Calculate with decimals

Keep equal place values together

Before calculating, decide what the quantities mean and which operation finds the amount wanted. Addition and subtraction combine or remove matching place-value units; multiplication counts equal groups and division finds equal shares.

Check your understanding 1 Check the remaining amount

A ribbon is 6.2 m long. A piece of length 1.48 m is cut off. Estimate whether the remainder is between 4 m and 5 m, then find its exact length. Check by adding back the cut piece.

Your estimate, subtraction and check, or work on paper

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Removing about 1.5 m from 6.2 m leaves about 4.7 m, so a result between 4 m and 5 m is reasonable. Write 6.2 as 6.20 to align the hundredths. The extra zero preserves the length: twenty hundredths is the same amount as two tenths.

The hundredths column has zero, so exchange one tenth for ten hundredths. There is then only one tenth, so exchange one whole for ten tenths. You now have five ones, eleven tenths and ten hundredths. Removing one one, four tenths and eight hundredths leaves four ones, seven tenths and two hundredths: 4.72 m. Check 4.72 + 1.48 = 6.20 m.

For more practice with these exchanges, revisit Add and subtract decimals.

Multiply a decimal by a whole number

Four ribbons are each 2.35 m long. Four groups of 235 hundredths give 940 hundredths:

That is 9.4 m. The decimal place is justified by the hundredths being counted, not by an unexplained rule for moving a dot. Zero groups would give zero, so multiplying a decimal by zero also gives zero.

Divide into equal shares

For 7.68÷3, share 768 hundredths equally: each share has 256 hundredths, or 2.56.

For 7÷4, each share first gets one whole, with three wholes left. Exchange those for thirty tenths. Each share gets seven tenths, leaving two tenths. Exchange those for twenty hundredths; each share gets five hundredths. The answer is 1.75. Check 1.75 × 4 = 7.

For 8÷4, the value is 2. If a decimal form is requested, write 2.0. The value and the requested way of writing it are separate.

Check your understanding 2 A total split equally

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

Length in metres

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2.16 m. The pieces check: 2.16 × 3 = 6.48 m.

Decide when an answer must be approximate

For 5÷3, the division keeps producing sixes: 1.666…. To two decimal places it is 1.67. Write 5÷3 ≈ 1.67, because 1.67 × 3 = 5.01, not exactly five.

Round the final answer to the accuracy requested. Rounding an intermediate value too early can change a later total.

Check your understanding 3 The same calculation, two kinds of answer

Five metres of cord are shared equally among three people. Give an exact division statement and the length to two decimal places. Explain what your rounded answer loses.

Your exact statement, rounded length and explanation

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Each exact share is 5÷3 m, or 1.666… m. To two decimal places it is 1.67 m. Three rounded shares total 5.01 m, so the rounded description cannot be used as an exact cutting instruction for all three pieces.

Check your understanding 4 Keep the exact quotient until the last step

Find 2.01÷2 exactly, then round the answer to two decimal places. Explain why rounding each equal share changes the total.

Your exact quotient, rounded answer and check

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The exact quotient is 1.005. Check 1.005 × 2 = 2.01. It is halfway between 1.00 and 1.01, so to two decimal places it rounds to 1.01. Two rounded shares total 2.02, which is not exactly 2.01. Preserve the exact quotient for a later calculation unless the question asks you to use the rounded value.