written division. Record the available units, quotient digit and units left at every place. Exchange each unit left for ten of the next smaller units. Continue through hundredths.

Write the exact quotient as a decimal, then round it to 1 decimal place. Show that number of decimal places in the rounded quotient.

Available ones

Quotient ones digit

Ones left

Available tenths

Quotient tenths digit

Tenths left

Available hundredths

Quotient hundredths digit

Hundredths left

Exact decimal quotient

Rounded quotient

Show the exchanges and decimal quotientHide answer

11 ones: 2 each, 3 left.
30 tenths: 7 each, 2 left.
20 hundredths: 5 each, none left.

The exact quotient is 2.75. Check 2.75 × 4 = 11. To one decimal place, report 2.8. Keep the exact quotient until this final step.

Multiply the parts, then round the total

1.06 = 1 + 0.06. Multiply each place part that is not zero by 4. The empty tenths place contributes nothing.

Exact products before the final rounding

Whole part × 4: 1 part; 4 product. Hundredths × 4: 0.06 part; 0.24 product. Add, then round: 4.24 exact total; 4.2 1 place.

Keep each product exact, add the products, then round their total.

Figure data

This table is included in print while it is open.

Values shown on the cards
Set	Values
Whole part × 4	1 part; 4 product
Hundredths × 4	0.06 part; 0.24 product
Add, then round	4.24 exact total; 4.2 1 place

Add 4 + 0.24 = 4.24. To one decimal place, the final report is 4.2. Keep the parts and their products exact until the total is complete.

Calculate 3.04 × 4 using place-value parts. Split 3.04 into all its nonzero place-value parts and multiply each part by 4.

Write each part and its product in any order. Add the products to get the exact total, then round it to 1 decimal place. Show that number of decimal places in the rounded total.

Part 1

Part 1 product

Part 2

Part 2 product

Exact total

Rounded total

Show both parts and the final reportHide answer

The parts are 3 and 0.04, in either order. Their products with 4 are 12 and 0.16. The exact total is 12.16, which rounds to 12.2 to one decimal place.

A zero tenths place needs no separate part. Keep the four hundredths as 0.04, not 0.4.

Keep original parts for a total

These original parts add to 1.26. Follow the two paths to see what changes when the parts are rounded first.

Two ways to report the same parts

Original parts: 0.62; 0.64. Add, then round: 1.26 exact sum; 1.3 1 place. Round, then add: 0.6+0.6 rounded; 1.2 sum.

Adding the originals first gives a different report from adding their rounded parts.

Figure data

This table is included in print while it is open.

Values shown on the cards
Set	Values
Original parts	0.62; 0.64
Add, then round	1.26 exact sum; 1.3 1 place
Round, then add	0.6+0.6 rounded; 1.2 sum

Both rounded parts are 0.6, so adding them gives 1.2. Adding the originals first gives 1.26, which rounds to 1.3. Keep the original parts when the question asks you to round the final total.

Choose two different positive parts that add to 0.80. Each part must be less than 1.00 and a whole number of hundredths.

Round each part to 1 decimal place, then add the rounded parts. This must differ from rounding the exact total directly to 1 decimal place.

First part

Second part

Show one pair and both checksHide answer

0.25 and 0.55 work. Their exact total is 0.80, which rounds to 0.8. The parts round to 0.3 and 0.6; adding those gives 0.9.

This time early rounding made the report larger. The worked picture made it smaller. Check both paths for each new pair; the error can go either way.

Which starting numbers could give this rounded answer?

Find every original digit

Which digits make 1.□6 ÷ 4 round to 0.4, to one decimal place? Calculate each quotient before rounding it.

Check the edges of the working digits

Digit 3: 1.36 ÷ 4: 0.34 exact; 0.3 1 place. Digit 4: 1.46 ÷ 4: 0.365 exact; 0.4 1 place. Digit 7: 1.76 ÷ 4: 0.44 exact; 0.4 1 place. Digit 8: 1.86 ÷ 4: 0.465 exact; 0.5 1 place.

Check the smallest and largest digits that work. Try the next digit outside each end too.

Figure data

This table is included in print while it is open.

Values shown on the cards
Set	Values
Digit 3: 1.36 ÷ 4	0.34 exact; 0.3 1 place
Digit 4: 1.46 ÷ 4	0.365 exact; 0.4 1 place
Digit 7: 1.76 ÷ 4	0.44 exact; 0.4 1 place
Digit 8: 1.86 ÷ 4	0.465 exact; 0.5 1 place

Values shown on the cards
Set	Values
Digit 3: 1.36 ÷ 4	0.34 exact; 0.3 1 place
Digit 4: 1.46 ÷ 4	0.365 exact; 0.4 1 place
Digit 7: 1.76 ÷ 4	0.44 exact; 0.4 1 place
Digit 8: 1.86 ÷ 4	0.465 exact; 0.5 1 place

Digits 4 and 7 work. The digits between them work too: 1.56 ÷ 4 = 0.39 and 1.66 ÷ 4 = 0.415, both rounding to 0.4. So the complete set is 4, 5, 6, 7.

As the digit rises, the quotient rises. Digit 3 already rounds below 0.4; digit 8 already rounds above it. No digit beyond those edges can work.

Write every digit for the box so that 3.square2 ÷ 5, rounded to 1 decimal place, gives 0.7.

Calculate with the original numbers, then round the answer. Separate the digits with commas.

Every possible digit

Show all the working digitsHide answer

3, 4, 5, 6, 7 all work. Their exact quotients are 0.664, 0.684, 0.704, 0.724 and 0.744. Each rounds directly to 0.7.

The neighbouring digit 2 gives 3.22 ÷ 5 = 0.644, rounding to 0.6. Digit 8 gives 3.82 ÷ 5 = 0.764, rounding to 0.8. The quotient increases with the digit, so those edges exclude every other digit.

Find the smallest and largest starting values

An original has up to two decimal places. Multiply it by 3; the product rounds to 1.4, to one decimal place. Find the smallest and largest possible originals.

Check the two extremes and their neighbours

0.44 × 3: 1.32 exact; 1.3 1 place. 0.45 × 3: 1.35 exact; 1.4 1 place. 0.48 × 3: 1.44 exact; 1.4 1 place. 0.49 × 3: 1.47 exact; 1.5 1 place.

Both end values work. One hundredth further out gives a different rounded answer.

Figure data

This table is included in print while it is open.

Values shown on the cards
Set	Values
0.44 × 3	1.32 exact; 1.3 1 place
0.45 × 3	1.35 exact; 1.4 1 place
0.48 × 3	1.44 exact; 1.4 1 place
0.49 × 3	1.47 exact; 1.5 1 place

Every hundredth from 0.45 to 0.48 works. The picture checks both ends and the originals just outside them. 0.45 × 3 = 1.35 is halfway and rounds up. The required answers are the two extremes, rather than any two working originals.

An original number is divided by 3. The answer, rounded to 2 decimal places, is 0.67.

Find the smallest and largest possible original numbers from 0 to 9.99, using up to 2 decimal places. Calculate with the original number before rounding the answer. At halfway, choose the higher number.

Smallest original number

Largest original number

Show both extremes and their neighboursHide answer

The extremes are 2.00 and 2.02. Their quotients are 0.666… and 0.67333…, both rounding to 0.67.

Just outside, 1.99 ÷ 3 = 0.66333… rounds to 0.66; 2.03 ÷ 3 = 0.67666… rounds to 0.68. The quotient increases with the original, so those neighbours exclude every other hundredth outside the two extremes.

Fill the cups equally

Six bottles each contain 0.45 ℓ of juice. All the juice is shared equally among four cups. Find the amount in one cup, to two decimal places. Choose your calculation before working it out.

Juice per cup to two decimal places

Would four portions of exactly 0.68 ℓ use exactly the available juice?

YesNo

Show workingHide answer

0.45 × 6 = 2.70 ℓ in all the bottles.

2.70 ÷ 4 = 0.675 ℓ exactly in one cup, or 0.68 ℓ to two decimal places. The next digit is five, so the hundredths round up.

Four portions of exactly 0.68 ℓ would total 2.72 ℓ. The rounded answer describes the share; the exact equal share remains 0.675 ℓ.

For a grown-up

Pip’s synthetic voice uses the University of Edinburgh’s Alba speech dataset, via Piper’s Alba voice, under CC BY 4.0. Audio is optional.

Syllabus and review details

Primary 4 Mathematics 2024 · 2021-updated-2025-10
Primary 6 Foundation Mathematics 2026 · 2021-updated-2025-10

Last reviewed: 2026-11-07

Reading settings

Keep equal place values together

Check the remaining amount

Multiply a decimal by a whole number

Check the units in both products

Divide into equal shares

A total split equally

Check each equal share

Find one equal group

How much is in each group?

Find the whole total

What total was shared?

Write the answer as a decimal

Share the remainder too

Check a rounded answer

The same calculation, two kinds of answer

Round the final answer

Check the final rounding

Exchange the leftovers, then round

Share a new whole number into decimal parts

Multiply the parts, then round the total

Record the parts and round a new product

Keep original parts for a total

Build a pair whose rounding paths disagree

Find every original digit

Find the complete set for a new quotient

Find the smallest and largest starting values

Recover the extremes before division

Fill the cups equally

Lesson done

Keep equal place values together

Check the remaining amount

Multiply a decimal by a whole number

Check the units in both products

Divide into equal shares

A total split equally

Check each equal share

Find one equal group

How much is in each group?

Find the whole total

What total was shared?

Keep sharing into smaller units

Write the answer as a decimal

Share the remainder too

Check a rounded answer

The same calculation, two kinds of answer

Round the final answer

Check the final rounding

Exchange the leftovers, then round

Share a new whole number into decimal parts

Multiply the parts, then round the total

Record the parts and round a new product

Keep original parts for a total

Build a pair whose rounding paths disagree

Find every original digit

Find the complete set for a new quotient

Find the smallest and largest starting values

Recover the extremes before division

Fill the cups equally

Questions