Multiplying decimals

When you reach for it

Shuyi and Maximo run a taco truck. Every order of tacos gets 0.7 pound of filling, and six orders are waiting at the window. How much filling should Shuyi scoop? A taco costs 2.65 dollars, and Darshan wants four. How much does he pay?

Both questions ask for the same thing: a number of equal groups, where each group is a decimal. That is multiplication, just as 6 groups of 7 is 6 \times 7. The only new part is the decimal point. You reach for this lesson whenever one amount with a decimal point is repeated a whole number of times: a price times how many you buy, a weight per order times how many orders, a length per step times how many steps.

You already have two tools that do the job. Adding 0.7 six times works (see Adding and subtracting decimals), but it is slow. And you can multiply whole numbers in columns (see The standard multiplication algorithm). This lesson joins them: it counts tenths and hundredths the way you count anything else, and it shows why the point lands where it does.

This lesson takes three days. Part 1 multiplies tenths by a whole number with base-ten pictures. Part 2 multiplies hundredths, first with an area model and then in columns. Part 3 puts the factors in either order, uses an estimate to check the point, and handles products that end in zero.

The procedure

Part 1: tenths times a whole number

Start with Shuyi’s filling: 6 orders of 0.7 pound each. One way is to add:

Multiplication is the short way to write that sum: 6 \times 0.7 means 6 groups of 0.7.

Now look at what 0.7 is. It is 7 tenths (see Tenths and hundredths). So 6 groups of 0.7 are 6 groups of 7 tenths. Count the tenths the way you would count anything:

The unit stays the same. Six groups of 7 apples are 42 apples, and six groups of 7 tenths are 42 tenths. All that is left is to write 42 tenths as a decimal. Ten tenths make one whole, so 40 tenths are 4 wholes, and 2 tenths are left:

The picture shows the same thing. Each thin strip is one tenth of a square. Six groups of seven strips are 42 strips. Pushed together, they fill 4 whole squares with 2 strips to spare. Shuyi scoops 4.2 pounds.

The same counting works when the decimal has ones as well as tenths. Take 3 \times 2.6. The number 2.6 is 26 tenths, so

There is a second way to see it. Split 2.6 into 2 + 0.6, multiply each part by 3, and add. This is the The distributive property:

Both ways give 7.8. The first way is faster. The second shows where each part of the answer comes from: the 6 from the ones, the 1.8 from the tenths.

Part 2: hundredths times a whole number

Darshan wants 4 tacos at 2.65 dollars each. The cost is 4 \times 2.65.

The number 2.65 has a hundredths digit, so count in hundredths. Two dollars and sixty-five cents is 265 cents, and a cent is one hundredth of a dollar (see Money written as decimals). So

and 1060 hundredths are 10 ones and 60 hundredths, which is 10.60. Darshan pays 10.60 dollars.

The area model shows the same product as a rectangle (see The area model). Make the rectangle 4 tall and 2.65 wide. Then cut the width into its place values, 2, 0.6 and 0.05. Each small rectangle is one easy product.

The middle one is 4 \times 6 tenths = 24 tenths = 2.4. The last one is 4 \times 5 hundredths = 20 hundredths = 0.20. Adding the three areas, with the points lined up,

The area model is the distributive property drawn as a picture. It is clear, but it gets long when the numbers grow. The column method does the same work in a short form.

Write the whole number under the decimal, lined up on the right, as you would for whole numbers. Then multiply each digit of 2.65 by 4, starting at the right, and carry as usual:

Each carry is a trade you already know. Twenty hundredths are 2 tenths, so the 2 moves to the tenths column. Twenty-six tenths are 2 ones and 6 tenths, so the 2 moves to the ones column.

Where does the point go? Every digit of the answer is in the same place as the digit it came from: hundredths times 4 give hundredths, tenths times 4 give tenths. So the answer has the same number of places after the point as the decimal had, here two. This is why the column method lines the numbers up on the right and not by their points: the whole number has no tenths or hundredths, it only tells you how many groups there are.

This also answers a quick question: when the whole number is 10 or 100, every digit shifts one or two places, a shortcut taught in Multiplying a decimal by ten, one hundred or one thousand.

Part 3: either order, estimating, and zeros at the end

Either order. Sometimes the decimal comes first: 0.7 \times 6. The order of the factors does not change a product. This is the commutative property of multiplication (see Properties of multiplication), so

You can also read 0.7 \times 6 as “seven tenths of 6.” One tenth of 6 is 0.6, since ten of those make 6. Seven of them make 7 \times 0.6 = 4.2. The same answer, as it must be.

An estimate places the point. Before or after you multiply, round the decimal to a nearby whole number (see Estimating by rounding). For 4 \times 2.65, the price 2.65 is close to 3, so the cost is close to 4 \times 3 = 12 dollars. The column method gave the digits 1060. Of 1.060, 10.60 and 106.0, only 10.60 is close to 12. The estimate cannot tell you every digit, but it tells you at once if the point is in the wrong place.

Zeros at the end. The product 4 \times 2.65 came out as 10.60. A zero in the last place after the point adds no hundredths, so

Both are correct. On a bill you keep two places, 10.60 dollars, because money is counted in cents. For a weight, 10.6 pounds is the plainer way to write it. There are two rules for this. Place the point first, then drop zeros, never the other way around: dropping the 0 from 1060 before placing the point would give 106 hundredths, the wrong number. And only zeros after the point may go: the 0 in the 10 is holding the ones place.

Bigger whole numbers. When the whole number has two digits, multiply as you would for whole numbers, with one partial product for each digit (see The standard multiplication algorithm), and place the point by the same rule. The worked examples below do this twice.

Worked examples

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

Boundaries

• The whole number may be any whole number, including 0. Whole numbers are 0, 1, 2, 3 and so on. Zero groups of anything is nothing, so 0 \times 2.65 = 0. One group is the number itself, so 1 \times 2.65 = 2.65.
• The point rule here is for a decimal times a whole number. The product has as many places after the point as the decimal factor, because the whole number only counts groups. When both factors have digits after the point, as in 0.3 \times 0.4, tenths times tenths are not tenths, and the rule changes. That case is the next lesson.
• Drop zeros only after the point is placed, and only after the point. 10.60 = 10.6, but 1060 hundredths is not 106 hundredths, and the zero in 10 cannot go.
• Money keeps two places. 10.6 and 10.60 are the same number, but a price is written to the cent, 10.60 dollars.
• An estimate checks the point, not every digit. 4 \times 2.65 = 10.70 would pass the estimate “about 12” and still be wrong. The column method, or the count of hundredths, finds the digits; the estimate only guards the place of the point.

Where this goes next

Every product in this lesson has one whole number factor. The next lesson, Where the point goes in a product, lets both factors carry a point, such as 0.3 \times 0.4 or 2.6 \times 1.5, and explains how many places the product gets then. Later, Multiplying multi-digit decimals with the standard algorithm runs the column method on longer decimals of both kinds.

Price times quantity, weight per order times orders, and distance per lap times laps come back again and again in Decimal problems in words. And the same move, a decimal times a number, is how a percent of an amount is found in Finding a percent of a quantity and how the distance around a circle is found in Circumference.

Exercises

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

Part 1: tenths times a whole number

1. Count tenths to find each product: (a) 3 \times 0.2, (b) 5 \times 0.7, (c) 9 \times 0.8.
2. Write 4 \times 0.9 as a repeated addition, then find the product.
3. Find each product: (a) 2 \times 3.4, (b) 6 \times 1.3, (c) 7 \times 2.5.
4. How many tenths are 8 groups of 0.7? Write the product as a decimal.
5. Shuyi puts 0.3 pound of beans in each taco bowl. How many pounds of beans go into 7 bowls?
6. Maximo chops 1.6 pounds of onions each day for 5 days. How many pounds does he chop in all?
7. Use the distributive property to find 6 \times 4.3, splitting 4.3 into 4 + 0.3.
8. What whole number times 0.6 gives 4.8? Explain using tenths.

Part 2: hundredths times a whole number

9. Find each product: (a) 3 \times 0.21, (b) 4 \times 0.16, (c) 2 \times 0.38.
10. Use an area model to find 3 \times 2.14. Write the three small products.
11. Use the column method to find 5 \times 1.63.
12. Use the column method to find 7 \times 4.08.
13. Darshan buys 3 tacos at 2.95 dollars each. How much does he pay?
14. Each taco gets 0.06 pound of salsa. How much salsa do 9 tacos get?
15. A family buys 4 burritos at 5.69 dollars each. What is the cost?
16. The truck buys 8 sacks of onions at 12.35 dollars a sack. What is the cost?

Part 3: either order, estimating, and zeros at the end

17. Estimate 0.92 \times 6, then find the exact product.
18. The digits of 7 \times 3.18 are 2226. Use an estimate to place the point.
19. Find each product, then write it without zeros at the end: (a) 5 \times 0.62, (b) 4 \times 0.75, (c) 6 \times 1.15.
20. Each order of tacos uses 0.4 pound of tortillas. How many pounds do 15 orders use?
21. On Monday the truck sold 23 fish tacos at 2.25 dollars each. Estimate the money taken in, then find it exactly.
22. Shuyi buys 3 pounds of cheese at 4.29 dollars a pound and 2 bags of tortillas at 3.15 dollars a bag. She pays with a twenty-dollar bill. How much change does she get?

Reasoning

23. Explain why 3 \times 0.4 is 1.2 and not 0.12.
24. Maximo multiplies 6 \times 0.35, gets the digits 210, and writes 21.0. Use an estimate to show the point is in the wrong place, then give the right product in its simplest form.
25. Explain why 8 \times 0.9 is less than 8. Then find it to check.