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B2.4 · Add and subtract whole and decimal numbers

Learn to add and subtract whole and decimal numbers through clear examples and targeted practice.

Ontario Grade 6 Mathematics

Number

Ontario Grade 6 Mathematics · B2.4

Addition joins amounts. Subtraction takes an amount away or finds the difference between two amounts. A digit’s place tells its value. In 24.624.6, the 22 means 22 tens, the 44 means 44 ones, and the 66 means 66 tenths. A tenth is one of ten equal parts of one. You will use place value to add and subtract whole numbers and decimals. A number line can also show addition as a move to the right and subtraction as a move to the left.

What you will learn

  • Add whole numbers by matching place values.
  • Add and subtract decimals by lining up decimal points.
  • Regroup when a place has too many or too few units.
  • Check answers with an estimate or the opposite operation.

1. Add whole numbers with place value

Imagine two groups of tiles: 128128 red tiles and 247247 blue tiles. To find how many tiles there are altogether, join the groups. Add ones to ones, tens to tens, and hundreds to hundreds. The answer to an addition question is called the sum.
Start with the ones. Eight ones and seven ones make fifteen ones. Trade ten ones for one ten. Five ones remain. Now add the tens, including the new ten: two tens plus four tens plus one ten make seven tens. Finally, add the hundreds. The total is three hundreds, seven tens, and five ones.
This trade is called regrouping. Regrouping means exchanging units for an equal amount in another place. For example, ten ones have the same value as one ten. The value stays the same; it is just shown in a different way.
A number line can show the total, too. Begin at the first amount and move right by the second amount. The point where you land is the sum. A number line is useful for seeing that addition increases an amount, while place-value columns help keep the calculation organized.
10 ones=1 ten10\text{ ones}=1\text{ ten}
  • Addition joins amounts.
  • Keep matching place values in the same column.
  • Regroup ten units as one unit in the place to the left.

2. Add decimal numbers

A decimal number has a decimal point. Digits to the right of the point show parts of one. The first place is tenths, and the next is hundredths. A hundredth is one of one hundred equal parts of one. For example, 3.753.75 means three ones, seven tenths, and five hundredths.
Line up decimal points before adding. This puts ones with ones, tenths with tenths, and hundredths with hundredths. You may write a zero at the end of a decimal to show an empty place. It does not change the value: 2.62.6 and 2.602.60 are equal.
Add from the smallest place shown, moving left. If there are ten or more units in a place, regroup. Ten hundredths make one tenth, and ten tenths make one one. Keep the decimal point in the answer directly below the decimal points in the numbers you added.
An estimate is a nearby value that helps you judge an answer. For example, 3.753.75 is close to 44, and 2.602.60 is close to 33. Their sum should be close to 77. An estimate is not the exact answer; it is a quick check of the answer’s size.
10 hundredths=1 tenth10\text{ hundredths}=1\text{ tenth}
  • Line up decimal points before adding.
  • A zero at the end of a decimal can show an empty place without changing the value.
  • Regroup ten units in a place as one unit in the place to its left.

3. Subtract whole and decimal numbers

Subtraction can show what remains after some is taken away. It can also compare two amounts. The answer is called the difference. On a number line, subtraction can be shown by starting at the first number and moving left by the amount being taken away.
Line up matching places before subtracting. If a place does not have enough units, regroup one unit from the place to its left. For example, one ten can be regrouped as ten ones. With decimals, one one can be regrouped as ten tenths, and one tenth as ten hundredths.
For decimal subtraction, line up the decimal points. If one number has fewer decimal places, you may add zeros at its end. For example, write 8.28.2 as 8.208.20. The zero shows that there are no hundredths, and the value remains the same.
After subtracting, check by adding the difference to the amount taken away. The sum should equal the starting amount. You can also estimate to see whether the answer is about the right size.
difference+amount taken away=starting amount\text{difference}+\text{amount taken away}=\text{starting amount}
  • Subtraction finds what remains or the difference between amounts.
  • Regroup when there are not enough units in a place to subtract.
  • Check subtraction by adding the difference and the amount taken away.

4. Use a steady routine

First, decide whether the question joins amounts or takes one amount away. Next, line up matching place values. For decimals, line up the decimal points. Then calculate, regrouping when needed.
Finally, check the result. Ask whether it is close to a useful estimate. For subtraction, add the difference and the amount taken away. This check can reveal a place-value error even when the written calculation looks neat.
As you practise, say what each digit represents. This can help you notice if a digit has slipped into the wrong place. A hundredths digit must stay in the hundredths place when you add or subtract.
  • Choose addition or subtraction from what the question asks.
  • Match places, calculate, and check.
  • Use an estimate to judge whether an answer makes sense.

Worked example

Add whole numbers

A class has 128 red tiles and 247 blue tiles. How many tiles are there altogether?
  1. Choose addition
    The question asks for both groups together, so add the two tile counts.
  2. Add matching places
    Add the ones first: eight plus seven makes fifteen ones. Regroup ten ones as one ten, leaving five ones. Add the tens, including the regrouped ten: two tens plus four tens plus one ten make seven tens. The hundreds make three hundreds.
    128+247=375128+247=375
  3. Check with an estimate
    The numbers are close to one hundred thirty and two hundred fifty. Their total should be near three hundred eighty, and three hundred seventy-five is close.
    130+250=380130+250=380
Answer: There are 375 tiles altogether.
Check: The exact total, 375, is close to the estimate of 380.

Worked example

Add decimal numbers

One piece of ribbon is 3.75 metres long. Another piece is 2.6 metres long. What is their combined length?
Adding 3.75 and 2.60
Adding 3.75 and 2.60The points show the starting length and the combined length, within the displayed range from 0 to 7 metres.01234567+2.5999999999999996 metres

The points show the starting length and the combined length, within the displayed range from 0 to 7 metres.

  1. Line up the decimal points
    Write 2.62.6 as 2.602.60. The zero shows there are zero hundredths and does not change the number. Now tenths and hundredths are in matching places.
    3.75+2.603.75+2.60
  2. Add and regroup
    The hundredths add to five. The tenths add to thirteen, so regroup ten tenths as one one and leave three tenths. The ones then add to six. The answer has six ones, three tenths, and five hundredths.
    3.75+2.60=6.353.75+2.60=6.35
  3. Check with an estimate
    The lengths are close to four metres and three metres. A total close to seven metres makes sense.
    4+3=74+3=7
Answer: The combined length is 6.35 metres.
Check: The answer is close to the estimate of 7 metres.

Worked example

Subtract decimal numbers

A container holds 8.2 litres of water. Then 3.47 litres are poured out. How much water remains?
Subtracting 3.47 from 8.20
Subtracting 3.47 from 8.20The remaining amount and starting amount are marked within the displayed range from 0 to 9 litres.0123456789+3.469999999999999 litres

The remaining amount and starting amount are marked within the displayed range from 0 to 9 litres.

  1. Set up the subtraction
    The poured-out amount is taken from the starting amount. Write 8.28.2 as 8.208.20 to line up the decimal points and places.
    8.20−3.478.20-3.47
  2. Regroup and subtract
    There are no hundredths to subtract, so regroup one tenth as ten hundredths. One tenth remains, which is not enough to subtract four tenths. Regroup one one as ten tenths; seven ones remain. Subtract each place to get four ones, seven tenths, and three hundredths.
    8.20−3.47=4.738.20-3.47=4.73
  3. Check by adding
    Add the amount remaining to the amount poured out. The sum should return to the starting amount.
    4.73+3.47=8.204.73+3.47=8.20
Answer: There are 4.73 litres of water remaining.
Check: Adding 4.73 litres and 3.47 litres gives the original 8.20 litres.

Common mistakes and how to avoid them

Lining up the last digits instead of the decimal points.
Correction: Line up the decimal points so ones, tenths, and hundredths stay in matching places.
Thinking that a zero at the end of a decimal changes its value.
Correction: A zero at the end can show an empty place. For example, 2.62.6 and 2.602.60 have the same value.
Subtracting a larger digit from a smaller digit without regrouping.
Correction: Regroup one unit from the place to the left, then subtract in the current place.

Lesson summary

  • Addition joins amounts; subtraction finds what remains or the difference.
  • Line up matching place values. For decimals, line up the decimal points.
  • Regroup when needed, then check with an estimate or the opposite operation.

Check your understanding

Question 1

What is 368+457368+457?
  1. 715
  2. 825
  3. 815
  4. 725
Show answer and explanation
825
Add the ones to make fifteen, then regroup one ten. The tens, including that regrouped ten, make twelve tens, so regroup one hundred. The total is 825.

Question 2

What is 5.6+2.355.6+2.35?
  1. 7.95
  2. 7.41
  3. 2.91
  4. 8.05
Show answer and explanation
7.95
Write 5.65.6 as 5.605.60 so the decimal points line up. Add the matching places to get 7.95.

Question 3

What is 9.00−4.689.00-4.68?
  1. 5.32
  2. 4.42
  3. 4.32
  4. 4.68
Show answer and explanation
4.32
Regroup to subtract the hundredths and tenths. The difference is 4.32. Check by adding 4.324.32 and 4.684.68 to get 9.009.00.

Key terms

Place value
The value a digit has because of its position, such as ones, tenths, or hundreds.
Regroup
Trade units in one place for an equal amount in another place, such as ten ones for one ten.
Sum
The answer to an addition question.
Difference
The answer to a subtraction question.
Estimate
A nearby value used to judge whether an exact calculation makes sense.

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