Adding and Subtracting Decimals

One instruction carries this whole topic, and it is not the one children apply by default: line up the decimal points, not the ends of the numbers.

Decimals and percentages · topic 3 of 7Grades 4 to 6Number7 min read7 practice questions

The one instruction

Line up the decimal points, one directly under the other.

That really is the whole method. Once the points are aligned every column holds digits of the same value, tenths above tenths and hundredths above hundredths, and from there it is the column arithmetic your child has been doing since Grade 2.

112.403.75+16.15
The points sit in one straight line. Tenths land under tenths, hundredths under hundredths, and the point in the answer goes in the same column as the ones above it.

The rule

Write the numbers with their decimal points in a vertical line. Fill any empty places with zeros. Add or subtract column by column, starting from the right.

Put the decimal point in the answer straight below the others, before you write a single digit.

Filling the gaps

Numbers rarely turn up with matching decimal places. 12.4 and 3.75 do not match, so the first job is to make them.

Write 12.4 as 12.40. Nothing has been added to it, since a zero on the right-hand end never changes a decimal, and now every column has a digit sitting in it waiting to be used.

It matters more for subtraction than for addition. Trying to work out 6 − 2.35 with two empty columns under the 6 usually ends in guesswork, because there is nothing there to exchange from. Written as 6.00 the whole thing becomes an ordinary subtraction.

Estimate first

Ten seconds spent estimating catches nearly every wrong answer on this topic, and it is particularly good at catching the ones where the decimal point has ended up in the wrong place.

  • 12.4 + 3.75 is roughly 12 + 4 = 16. An answer of 1.615 or 161.5 is instantly wrong.
  • 6 − 2.35 is roughly 6 − 2 = 4. An answer of 3.65 passes. An answer of 4.35 does not.

Treat the estimate as step one of the method rather than as something to do if the clock allows, because the version that gets skipped is always the one at the end. A child who has already decided roughly what the answer should look like almost never hands in one that is wildly off.

Worked examples

1

Different lengths

Work out 12.4 + 3.75
  1. Estimate: about 12 + 4 = 16.
  2. Write 12.4 as 12.40 so both have two decimal places.
  3. Line up the points and add from the right: 0 + 5 = 5, then 4 + 7 = 11, so write 1 and carry 1.
  4. Carry on into the whole numbers: 2 + 3 + 1 = 6, then 1. The answer is 16.15, close to the estimate.
12.4 + 3.75 = 16.15
2

Subtracting from a whole number

Work out 6 − 2.35
  1. Estimate: about 6 − 2 = 4, so expect something a bit under 4.
  2. Write 6 as 6.00. Every whole number has a decimal point, it is just usually invisible.
  3. Line up the points and subtract from the right, exchanging where needed.
  4. The result is 3.65, which sits just under 4 as expected.
6 − 2.35 = 3.65
3

A money problem

A book costs 8.50 dollars and a pen costs 1.75 dollars. You pay with a 20 dollar note. How much change?
  1. Two steps here, so do the total first. 8.50 + 1.75 = 10.25.
  2. Estimate the change: about 20 − 10 = 10 dollars.
  3. Write 20 as 20.00 and subtract 10.25.
  4. 20.00 − 10.25 = 9.75, which sits just under the estimate.
The change is 9.75 dollars

Common mistakes

1. Lining up the right-hand ends instead of the points

12.4 gets written above 3.75 with the 4 sitting over the 5, which puts tenths above hundredths and guarantees a nonsense answer. The habit comes from whole numbers, where lining up the right-hand ends and lining up the columns are the same action, so nobody ever had to distinguish between them. As soon as a decimal point appears, only the point can be trusted.

2. Losing the decimal point in the answer

Every digit comes out right and the answer is written as 1615 instead of 16.15. The fix is to put the point into the answer line before doing any arithmetic at all, directly beneath the ones above it, and then fill the digits in around it. Something already on the page cannot be forgotten in the rush at the end.

3. Leaving a column empty

2.35 gets subtracted from a bare 6, with two empty columns underneath and nothing to exchange from. Fill every gap with a zero before starting. An empty column invites a guess and a zero does not.

Practice questions

Seven questions. One attempt each, and the explanation appears once the answer is in.

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For parents

Give them squared paper and insist on one digit per square. The columns then line up by themselves, without anybody having to nag about neatness, and a surprising share of the errors on this topic simply stop happening. It is a cheaper fix than any amount of explaining, and it works on the children who understand the method perfectly and still get the answer wrong.

Keep going

Maths CheckTwelve questions, five minutes, then a printable report naming what to work on next. Free, no sign up.OpenShow me the workingLong division, long multiplication and column methods, walked step by step on your own numbers.Open

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