Grade 5 · 5.NBT.B.7 · 6.NS.B.3

Adding and subtracting decimals

The rain gauge caught 3.4 cm of rain on Monday and 1.25 cm on Tuesday. Before you add them, which digit goes with which?

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When you reach for it

The school weather station has a rain gauge: a clear tube marked in centimeters. Each morning one of the class reads how deep the water stands, to the nearest hundredth of a centimeter, writes it in the log and empties the tube. A week of the log might read 3.4, 1.25, 0, 0.6 and 2.08.

Two questions come up every week. How much rain fell in all? That is a sum. How much more fell on one day than another, or how much more is needed to reach some amount? That is a difference. Both are the whole-number addition and subtraction you already know, with one new demand: the digits have to be matched by place, not by where they happen to end.

You reach for this skill whenever decimal amounts of the same unit are combined or compared: rainfall, lengths, masses, times on a stopwatch, and the prices in money written as decimals. It rests on decimal place value, on comparing decimals and on multi-digit subtraction.

The procedure

The lesson is taught in four parts, one a day.

Part 1: line up the places

In 3.4 the 4 means four tenths. In 1.25 the 2 means two tenths and the 5 means five hundredths. Adding means putting together amounts of the same kind: ones with ones, tenths with tenths, hundredths with hundredths, just as 3 tens and 5 tens make 8 tens. That is why tenths and hundredths written as fractions are first rewritten in one unit before they are added.

The decimal point is the marker that keeps the kinds apart. The ones digit always sits just to its left and the tenths digit just to its right. So if the two numbers are written one above the other with the decimal points in one column, every column holds one kind of unit, and adding down a column adds like to like.

Writing the numbers so that their last digits line up does not do this. It puts

\begin{array}{r} 3.4 \\ +\ 1.25 \\ \hline \end{array}

the 4 tenths above the 5 hundredths and the 3 ones above the 2 tenths, so it adds things of different sizes.

Once the points are lined up, one number may stop before the other. 3.4 has no hundredths digit. You may write a 0 there, because

3.4 = 3\tfrac{4}{10} = 3\tfrac{40}{100} = 3.40

Four tenths is the same amount as forty hundredths. A zero written after the last digit to the right of the point never changes the value; it only says “no hundredths”. Now add each column, starting on the right, and put the point of the answer in the same column:

\begin{array}{r} 3.40 \\ +\ 1.25 \\ \hline 4.65 \end{array}

Five hundredths, six tenths, four ones: 4.65 cm of rain over the two days.

Figure 13.4 + 1.25 with the decimal points in one column. The gray zero is written in so every column has a digit; it changes nothing, because 3.4 has no hundredths.

Subtraction lines up the same way, for the same reason: tenths can only be taken from tenths. A whole number has a decimal point too, just after its ones digit, even when it is not written: 5 = 5.0 = 5.00. Writing it in is how a whole number joins the columns.

Part 2: ten of one place make one of the next

Adding down a column can give ten or more. With whole numbers ten ones become one ten. The same trade works on both sides of the point, because each place is worth ten of the place to its right:

10 \text{ hundredths} = 1 \text{ tenth}, \qquad 10 \text{ tenths} = 1 \text{ one}

So when the hundredths add to 13, write 3 in the hundredths column and carry 1 tenth to the tenths column. When the tenths add to 12, write 2 and carry 1 one across the decimal point to the ones. Nothing about the point makes this trade different: it is the same ten for one that happens between ones and tens.

Figure 20.68 + 0.25 on a hundredths grid. Two of the dotted squares finish the seventh column, so ten hundredths become one more tenth. In all 93 squares are marked: 0.93.

On a hundredths grid each small square is one hundredth and each full column of ten is one tenth. The grid above shows 0.68 + 0.25. The seventh column already had 8 shaded squares; two of the dotted squares finish it, which is the carry made visible.

Part 3: subtracting, and borrowing across the point

To subtract, line up the points, fill empty places with zeros, and subtract each column from the right. When the top digit in a column is too small, take one from the next place to the left and trade it in: one tenth becomes ten hundredths, one one becomes ten tenths. This is the regrouping of regrouping in subtraction, and it crosses the point the same way carrying does.

The zeros matter most here. In 8 - 5.36 the 8 has no tenths or hundredths to take from, so write it as 8.00 and regroup down from the ones, as in subtracting across zeros: 8 ones become 7 ones, 9 tenths and 10 hundredths. Without the zeros there would be no columns to subtract in.

Part 4: estimate first, check after

The column method is exact, but it is easy to put a point in the wrong place. Two checks catch that.

Estimate before you work. Round each number to the nearest whole number and add or subtract those. The exact answer should be close to the estimate; if one is 5 and the other 46.5, a point has slipped. This is estimating sums and differences with decimals in it.

Check a difference by adding. If a - b = d, then d + b = a. Add your answer to the number you took away; you must get back the number you started with.

With several numbers to add, the commutative property of addition lets you change the order and the associative property lets you change the grouping, so you may first pair numbers that make a whole: 0.75 + 0.25 = 1.

Worked examples

Examples 1 and 2 go with Part 1, 3 and 4 with Part 2, 5 and 6 with Part 3, and 7 and 8 with Part 4.

Example 1

Adding tenths and a small reading

Agnes logged 2.3 cm of rain on Monday, 1.6 cm on Tuesday and 0.05 cm on Wednesday. Find the total for Monday and Tuesday, then for all three days.

Solution

Monday and Tuesday: both numbers have tenths, so the points line up with no zeros needed. Tenths: 3 + 6 = 9. Ones: 2 + 1 = 3.

\begin{array}{r} 2.3 \\ +\ 1.6 \\ \hline 3.9 \end{array}

That is 3.9 cm.

All three days: 0.05 is five hundredths, not five tenths, because the 0 just after the point holds the tenths place. Write 3.9 as 3.90 so both numbers reach the hundredths. Hundredths: 0 + 5 = 5. Tenths: 9 + 0 = 9. Ones: 3 + 0 = 3.

\begin{array}{r} 3.90 \\ +\ 0.05 \\ \hline 3.95 \end{array}

The three days made 3.95 cm. The tiny Wednesday reading changed only the hundredths, which is right for an amount smaller than one tenth.

Example 2

Taking away when the lengths differ

Ravi logged 4.75 cm on Thursday and 1.3 cm on Friday. How much more rain fell on Thursday? Then add 2.8 + 0.14.

Solution

How much more is a difference: 4.75 - 1.3. Line up the points and write 1.3 as 1.30. Hundredths: 5 - 0 = 5. Tenths: 7 - 3 = 4. Ones: 4 - 1 = 3.

\begin{array}{r} 4.75 \\ -\ 1.30 \\ \hline 3.45 \end{array}

Thursday had 3.45 cm more. Check by adding: 3.45 + 1.30 = 4.75.

For 2.8 + 0.14, write 2.80. Hundredths: 0 + 4 = 4. Tenths: 8 + 1 = 9. Ones: 2 + 0 = 2. The sum is 2.94.

Example 3

Ten hundredths become a tenth

Zainab’s gauge held 0.68 cm one morning and 0.25 cm the next. Find the total.

Solution

Line up the points. Hundredths: 8 + 5 = 13. Thirteen hundredths is one tenth and three hundredths, so write 3 and carry 1 to the tenths. Tenths: 6 + 2 + 1 = 9. Ones: 0.

\begin{array}{r} 0.68 \\ +\ 0.25 \\ \hline 0.93 \end{array}

The total is 0.93 cm. The hundredths grid shows the same thing: 68 squares and 25 squares fill nine whole columns and three squares more, 93 hundredths.

Example 4

Carrying across the point

Matteo adds two readings, 1.86 cm and 2.47 cm. Then he adds the last day of the month, 0.58 cm, to a month total of 9.6 cm. Find both sums.

Solution

1.86 + 2.47. Hundredths: 6 + 7 = 13, write 3, carry 1 tenth. Tenths: 8 + 4 + 1 = 13, write 3, carry 1 one across the point. Ones: 1 + 2 + 1 = 4.

\begin{array}{r} 1.86 \\ +\ 2.47 \\ \hline 4.33 \end{array}

The two readings make 4.33 cm.

9.6 + 0.58. Write 9.60. Hundredths: 0 + 8 = 8. Tenths: 6 + 5 = 11, write 1, carry 1 one. Ones: 9 + 0 + 1 = 10, write 0, carry 1 ten.

\begin{array}{r} 9.60 \\ +\ 0.58 \\ \hline 10.18 \end{array}

The month’s total is 10.18 cm. The answer has a tens digit that neither number had, exactly as 96 + 6 grows a hundreds digit.

Example 5

Borrowing twice

Kiri logged 5.32 cm in the first week and 2.86 cm in the second. How much more fell in the first week? Then find 6.2 - 3.45, where the top number is the shorter one.

Solution

Find 5.32 - 2.86 with the points lined up.

Hundredths: 2 is less than 6, so take 1 tenth from the 3 tenths, leaving 2 tenths, and trade it for 10 hundredths: 12 - 6 = 6.

Tenths: now 2, less than 8, so take 1 one across the point from the 5 ones, leaving 4, and trade it for 10 tenths: 12 - 8 = 4.

Ones: 4 - 2 = 2.

\begin{array}{r} 5.32 \\ -\ 2.86 \\ \hline 2.46 \end{array}

The first week had 2.46 cm more. Check: 2.46 + 2.86 = 5.32.

6.2 - 3.45: the top number stops at tenths, so write it as 6.20. That zero gives the hundredths column a top digit to borrow into. Hundredths: 0 is less than 5, so take 1 tenth from the 2 tenths, leaving 1, and trade it for 10 hundredths: 10 - 5 = 5. Tenths: 1 is less than 4, so take 1 one from the 6, leaving 5, and trade it for 10 tenths: 11 - 4 = 7. Ones: 5 - 3 = 2.

\begin{array}{r} 6.20 \\ -\ 3.45 \\ \hline 2.75 \end{array}

The difference is 2.75. Check: 2.75 + 3.45 = 6.20.

Example 6

Subtracting from a whole number

The class has set itself a target of 8 cm of rain logged this month, and so far the log adds to 5.36 cm. How much more rain is needed? Then find 4.03 - 1.58.

Solution

How much more is 8 - 5.36. The whole number 8 has its point after the 8; write it as 8.00 so it has tenths and hundredths columns to subtract from.

Hundredths: 0 is less than 6, and the tenths place holds 0 too, so go to the ones. Take 1 one from the 8, leaving 7; it is 10 tenths. Take 1 of those tenths, leaving 9; it is 10 hundredths. The top number now reads 7 ones, 9 tenths, 10 hundredths, still 8. Hundredths: 10 - 6 = 4. Tenths: 9 - 3 = 6. Ones: 7 - 5 = 2.

\begin{array}{r} 8.00 \\ -\ 5.36 \\ \hline 2.64 \end{array}

The class needs 2.64 cm more. Check: 2.64 + 5.36 = 8.00.

4.03 - 1.58: hundredths 3 is less than 8, and there are 0 tenths, so regroup from the ones: 4 ones become 3 ones, 9 tenths and 13 hundredths. Hundredths: 13 - 8 = 5. Tenths: 9 - 5 = 4. Ones: 3 - 1 = 2. The difference is 2.45, and 2.45 + 1.58 = 4.03.

Example 7

A week in the log

Selin’s week reads 0.75, 1.6, 0, 2.08 and 0.25 cm. Estimate the total, then find it.

Solution

Estimate: to the nearest whole number the readings are 1, 2, 0, 2 and 0, so the total is about 5 cm.

Exact: the 0 adds nothing. The commutative and associative properties let us add in any order, so pair the two readings that make a whole: 0.75 + 0.25 = 1.00. Then

1 + 1.6 = 2.6, \qquad 2.6 + 2.08 = 4.68

where the last sum is 2.60 + 2.08: hundredths 8, tenths 6, ones 4.

The week’s total is 4.68 cm, and 4.68 \approx 5, so the answer is in the right place. Adding all four non-zero readings in one column gives the same: hundredths 5 + 0 + 8 + 5 = 18, write 8 carry 1; tenths 7 + 6 + 0 + 2 + 1 = 16, write 6 carry 1; ones 1 + 2 + 1 = 4.

Example 8

Checking someone else’s difference

At noon Matteo read 2.6 cm in the gauge, and by evening it read 4.13 cm. He wrote that 3.87 cm fell in the afternoon. Is he right?

Solution

Estimate: 4.13 - 2.6 \approx 4 - 3 = 1. An answer near 4 cannot be right.

Exact: write 2.6 as 2.60 under 4.13. Hundredths: 3 - 0 = 3. Tenths: 1 is less than 6, so take 1 one, leaving 3 ones, and make 11 tenths: 11 - 6 = 5. Ones: 3 - 2 = 1.

\begin{array}{r} 4.13 \\ -\ 2.60 \\ \hline 1.53 \end{array}

The afternoon brought 1.53 cm. Check: 1.53 + 2.60 = 4.13. Matteo’s 3.87 is 4.13 - 0.26: he lined up the last digits, so the 2 ones of 2.6 were taken from the tenths.

Boundaries

The units must match. A sum of decimals means something only when the numbers count the same unit. A reading of 4 mm is 0.4 cm, and it has to be changed to centimeters, using metric units of length, before it is added to 1.25 cm.

Zeros go only at the right-hand end. A zero after the last digit to the right of the point leaves the value alone: 0.6 = 0.60. A zero slipped in between the point and the digits after it moves those digits to new places and changes the number: 0.6 and 0.06 differ by a factor of ten.

In this lesson the larger number comes first. Every difference here takes a smaller number from a larger one, so the answer is zero or more. Taking 4.75 from 1.3 asks for a number below zero, which waits for adding and subtracting rational numbers.

More places work the same way. The method stops at hundredths only because the readings do. Thousandths line up and regroup by the same ten for one, one place further right.

An estimate catches big slips, not small ones. Rounding to whole numbers will flag a point in the wrong place, but not a hundredth gone astray. Adding back the number taken away catches both.

Where this goes next

Most problems that need decimal sums arrive in words, and decimal word problems is where you decide which operation a story asks for. Money follows these rules exactly, with two places after the point for cents: money written as decimals adds a price of 3.75 dollars and one of 2.50 dollars this way. Adding the same decimal again and again leads to multiplying a decimal by a whole number, and on to multiplying decimals and dividing decimals in columns. Once numbers below zero join in, the same lining up runs through adding and subtracting rational numbers.

Exercises

Exercises 1 to 6 go with Part 1, 7 to 12 with Part 2, 13 to 18 with Part 3 and 19 to 23 with Part 4. The reasoning group draws on all four.

Part 1: line up the places

Add or subtract.

3.2 + 4.5
0.6 + 2.13
5.47 + 1.2
7.9 - 3.4
6.58 - 2.3
Kiri logged 1.4 cm of rain one day and 0.05 cm the next. What was the two-day total?

Part 2: carrying

Add.

0.47 + 0.38
2.75 + 1.86
3.9 + 0.46
8.65 + 1.35
A hundredths grid has 0.56 shaded. Zainab shades 0.3 more. How many small squares did she shade, and what decimal is now shaded?
Selin’s log shows 4.68 cm of rain one week and 3.57 cm the next. Find the total.

Part 3: subtracting

Subtract, and check each answer by adding.

5.62 - 2.48
7.3 - 4.65
9 - 3.72
6.04 - 2.39
10 - 0.85
The gauge holds 12 cm of water before it overflows. It now holds 7.46 cm. How much more rain can it take?

Part 4: estimate, check, and longer sums

Estimate 2.85 + 0.9 + 3.12 by rounding each number to the nearest whole number, then find the exact sum.
Find 1.25 + 3.4 + 0.75 + 0.6 by choosing a good order.
Kiri’s log for one week reads 0.8, 1.36, 0, 2.5 and 0.04 cm. Find the total, and say how much more than 4 cm it is.
Ravi says 6.1 - 2.75 = 4.45. Estimate to test his answer, then find the difference and check it by adding.
Agnes’s log adds to 2.68 cm so far this month. How much more rain would bring it to 5 cm?

Reasoning

Explain why 3.4 and 3.40 are the same number, but 3.4 and 3.04 are not.
Ravi adds 4 + 0.35 by writing the 4 under the 5, and gets 0.39. Use place value to explain what went wrong, and find the sum.

Answers

7.7. Tenths 2 + 5 = 7, ones 3 + 4 = 7.
2.73. Write 0.60; hundredths 3, tenths 6 + 1 = 7, ones 2.
6.67. Write 1.20; hundredths 7, tenths 4 + 2 = 6, ones 5 + 1 = 6.
4.5. Tenths 9 - 4 = 5, ones 7 - 3 = 4.
4.28. Write 2.30; hundredths 8 - 0 = 8, tenths 5 - 3 = 2, ones 6 - 2 = 4.
1.45 cm. Write 1.40 + 0.05; the 5 is hundredths.
0.85. Hundredths 7 + 8 = 15, write 5 carry 1; tenths 4 + 3 + 1 = 8.
4.61. Hundredths 5 + 6 = 11, carry 1; tenths 7 + 8 + 1 = 16, carry 1; ones 2 + 1 + 1 = 4.
4.36. Write 3.90; hundredths 6; tenths 9 + 4 = 13, carry 1; ones 3 + 1 = 4.
10.00, which is 10. Every column makes ten: hundredths 5 + 5 = 10, tenths 6 + 3 + 1 = 10, ones 8 + 1 + 1 = 10.
30 squares, since 0.3 is 3 tenths, 30 hundredths. Now 0.86 is shaded: 56 + 30 = 86 hundredths.
8.25 cm. Hundredths 8 + 7 = 15, carry 1; tenths 6 + 5 + 1 = 12, carry 1; ones 4 + 3 + 1 = 8.
3.14. Regroup a tenth: 12 - 8 = 4, then tenths 5 - 4 = 1, ones 5 - 2 = 3. Check: 3.14 + 2.48 = 5.62.
2.65. Write 7.30; regroup a tenth, 10 - 5 = 5; regroup a one, 12 - 6 = 6; ones 6 - 4 = 2. Check: 2.65 + 4.65 = 7.30.
5.28. Write 9.00, which is 8 ones, 9 tenths, 10 hundredths. Check: 5.28 + 3.72 = 9.00.
3.65. With 0 tenths, regroup from the ones: 5 ones, 9 tenths, 14 hundredths. Check: 3.65 + 2.39 = 6.04.
9.15. Write 10.00, which is 9 ones, 9 tenths, 10 hundredths. Check: 9.15 + 0.85 = 10.00.
4.54 cm. 12.00 - 7.46 = 4.54. Check: 4.54 + 7.46 = 12.00.
Estimate 3 + 1 + 3 = 7. Exact 6.87: 2.85 + 0.90 = 3.75, then 3.75 + 3.12 = 6.87, close to 7.
6. Pair 1.25 + 0.75 = 2 and 3.4 + 0.6 = 4, then 2 + 4 = 6.
4.7 cm, which is 0.7 cm more than 4 cm. 0.80 + 1.36 = 2.16, 2.16 + 2.50 = 4.66, 4.66 + 0.04 = 4.70.
He is wrong: 6 - 3 = 3, far from 4.45, and 4.45 + 2.75 = 7.2, not 6.1. The difference is 3.35, since 6.10 - 2.75 = 3.35 and 3.35 + 2.75 = 6.10.
2.32 cm. 5.00 - 2.68 = 2.32. Check: 2.32 + 2.68 = 5.00.
3.40 is 3 ones, 4 tenths and 0 hundredths, and 40 hundredths is the same as 4 tenths, so the extra zero changes nothing. In 3.04 the zero sits in the tenths place and pushes the 4 into the hundredths place, so 3.04 is 3 ones and 4 hundredths, smaller than 3.4 by 0.36.
His 4 landed in the hundredths column, so he added 4 hundredths instead of 4 ones: 0.35 + 0.04 = 0.39. The point of 4 is just after the 4, so write 4.00 over 0.35 with the points lined up. The sum is 4.35.