# Multiplying Decimals by Decimals

This is a complete lesson with instruction & exercises for 5th grade about multiplying decimals by decimals. The interpretation for multiplying a decimal by a decimal is to think of it as taking a fractional part of a decimal number (the symbol × translates to "of"). The lesson compares multiplication by a decimal to scaling & shrinking a stick. Lastly, it shows the common shortcut to decimal multiplication (multiply as if there were no decimal points; the answer has as many decimals as the factors have in total.)

In the video below, I explain the rule for multiplying decimals (put as many decimal digits in the answer as there are in the factors.) I explain where this rule comes from, using fraction multiplication. The lesson continues below the video.

 You have learned to think of multiplication by a whole number, such as 3 × 4 or 8 × 0.6, as repeated addition. However, this concept does not work when neither of the factors is a whole number, as in 0.83 × 1.43 or 2/3 × 7/11. Instead, when you multiply decimals or fractions, think of it as finding "a certain part of” the other factor. In this sense, the symbol “×” translates to “of.” Example. 0.1 × 80 means finding one-tenth “of” 80. That is simply 8. Example. 0.4 × 80 means finding four-tenths “of” 80. Since one-tenth of 80 is 8, then 0.4 of 80 is four times as much, or 32. Example. 0.02 × 3,000 means finding two-hundredths of 3,000. Since one-hundredth of 3,000 is 30, then 0.02 of 3,000 is two times as much, or 60.

1. Write as a multiplication using a decimal, and solve. Remember, "of" translates into "×". Use the
top problem in each box to help you solve the bottom one.

 a. one-tenth of 50 _______ × ______ =  _______b. three-tenths of 50 _______ × ______ =  _______ c. one-tenth of 700 ______ × _______ =  _______d. four-tenths of 700 ______ × _______ =  _______ e. one-hundredth of 4,000 _______ × _______ =  _______f. six-hundredths of 4,000 _______ × _______ =  _______

2. Solve. Use the top problem in each box to help you solve the bottom one.

 a. Find 0.1 × 30   ________     Find 0.4 × 30   ________ b. Find 0.1 × 400   _________     Find 0.6 × 400   _________ c. Find 0.01 × 600   _________     Find 0.07 × 600   _________ d. Find 0.1 × 520   ________     Find 0.3 × 520   ________ e. Find 0.001 × 5,000   _________    Find 0.002 × 5,000   _________ f.  Find 0.01 × 800   _________    Find 0.11 × 800   _________

3. Answer. You do not have to calculate.

a. You have learned that 0.1 × 246 means one-tenth of 246.
Will the result of  0.1 × 246 be more or less than 246?

b. Also, 0.1 × 0.8 means one-tenth of 0.8.
Will the result of  0.1 × 0.8  be more or less than 0.8?

c. Will the result of  1.9 × 928  be more or less than 928?

 Scaling means expanding or shrinking something by some factor. This red stick is 40 pixels long. Let’s scale it to be four times as long: → We can write a multiplication "equation": 4 × = Using pixels, 4 × 40 px = 160 px. Now let’s scale the red stick to be 0.4 (four-tenths) as long as it is at first: → Notice, it shrank! We can write: 0.4 × = In pixels, 0.4 × 40 px = 16 px. The number we multiply by (4 and 0.4 above) is called the scaling factor. If the scaling factor is more than 1, such as 2.3, the resulting stick is longer than the original one. If the scaling factor is less than 1, such as 0.5 or 0.66, the resulting stick is shorter.

4. The stick is being shrunk. How long will it be in pixels? Compare the problems.

 a.  0.1 × = 0.1 × 40 px = ________ px b.  0.3 × = 0.3 × 40 px = ________ px c.  0.6 × = 0.6 × 40 px = ________ px d.  0.2 × = 0.2 × 40 px = ________ px e.  0.5 × = 0.5 × 40 px = ________ px f.  0.9 × = 0.9 × 40 px = ________ px

 Let’s expand this stick (40 px) to be 1.2 times as long: → We can write a multiplication:  1.2 × = To calculate how long it is in pixels, let’s first figure out what 0.2 of 40 is. Since one-tenth of 40 is 4, then 0.2 of 40 is double that, or 8. Then, 1.2 × 40 px would be 1 × 40 px  and  0.2 × 40 px, or 40 + 8 = 48 pixels.

5. The red stick is 50 pixels long. It is being expanded or shrunk. Fill in the blanks.

 a.  0.5 × = 0.5 × 50 px = ________ px b.  0.3 × = 0.3 × 50 px = ________ px c.  1.5 × = 1.5 × 50 px = ________ px d.  1.3 × = 1.3 × 50 px = ________ px

6. Tell if the resulting stick after being "multiplied" will be shorter or longer than the original.

 a. 3.1 × will be longer/shorter than b. 0.3 × will be longer/shorter than c.  0.9 × will be longer/shorter than d.  1.2 × will be longer/shorter than Towards a shortcut

Half of 5 is 2.5, or  0.5 × 5 = 2.5. This resembles the familiar multiplication 5 × 5 = 25!

One-tenth of 20 is 2, so three-tenths of 20 is 6. We can write 0.3 × 20 = 6.
This resembles the familiar multiplication 3 × 2 = 6!

The shortcut to decimal multiplication is:
 1) Multiply as if there were no decimal points. 2) Place the decimal point in the answer.

But where will we put the decimal point? Let’s explore that a little bit!

7. Fill in Anita’s reasoning.

 a. To calculate 0.8 × 0.8, I first multiply 8 × 8 = 64. The answer to 0.8 × 0.8 has to be slightly     smaller than 0.8, because scaling anything by 0.8 is close to the original, but somewhat smaller.     So, 0.8 × 0.8 can’t be 64, and it cannot  be 6.4, but it is _________! b. 0.1 × 5.6 has to be 1/10 of the size of 5.6. So, it cannot be 56. Could it be 5.6?  No, because     1 × 5.6 = 5.6. So, 0.1 × 5.6 has to equal __________. c. 0.4 × 0.06 has to be smaller than 0.06. It can be neither 24, nor 2.4. Is it 0.24 or 0.024?  ________

 The shortcut to decimal multiplication 1) Multiply as if there were no decimal points. 2) Place the decimal point in the answer. The number of decimal digits in     the answer is the SUM of the number of decimal digits in the factors.
Example 1.   0.05 × 0.7

5 × 7 is 35. The factor 0.05 has two and
0.7 has one decimal digit. The answer
has to have three, so the answer is 0.035.

Example 2.  12 × 2 × 0.3 × 0.2

12 × 2 × 3 × 2 = 144. The factors have 0, 0, 1,
and 1 decimal digits—a total of 2. The answer
has to have 2 decimal digits, so the answer is 1.44.

8. Multiply first as if there were NO decimal points. Then add the decimal point to the answer.

 a.  0.5 × 0.3 = ________ b.  0.9 × 0.6 = ________ c.  0.4 × 0.08 = ________d.  0.7 × 0.02 = ________ e. 0.1 × 0.3 = ________f.  0.1 × 2.7 = ________ g.  0.2 × 0.1 = ________h.  0.8 × 0.1 = ________ i.  0.9 × 0.01 = ________ j.   0.9 × 0.1  = ________ k.  0.7 × 0.3 = ________l.   7 × 0.03  = ________

9. Multiply.

 a.  0.4 × 0.8   = ________ b.  0.7 × 1.1   = ________c.  0.02 × 0.9 = ________ d.  0.02 × 0.5 = ________ e.  0.002 × 9  = ________f.   1.1 × 0.3   = ________ g.  2.1 × 0.2 × 0.5 = _________ h.   0.4 × 4 × 0.2   = _________i.   6 × 0.06 × 0.2  = _________

 The answer to a decimal multiplication may end in one or more zeros. That is no problem. However, after placing the decimal point, you may simplify the final answer by dropping the ending decimal zeros. 50 × 0.006 50 × 6 = 300. The factors have 0 and 3 decimal digits, so the answer has to have 3. Therefore, the answer is 0.300, but it simplifies to 0.3. 400 × 0.05 400 × 5 = 2000. The factors have 0 and 2 decimal digits, so the answer has to have 2. The answer is 20.00. You can simplify that to 20.

10. Solve.

 a.  0.4 × 0.5 = _______ b.  20 × 0.06 = _______ c.  40 × 0.05 = _______ d.  3 × 0.2 × 0.5 = _______ e.  300 × 0.009 = ________ f.  40 × 0.05 = __________ g.  0.6 × 0.2 × 0.5 = ________ h.  600 × 0.004 = __________ i.   0.4 × 0.5 × 60 = ________

 Potatoes cost \$1.20 / kg. If you buy 23 kilograms, you multiply 23 × \$1.20 to find the total price. If you buy 0.8 kg, you do the same: multiply the price by 0.8. To find  0.8 × \$1.20, first multiply without the decimal point: 8 × 120 = 960. The factors have 1 and 2 decimal digits, so the answer must have three decimal digits: 0.960. We can omit the final zero, and give the answer as \$0.96. Note that the answer also makes sense: 0.8 kg of potatoes should cost a bit less than 1 kg of potatoes, which was \$1.20.

11. Find the total cost. Write a multiplication.

a. Ribbon costs \$1.10 per meter, and you buy 0.4 meters.

b. Nuts cost \$8 per pound. You buy 0.3 pounds.

c. A phone call costs \$7 per hour. You talk for 1.2 hours.

d. Lace costs \$2.20 per meter, and you buy 1.5 meters.

You can make worksheets for decimal multiplication here.

This lesson is taken from Maria Miller's book Math Mammoth Decimals 2, and posted at www.HomeschoolMath.net with permission from the author. Copyright © Maria Miller. It addresses the Common Core Standard for 5th grade 5.NBT.7.

#### Math Mammoth Decimals 2

A self-teaching worktext for 5th-6th grade that covers the four operations with decimals up to three decimal digits, concentrating on decimal multiplication and division. The book also covers place value, comparing, rounding, addition and subtraction of decimals. There are a lot of mental math problems.

Download (\$6.25). Also available as a printed copy.

=> Learn more and see the free samples!