Wash-and-fold costs dollars a pound and the bag weighs pounds. When both numbers have a decimal point, where does the point go in the product, and why there?

Grade 5 · 5.NBT.B.7

Where the point goes in a product

Wash-and-fold costs 1.3 dollars a pound and the bag weighs 2.5 pounds. When both numbers have a decimal point, where does the point go in the product, and why there?

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Multiplying decimalsGrade 5 Converting between fractions and decimalsGrade 5

Where you are 8 in Decimal arithmetic

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Arithmetic5

Decimal arithmetic8

▴ 2 above Dividing decimals Multiplying multi-digit decimals with the standard algorithm

Where the point goes in a product

Decimal problems in words Dividing by a decimal ▾ Show all 8 (1 more)

Why this exists

Wale runs a laundromat. Customers drop off a bag of laundry, and Wale washes, dries and folds it for 1.3 dollars a pound. Gauri’s bag weighs 2.5 pounds. What does she pay?

The cost is the price of one pound times the number of pounds, 1.3 \times 2.5. If the bag weighed a whole number of pounds, Multiplying decimals would finish the job: 3 pounds would cost 3 \times 1.3 = 3.9 dollars. But a scale does not stop at whole pounds. Now both factors have a decimal point.

Multiplying the digits is not the hard part: 13 \times 25 = 325. The hard part is where the point goes. Is Gauri’s bill 32.5 dollars, 3.25 dollars, or 0.325 dollars? Only one of those is right, and this lesson shows how to know which, using pictures of tenths and hundredths first and then a pattern that works every time.

The lesson takes three days. Day 1 multiplies tenths by tenths on a hundredths grid. Day 2 uses area models for bigger factors and finds the pattern in the number of decimal places. Day 3 places the point by an estimate and by the count, handles zeros in front, and solves laundromat problems.

Definition

The digits after a decimal point are its decimal places. 2.5 has one decimal place (tenths), 0.45 has two (hundredths), and 0.056 has three (thousandths). A whole number such as 6 has zero decimal places.

Where the point goes. Multiply two decimals as if the points were not there. Then the product has as many decimal places as the two factors have together, counted before any zero at the end of the product is dropped.

For 1.3 \times 2.5, each factor has one decimal place, so the product has 1 + 1 = 2. The digits 325 become 3.25, and Gauri pays 3.25 dollars.

Intuition

Day 1: a tenth of a tenth is a hundredth

Start with a square whose side is 1. Cut it into 10 equal columns. Each column is one tenth of the square, 0.1. Now cut one column into 10 equal pieces. Each piece is a tenth of a tenth.

Figure 1One column is a tenth of the square. Cut that column into ten and one piece is a tenth of a tenth. It takes 100 such pieces to fill the square, so 0.1 \times 0.1 = 0.01.

How big is that piece? Cut every column the same way and the square becomes a 10 by 10 grid of 100 small squares. One small square is one hundredth of the whole, 0.01 (one hundredth is \displaystyle \frac{1}{100}). So

0.1 \times 0.1 = 0.01

A tenth of a tenth is a hundredth. This is the one new fact behind the whole lesson.

Multiplication can be read as the area of a rectangle: width times height. On the grid, a rectangle 0.7 wide and 0.3 tall covers 7 columns of tenths and 3 rows of tenths. Count the small squares inside it: 7 \times 3 = 21. Each small square is a hundredth, so the rectangle is 21 hundredths.

Figure 20.7 \times 0.3 is the shaded rectangle: 7 columns by 3 rows of hundredths, 21 hundredths in all, so 0.7 \times 0.3 = 0.21.

0.7 \times 0.3 = 21 \text{ hundredths} = 0.21

The digits came from 7 \times 3 = 21, the number of small squares. The place came from the size of each square, a hundredth. Turn the picture on its side and the rectangle is 0.3 wide and 0.7 tall, with the same 21 squares inside, so 0.3 \times 0.7 = 0.21 too. That is the commutative property at work.

Notice that 0.21 is less than both factors. Taking 0.3 of something takes only part of it, so 0.3 of 0.7 must be less than 0.7.

Day 2: area models and the pattern

A factor bigger than 1 needs more than one square, so a grid becomes clumsy. An area model keeps the idea and drops the small squares. Split each factor by place value, draw a rectangle with one part for every pair of pieces, and multiply each pair. The The distributive property says the parts add up to the whole product.

For Gauri’s bill, split 2.5 into 2 + 0.5 and 1.3 into 1 + 0.3.

Figure 3An area model for 1.3 \times 2.5 (not to scale). The four parts add to 2 + 0.5 + 0.6 + 0.15 = 3.25. The shaded part, tenths times tenths, is the only one with hundredths.

The four parts are 1 \times 2 = 2, 1 \times 0.5 = 0.5, 0.3 \times 2 = 0.6, and 0.3 \times 0.5 = 0.15 (three tenths of five tenths is 15 hundredths, from Day 1). Add them:

2 + 0.5 + 0.6 + 0.15 = 3.25

Now look at the digits. 13 \times 25 = 325, and the answer is 3.25. The area model found the same digits as the whole-number product, with the point two places in from the right. Why two? Because 1.3 is 13 tenths and 2.5 is 25 tenths, and tenths times tenths are hundredths:

13 \text{ tenths} \times 25 \text{ tenths} = 325 \text{ hundredths} = 3.25

The same thing happens with every pair of units. Each row of the table below can be seen on a grid: a tenth of a hundredth means cutting one small square into ten slivers, and 1000 slivers fill the whole square.

Example	Unit of the product	Places
3 \times 0.4 = 1.2	tenths	0 + 1 = 1
0.7 \times 0.3 = 0.21	hundredths	1 + 1 = 2
3 \times 0.04 = 0.12	hundredths	0 + 2 = 2
0.4 \times 0.03 = 0.012	thousandths	1 + 2 = 3

Read down the “Places” column. The places of the factors add. That is the pattern this lesson is named for, and it holds however many digits each factor has.

Day 3: placing the point two ways

There are two honest ways to place the point once the digits are known.

By an estimate. Round the factors to easy numbers and ask how big the product must be. Gauri’s bill is about 1 \times 3 = 3 dollars, since 1.3 is near 1 and 2.5 is near 3. Of 32.5, 3.25 and 0.325, only 3.25 is near 3. Friendly numbers and deliberate over and under estimates picks rounded numbers that are easy to work with.

By the count. Add the decimal places of the factors and count that many digits in from the right of the whole-number product.

Each way checks the other. The count is exact. The estimate is rough, but it still catches a slip in the count: moving the point one place makes the answer 10 times too big or too small, and a rounded guess is never that far off.

Zeros in front. Sometimes the whole-number product has fewer digits than the count asks for. Then zeros go in front to hold the empty places. Example 7 shows one: 8 hundredths times 7 tenths is 56 thousandths, and 56 thousandths is written 0.056, with a zero in the tenths place.

Zeros at the end. If the whole-number product ends in zeros, count the places first and drop the end zeros afterward. 0.6 \times 0.5 is 30 hundredths, 0.30, which is the same number as 0.3 (Example 2).

Results

Units multiply. A tenth of a tenth is a hundredth, and a tenth of a hundredth is a thousandth:

0.1 \times 0.1 = 0.01 \qquad 0.1 \times 0.01 = 0.001

Decimal places add. For decimals that stop, the product of a number with a decimal places and a number with b decimal places has a + b decimal places, counted before any end zero is dropped. Its digits are the product of the factors written without points.

Why it is true

Every decimal that stops is a whole number of its smallest unit. 1.3 is 13 tenths, that is 13 \times 0.1, and 0.45 is 45 hundredths, 45 \times 0.01.

Take 0.45 \times 0.7. Write each factor as a count times a unit:

\begin{aligned} 0.45 \times 0.7 &= (45 \times 0.01) \times (7 \times 0.1) \\ &= (45 \times 7) \times (0.01 \times 0.1) \end{aligned}

The second line only reorders and regroups the four numbers being multiplied, which the commutative and associative properties of multiplication allow. Now there are two separate jobs. The counts multiply as whole numbers, 45 \times 7 = 315. The units multiply as in the Results: 0.01 \times 0.1 = 0.001, a thousandth. So the product is 315 thousandths, 0.315.

The unit decides the places. A tenth has 1 decimal place, a hundredth 2, a thousandth 3, and multiplying units adds those numbers, because cutting a piece into ten adds one more place each time: a hundredth cut into ten is a thousandth. That is why the decimal places of the factors add.

Worked examples

Examples 1 and 2 go with Day 1, Examples 3 to 5 with Day 2, and Examples 6 and 7 with Day 3.

Example 1

Tenths times tenths on a grid

Isamu folds towels at the laundromat. A hand towel is 0.7 meter long and 0.3 meter wide. What is its area?

Solution

The area of a rectangle is length times width, 0.7 \times 0.3 square meter. Picture a square meter as a hundredths grid. The towel covers 7 columns of tenths and 3 rows of tenths, which is 7 \times 3 = 21 small squares, as in Figure 2. Each small square is a tenth of a tenth, a hundredth of a square meter. So the area is 21 hundredths:

0.7 \times 0.3 = 0.21

The towel’s area is 0.21 square meter. That fits: the towel is much smaller than a square meter, and 0.21 is much less than 1.

Example 2

A product that ends in zero

Wale’s bulletin board is 0.6 meter wide and 0.5 meter tall. What is its area?

Solution

The area is 0.6 \times 0.5 square meter. On a hundredths grid, 6 columns by 5 rows is 6 \times 5 = 30 small squares, so the area is 30 hundredths, 0.30 square meter.

Thirty hundredths is the same as three tenths: every ten small squares in a column make one tenth, so 30 squares fill 3 whole columns. The zero at the end adds nothing, and 0.30 = 0.3.

There is a second way to see it. 0.5 is one half, and half of 0.6 is 0.3. The board is 0.3 square meter.

Example 3

Ones and tenths times ones and tenths

Gauri’s bag of laundry weighs 2.5 pounds, and wash-and-fold costs 1.3 dollars a pound. What does she pay?

Solution

The cost is 1.3 \times 2.5 dollars. Split both factors by place value, 1.3 = 1 + 0.3 and 2.5 = 2 + 0.5, and use the area model of Figure 3. The The distributive property multiplies every piece of one factor by every piece of the other:

	2	0.5
1	2	0.5
0.3	0.6	0.15

The only part that is new since Multiplying decimals is 0.3 \times 0.5: 3 tenths of 5 tenths is 15 hundredths, 0.15. Adding the four parts gives

2 + 0.5 + 0.6 + 0.15 = 3.25

Gauri pays 3.25 dollars. Check with the count: 13 \times 25 = 325, and the factors have 1 + 1 = 2 decimal places, so 325 becomes 3.25. The two methods agree.

Example 4

Hundredths times tenths

A scoop of laundry powder weighs 0.45 pound. Isamu uses 0.7 of a scoop for a small load. How much powder is that?

Solution

Seven tenths of 0.45 pound is 0.7 \times 0.45, which is the same as 0.45 \times 0.7. Split 0.45 into 0.4 + 0.05:

	0.4	0.05
0.7	0.28	0.035

The first part is tenths times tenths: 7 \times 4 = 28 hundredths, 0.28. The second is tenths times hundredths, and a tenth of a hundredth is a thousandth, so it is 7 \times 5 = 35 thousandths, 0.035 (Thousandths: reading, writing and expanded form). Add them:

0.28 + 0.035 = 0.315

Isamu uses 0.315 pound. The count agrees: 45 \times 7 = 315, and 2 + 1 = 3 decimal places make 0.315. The product has more decimal places than either factor because its unit, a thousandth, is smaller than either factor’s unit. It is less than 0.45, as it should be, since Isamu uses less than a full scoop.

Example 5

Four parts, and a zero in the middle

The rubber mat by the dryers is 2.4 meters long and 1.7 meters wide. What is its area?

Solution

The area is 2.4 \times 1.7 square meters. Split 2.4 = 2 + 0.4 and 1.7 = 1 + 0.7:

	2	0.4
1	2	0.4
0.7	1.4	0.28

Add the parts, ones first, then tenths, then hundredths:

\begin{aligned} 2 + 0.4 + 1.4 + 0.28 &= 2 + 1.8 + 0.28 \\ &= 3.8 + 0.28 = 4.08 \end{aligned}

The mat is 4.08 square meters. The tenths are 4 from 0.4, 4 from 1.4 and 2 from 0.28, which make 10 tenths: exactly one more whole and no tenths left over, so the tenths place holds a 0. The count gives the same answer: 24 \times 17 = 408, with 1 + 1 = 2 places, is 4.08. Writing 4.8 would lose the zero that holds the tenths place and claim 8 tenths that are not there.

Example 6

Place the point by an estimate

Wale weighs Isamu’s laundry at 6.4 pounds. A rush order costs 1.25 dollars a pound. Isamu works out 125 \times 64 = 8000. What is the bill?

Solution

The digits are known; the question is where the point goes. Estimate first. The price 1.25 is between 1 and 2 dollars a pound, so the bill is between 1 \times 6.4 = 6.4 dollars and 2 \times 6.4 = 12.8 dollars. Of the numbers 0.8, 8 and 80, only 8 lies in that range.

Now count. The factors have 2 + 1 = 3 decimal places, so count three digits in from the right of 8000: 8.000. Only now drop the end zeros, which are zero tenths, hundredths and thousandths: 8.000 = 8. Both ways give 8 dollars.

A third check uses place value. 1.25 = 1 + 0.25, and 0.25 is a quarter, so the bill is 6.4 plus a quarter of 6.4: 6.4 + 1.6 = 8.

The order matters here. The three zeros of 8000 are what the count lands on. Take them away first and only the digit 8 is left to count into, which would put the point at 0.008, a bill of less than one cent. The estimate of 6.4 to 12.8 dollars shows that cannot be right.

Example 7

Zeros in front

Gauri treats a stain with 0.7 ounce of stain remover. Stain remover costs 0.08 dollars an ounce. What does the treatment cost, and what is that to the nearest cent?

Solution

The cost is 0.08 \times 0.7 dollars. The units are hundredths times tenths, which make thousandths, and the counts are 8 \times 7 = 56. So the cost is 56 thousandths of a dollar.

The count says the same: 2 + 1 = 3 decimal places. But 56 has only two digits. A zero goes in front to fill the third place, 056, and the point goes in front of all three:

0.08 \times 0.7 = 0.056

The zero in the tenths place says there are no tenths. An estimate agrees: 0.7 is less than 1, so the cost is less than 0.08 dollars, and 0.056 is less than 0.08 while 0.56 is not.

Money is paid in whole cents, which are hundredths of a dollar. The digit after the hundredths is 6, so 0.056 rounds up to 0.06 (Rounding decimals). The treatment costs about 0.06 dollars, six cents.

Boundaries

The count needs decimals that stop. A decimal that repeats forever, such as 0.333\ldots, has no last digit, so it has no number of decimal places to count.

The count gives the exact product. A price or a measurement often needs fewer places, as in Example 7, and rounding is a separate step taken after the exact product is known.

End zeros are dropped only after the point is placed (Example 6). Before that they are digits of the product.

The product is less than a factor only when the other factor is less than 1. Multiplying by 0.7 shrinks a number; multiplying by 1.3 makes it bigger. Every number in this lesson is positive.

Ignoring the points is for multiplying only. Adding and subtracting need tenths lined up with tenths, because those operations combine places.

Where this goes next

The same count works for factors with more digits, such as a price with cents times a weight in hundredths of a pound. Multiplying multi-digit decimals with the standard algorithm multiplies those with the column method and places the point by this count. Multiplying a decimal by ten, one hundred or one thousand is the special case where one factor is 10, 100 or 1000, or 0.1 or 0.01. Division runs the idea backward: Dividing decimals shares a decimal into equal parts, and Decimal problems in words mixes all four operations in stories about money and measures.

Exercises

Exercises 1 to 8 go with Day 1, 9 to 16 with Day 2, and 17 to 22 with Day 3. The reasoning group draws on all three days.

Day 1: tenths times tenths

Use a hundredths grid to find 0.4 \times 0.6.
Find 0.8 \times 0.7.
Find 0.2 \times 0.9.
Find 0.5 \times 0.8, and write the answer with as few digits as possible.
1.2 is 12 tenths. Use that to find 1.2 \times 0.4.
On a hundredths grid, a rectangle 6 columns wide and 9 rows tall is shaded. Which multiplication does it show, and what is the product?
Isamu folds a pillowcase into a rectangle 0.9 meter by 0.4 meter. What is its area?
A square sign by Wale’s door is 0.5 meter on each side. What is its area?

Day 2: area models and the pattern

Use an area model to find 1.4 \times 2.3.
Use an area model to find 3.2 \times 1.6.
Split 0.36 into 0.3 + 0.06 to find 0.36 \times 0.4.
Find 4.12 \times 0.2.
Write each product as a decimal and name its unit: (a) 0.1 \times 0.1, (b) 0.01 \times 0.1, (c) 10 \times 0.01.
Given that 27 \times 14 = 378, find (a) 2.7 \times 1.4, (b) 0.27 \times 1.4 and (c) 27 \times 0.14.
The folding table is 1.8 meters long and 0.9 meter wide. What is its area?
Wash-and-fold costs 1.4 dollars a pound. What does a 3.5 pound bag cost?

Day 3: estimates, zeros and the laundromat

Isamu finds 48 \times 21 = 1008. Use an estimate to place the point in 4.8 \times 2.1, then check by counting decimal places.
Find 0.07 \times 0.6.
Find 0.04 \times 0.9.
Gauri’s comforter weighs 7.2 pounds, and bulky items cost 1.35 dollars a pound. Estimate the cost, then find it exactly.
Laundry powder sells for 4.2 dollars a pound. What do 1.5 pounds cost?
Wale’s own laundry weighs 6.2 pounds. Wash-and-fold is 1.15 dollars a pound, and a laundry bag costs 0.75 dollars more. What is the total?

Reasoning

Without multiplying, explain why 0.9 \times 0.8 is less than 0.9 and less than 0.8. Then find the product.
Wale multiplies a number with two decimal places by a number with one decimal place, and the answer he writes has only one decimal place. Explain how that can happen, and give an example.
Use a picture of a hundredths grid to explain why a tenth of a hundredth is a thousandth.

Answers

0.24. 4 columns by 6 rows is 4 \times 6 = 24 small squares, 24 hundredths.
0.56. 8 \times 7 = 56 hundredths.
0.18. 2 \times 9 = 18 hundredths.
0.4. 5 \times 8 = 40 hundredths, 0.40, and 40 hundredths fill 4 columns, so 0.40 = 0.4.
0.48. 12 tenths times 4 tenths is 12 \times 4 = 48 hundredths.
0.6 \times 0.9 = 0.54. The rectangle holds 6 \times 9 = 54 small squares, each a hundredth.
0.36 square meter. 0.9 \times 0.4 is 9 \times 4 = 36 hundredths.
0.25 square meter. 0.5 \times 0.5 is 5 \times 5 = 25 hundredths. (Half of a half is a quarter, and 0.25 is a quarter.)
3.22. Parts: 1 \times 2 = 2, 1 \times 0.3 = 0.3, 0.4 \times 2 = 0.8, 0.4 \times 0.3 = 0.12, and 2 + 0.3 + 0.8 + 0.12 = 3.22. Check: 14 \times 23 = 322 with 2 places.
5.12. Parts: 3 \times 1 = 3, 3 \times 0.6 = 1.8, 0.2 \times 1 = 0.2, 0.2 \times 0.6 = 0.12, and 3 + 1.8 + 0.2 + 0.12 = 5.12. Check: 32 \times 16 = 512 with 2 places.
0.144. 0.3 \times 0.4 = 0.12 and 0.06 \times 0.4 = 0.024 (hundredths times tenths are thousandths), and 0.12 + 0.024 = 0.144. Check: 36 \times 4 = 144 with 2 + 1 = 3 places.
0.824. Parts: 4 \times 0.2 = 0.8, 0.1 \times 0.2 = 0.02, 0.02 \times 0.2 = 0.004, and 0.8 + 0.02 + 0.004 = 0.824. Check: 412 \times 2 = 824 with 3 places.
(a) 0.01, one hundredth. (b) 0.001, one thousandth. (c) 0.1, one tenth: ten hundredths fill one column of the grid.
(a) 3.78, 1 + 1 = 2 places. (b) 0.378, 2 + 1 = 3 places. (c) 3.78, 0 + 2 = 2 places.
1.62 square meters. 1 \times 0.9 = 0.9 and 0.8 \times 0.9 = 0.72, and 0.9 + 0.72 = 1.62. Check: 18 \times 9 = 162 with 2 places.
4.9 dollars, written 4.90 dollars. Parts: 1 \times 3 = 3, 1 \times 0.5 = 0.5, 0.4 \times 3 = 1.2, 0.4 \times 0.5 = 0.2, which add to 4.9. Check: 14 \times 35 = 490 with 2 places gives 4.90.
10.08. The estimate 5 \times 2 = 10 rules out 1.008 and 100.8. The count: 1 + 1 = 2 places, so 1008 becomes 10.08.
0.042. 7 hundredths times 6 tenths is 42 thousandths; 3 places, so a zero goes in front: 0.042.
0.036. 4 \times 9 = 36 thousandths; 3 places, so 0.036.
About 1.4 \times 7 = 9.8 dollars, or between 7.2 and 14.4 dollars since 1.35 is between 1 and 2. Exactly 9.72 dollars: 135 \times 72 = 9720 with 2 + 1 = 3 places gives 9.720 = 9.72.
6.3 dollars, written 6.30 dollars. 42 \times 15 = 630 with 2 places gives 6.30. Check: 4.2 + 2.1 = 6.3, since 1.5 pounds is one pound and a half.
7.88 dollars. Wash-and-fold: 115 \times 62 = 7130 with 3 places gives 7.130 = 7.13 dollars. Then 7.13 + 0.75 = 7.88.
0.72. 0.9 \times 0.8 is 8 tenths of 0.9, only part of 0.9, so it is less than 0.9. Read the other way, it is 9 tenths of 0.8, only part of 0.8, so it is less than 0.8 too. On a grid, 9 columns by 8 rows is 72 squares, 0.72.
The whole-number product ended in a zero, and Wale dropped it after placing the point. The count gives three places, but if the last one or two are zeros, fewer show. Example: 0.25 \times 0.4: 25 \times 4 = 100 with 3 places is 0.100, and dropping the end zeros leaves 0.1, with one decimal place.
A hundredth is one small square of the grid. Cut that small square into 10 thin slivers. Do the same to all 100 small squares and the whole square holds 100 \times 10 = 1000 slivers, all the same size. So one sliver is a thousandth of the whole, and a tenth of a hundredth is a thousandth: 0.1 \times 0.01 = 0.001.