Chapter 6 · Decimal operations

Model and Estimate Decimal Sums and Differences

Add and subtract decimals on a hundredths grid by working with parts of the same size. Then round to estimate and check that an answer makes sense.

14 min read

You already know how to add and subtract whole numbers. Now you will do the same with decimals. You will find decimal sums, the answers when you add, and decimal differences, the answers when you subtract.

A picture that shows a number is called a model. Your model in this lesson is a hundredths grid. It is one whole square cut into 100 equal small squares.

  • Each small square is 1 hundredth of the whole. You write it as 0.010.01.
  • Each full line of 10 small squares, across or down, is 1 tenth of the whole. You write it as 0.10.1.

To show a decimal, you shade its parts. The grid below shows 0.360.36. It has 36 shaded squares. That is 3 full lines of ten, which are 3 tenths, and 6 more squares, which are 6 hundredths.

0.360.36 on a hundredths grid: 36 of the 100 small squares are shaded, which is 3 tenths and 6 hundredths.

Key concept

Work with parts of the same size

You can combine or remove parts only when they are the same size. So put hundredths with hundredths and tenths with tenths.

Ten hundredths fill one full line of the grid. That makes them the same amount as 1 tenth, so you can regroup them. To regroup is to trade parts for an equal amount in a different size. The trade works both ways: 10 hundredths for 1 tenth, or 1 tenth for 10 hundredths.

10 hundredths=1 tenth10 \text{ hundredths} = 1 \text{ tenth}

Worked example

Add with a model

Use a hundredths grid to find 0.36+0.270.36 + 0.27.

  1. Name the parts of each number so you can see which parts are the same size. The number 0.36 is 3 tenths and 6 hundredths. The number 0.27 is 2 tenths and 7 hundredths.

    0.36=0.3+0.060.27=0.2+0.070.36 = 0.3 + 0.06 \qquad 0.27 = 0.2 + 0.07

  2. Combine the hundredths, because they are the same size. Together, 6 small squares and 7 small squares make 13 small squares.

    6 hundredths+7 hundredths=13 hundredths6 \text{ hundredths} + 7 \text{ hundredths} = 13 \text{ hundredths}

  3. Regroup, because you now have more than 10 hundredths. Trade 10 of the 13 hundredths for 1 tenth. Those 10 squares fill one more full line, and 3 single squares are left.

    13 hundredths=1 tenth+3 hundredths13 \text{ hundredths} = 1 \text{ tenth} + 3 \text{ hundredths}

  4. Combine the tenths, because they are the same size. Count the 3 tenths, the 2 tenths, and the 1 new tenth from regrouping.

    3 tenths+2 tenths+1 tenth=6 tenths3 \text{ tenths} + 2 \text{ tenths} + 1 \text{ tenth} = 6 \text{ tenths}

  5. Write the total as a decimal. You have 6 tenths and 3 hundredths, which is 63 shaded squares in all.

    0.36+0.27=0.630.36 + 0.27 = 0.63

Answer

The sum of 0.360.36 and 0.270.27 is 0.630.63.

0.36+0.27=0.630.36 + 0.27 = 0.63: 63 of the 100 small squares are shaded, which is 6 full lines of ten and 3 more squares.

Subtracting uses the same grid, but now you take parts away. Start with the first number shaded. Then remove hundredths from the hundredths and tenths from the tenths.

Sometimes there are not enough single hundredths to remove. Then you regroup the other way and break 1 tenth into 10 hundredths. The amount does not change, because 1 full line and 10 single squares cover the same part of the whole.

The next grid shows 0.520.52, the starting number in the example that follows. It has 5 full lines of ten and 2 more squares. That is 5 tenths and 2 hundredths.

0.520.52 before anything is removed: 52 of the 100 small squares are shaded, which is 5 tenths and 2 hundredths.

Worked example

Subtract with a model

Start with the grid that shows 0.520.52. Use it to find 0.52−0.180.52 - 0.18.

  1. Name the parts to remove. The number 0.18 is 1 tenth and 8 hundredths. The grid has 5 tenths but only 2 single hundredths, so you cannot remove 8 hundredths yet.

    0.18=0.1+0.080.18 = 0.1 + 0.08

  2. Regroup to get more hundredths. Break 1 of the 5 tenths into 10 hundredths. Now you have 4 tenths and 12 hundredths, and the same 52 squares are still shaded.

    5 tenths+2 hundredths=4 tenths+12 hundredths5 \text{ tenths} + 2 \text{ hundredths} = 4 \text{ tenths} + 12 \text{ hundredths}

  3. Remove the 8 hundredths from the 12 hundredths, since they are the same size. That takes away 8 single squares.

    12 hundredths−8 hundredths=4 hundredths12 \text{ hundredths} - 8 \text{ hundredths} = 4 \text{ hundredths}

  4. Remove the 1 tenth from the 4 tenths, since they are the same size. That takes away 1 full line.

    4 tenths−1 tenth=3 tenths4 \text{ tenths} - 1 \text{ tenth} = 3 \text{ tenths}

  5. Write what is left as a decimal. You have 3 tenths and 4 hundredths, which is 34 shaded squares.

    0.52−0.18=0.340.52 - 0.18 = 0.34

Answer

The difference between 0.520.52 and 0.180.18 is 0.340.34.

A model gives you the exact answer. Sometimes you only need a number that is close. An estimate is a number close to the exact answer that you can find quickly.

To estimate a decimal sum or difference, follow two steps:

  1. Round each number.
  2. Add or subtract the rounded numbers.

You choose the place to round to. Rounding to the nearest whole number is quickest. Rounding to the nearest tenth takes a little longer, but the estimate is usually closer to the exact answer. When the numbers are less than 1, round to the nearest tenth. Rounding to the nearest whole number would turn each one into 0 or 1.

The table shows one sum and one difference estimated both ways. With rounded tenths, you still work with parts of the same size: ones with ones and tenths with tenths.

One sum and one difference, each estimated two ways
ProblemRound to the nearest whole numberRound to the nearest tenthExact answer
5.78+2.145.78 + 2.14
6+2=86 + 2 = 8
5.8+2.1=7.95.8 + 2.1 = 7.9
7.927.92
8.46−3.128.46 - 3.12
8−3=58 - 3 = 5
8.5−3.1=5.48.5 - 3.1 = 5.4
5.345.34

Worked example

Check answers with estimates

Priya adds 6.2+1.756.2 + 1.75 and gets 7.957.95. Marco subtracts 0.83−0.40.83 - 0.4 and gets 0.790.79. Use estimates to decide whether each answer is reasonable.

  1. Estimate Priya's sum. Her numbers are greater than 1, so round each one to the nearest whole number. Round 6.2 down to 6 and 1.75 up to 2. Then add the rounded numbers.

    6+2=86 + 2 = 8

  2. Compare her answer with the estimate. Her answer, 7.95, is very close to 8, so her sum is reasonable.

  3. Estimate Marco's difference. His numbers are less than 1, so round each one to the nearest tenth. Round 0.83 down to 0.8. Keep 0.4 as it is, because it has no hundredths. Then subtract the rounded numbers.

    0.8−0.4=0.40.8 - 0.4 = 0.4

  4. Compare his answer with the estimate. His answer, 0.79, is almost 8 tenths, but the estimate is only 4 tenths. That is too far apart, so his difference is not reasonable. There must be a mistake.

  5. Find the mistake and fix it. Marco took away 4 hundredths, but 0.4 means 4 tenths. Remove 4 tenths from the 8 tenths in 0.83. That leaves 4 tenths, and the 3 hundredths stay. The new answer is close to the estimate of 0.4.

    0.83−0.4=0.430.83 - 0.4 = 0.43

Answer

Priya's sum of 7.957.95 is reasonable because it is close to her estimate of 8. Marco's difference of 0.790.79 is not reasonable because it is far from his estimate of 0.40.4. The correct difference is 0.430.43.

Here is the big idea to keep. Decimals add and subtract the way whole numbers do when you work with parts of the same size: tenths with tenths and hundredths with hundredths. Ten hundredths and 1 tenth are the same amount, so you can trade one for the other whenever you need to.

An estimate tells you about where the answer should land. Round each number to the nearest whole number or tenth, then add or subtract. An answer close to your estimate is reasonable. An answer far from it needs another look.

Next, you will add and subtract decimals without a grid. You will line up the decimal points so that each column holds parts of the same size.

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Add and Subtract Decimals by Place Value

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