# Dividing Decimals—Mental Math

This is a complete lesson for 5th/6th grade with instruction and exercises, teaching students how to divide decimals using mental math (based on number sense). It starts out with some sharing divisions, then explains the basic strategy for those. Students also divide decimals with "measurement division", such as 0.45 ÷ 0.05, where we think how many times the divisor goes into the dividend. The lesson has pattern exercises, word problems, a cross-number puzzle, and more.

You can make worksheets for decimal division here.

1. First shade the parts. Then divide and write a division sentence.

 a. Shade 0.3. Divide it into 3 parts. ___ ÷   3   = ___
 b. Shade 0.64. Divide it into 2 parts. ___ ÷   2   = ___
 c. Shade 1.8. Divide it into 3 parts. ___ ÷ ___ = ___
 d. Shade 1.6. Divide it into 4 parts. ___ ÷ ___ = ___
 e. Shade 0.30.  Divide it into 10 parts. ___ ÷ ___ = ___
 f. Shade 0.1. Divide it into 10 parts. ___ ÷ ___ = ___

 A decimal divided by a whole number You can think of multiplication “backwards.” To solve 4.5 ÷ 5, think: What number multiplied by 5 will give me 4.5? Or, _____ × 5 = 4.5. The answer is 0.9. Or, think of “bananas” divided among a group of people. The only thing is, this time the “bananas” are tenths, hundredths, or thousandths! For example, 0.035 ÷ 5 is “35 thousandths divided by 5”. Replace the thousandths by bananas for a moment: “35 bananas divided by 5... equals 7 bananas.” The answer to the original problem is 7 thousandths, or 0.007. Another example: 0.12 ÷ 4 is “12 hundredths divided by 4”. This is essentially the division problem “12 divided by 4”, however, in terms of hundredths. The answer is 3 hundredths or 0.03.

2. Write the division problems with numbers, and solve.

 a. 9 tenths divided by 3 equals ... _______ ÷ ____ = _______ b. 72 thousandths divided by 9 equals ... _______ ÷ ____ = _______ c.  54 hundredths divided by 6 equals ... _______ ÷ ____ = _______ d.  240 thousandths divided by 60 equals ... _______ ÷ ____ = _______ e.  122 hundredths divided by 2 equals ... _______ ÷ ____ = _______

3. Divide. Think of dividing “bananas”: how many tenths, hundredths, or thousandths you are dividing.
Or, think of multiplication backwards.

 a.  0.024 ÷ 6 = ______ b.  0.24 ÷ 6 = _______c.   2.4 ÷ 6 = ________ d.  0.49 ÷ 7 = _______ e.  1.2 ÷ 3 = ________f.   0.056 ÷ 7 = _______ g.  5.40 ÷ 9 = _______ h.  0.20 ÷ 4 = ________i.   0.050 ÷ 10 = _______

 Trick!  Remember how 0.40 = 0.4? We can omit the decimal zero, but we can also write it. When dividing a decimal by a whole number, it often helps to “tag” a zero on the number before dividing. 0.8      ÷  100   (tag two zeros) →  0.800  ÷  100  =  0.008 (800 thousandths divided by 100   equals 8 thousandths.) 0.7    ÷  10   (tag one zero) →  0.70  ÷  10  =  0.07 (70 hundredths divided by 10  equals 7 hundredths.) 4    ÷  8   (tag one zero) →  4.0 ÷  8  =  0.5 (40 tenths divided by 8  equals 5 tenths.)

4. Divide. Tag a zero or zeros on the dividend.

 a.  0.3    ÷   5   = ________ b.  0.3    ÷  10  = ________c.  3       ÷   5   = ________ d.  0.06     ÷  12 = _______ e.  0.2        ÷  40 = _______f.   2          ÷  5   = _______ g.  0.3     ÷  50   = ________ h.  0.7     ÷  100 = ________i.  0.02    ÷  10   = ________

5. Jane shared \$2.00 equally among five friends.
How much did each one get?

6. If each heartbeat takes 0.8 seconds, how long do five heartbeats take?

Ten heartbeats?

7. Write two division problems and two multiplication problems with the same numbers—a fact family!

 a.    8  ×  0.04  =   0.32     _____ × _____ = ______ _____ ÷ _____ = ______ _____ ÷ _____ = ______ b. ____ × ____ = _____   ____ × ____ = _____   2   ÷   0.4  =   5    ____ ÷ ____ = _____ c. ______ × ______ = _______   ______ × ______ = _______    ______ ÷ ______ = _______     0.025  ÷   5   =   0.005

 Sometimes it helps to think how many times the divisor “goes” or “fits” into the dividend. Example 1.  0.24 ÷ 0.03 = ?  Think: “How many times will 3 hundredths go into 24 hundredths?” Just as 3 goes into 24 eight times, 3 hundredths goes into 24 hundredths 8 times. Example 2.  Mom cut 0.4-meter pieces from a 1.2-meter piece of material. How many pieces did she get? Think, “How many times does 0.4 go into 1.2?” The answer is of course easy: 3 times. We can also write a division from this situation: 1.2 ÷ 0.4 = 3.

8. Divide. Think: how many times does the divisor go into the dividend?

 a.  4.5 ÷ 0.5 = _______ b.  0.45 ÷ 0.05 = _______ c.  0.450 ÷ 0.005 = _______ d.  0.12 ÷ 0.06 = _______ e.  0.006 ÷ 0.002 = ______f.   0.63 ÷ 0.07 = ________ g.  2.1 ÷ 0.7 = ________ h.  1.5 ÷ 0.3 = ________i.   0.09 ÷ 0.01 = _______

9. Write a division sentence for each problem, and solve.

a.  How many 0.3 m pieces do you get from 1.8 m of cloth?    _______ ÷ _____ = ________

b.  How many 0.7 m pieces do you get from 4.2 m of wood?   _______ ÷ _____ = ________

c.  How many 0.05 m pieces do you get from 0.25 m of string?  _______ ÷ _____ = ________

 Example 3.  0.72 ÷ 0.008 =  ?  First, tag a zero on 0.72 so that it also has three decimals, just like 0.008 has three decimals. Now we get: 0.720 ÷ 0.008 =  ?  Now think, “How many times does 8 thousandths fit into 720 thousandths?” This is the same as asking, “How many times does 8 fit into 720?” The answer: 90 times. So, 0.720 ÷ 0.008 = 90  (not 0.90 or 0.090; just plain 90).

10. Divide. You may need to tag a zero or zeros on the dividend so that both numbers have the same
amount of decimal digits. Then think: How many times does the divisor go into the dividend?

 a.  0.20  ÷ 0.05 = _______ d.  0.3    ÷ 0.05 = _______ b.  1    ÷ 0.2 = _______ e.   5    ÷ 0.2 = _______ c.  0.4    ÷  0.02  = _______ f.  0.05   ÷ 0.001 = _______ g.  0.6   ÷ 0.05 = _______ j.   1      ÷ 0.02 = _______ h.  0.9   ÷ 0.01 = _______ k.   1     ÷ 0.01 = _______ i.  0.1     ÷  0.01  = _______ l.  0.03   ÷ 0.002 = _______

11. The asphalt crew does a 1.2-mile stretch of road each day.

a. How many days does it take for them to cover a distance of 6 miles?

b. How many days does it take for them to cover a distance of 60 miles?

12. Jack has \$1.45 in nickels in his pocket.

a. How many nickels does Jack have?

b. If you have not already, write a decimal division to match the problem.

13. How many 0.04-meter sticks can you cut from a 0.20-meter board?
Write a decimal division to match the problem.

 14. Which expressions match the problem?       There are two. (You do not have to calculate anything.)      One book that is 3 cm thick is lying in a box that       is 15 cm high. How many 1.5 cm thick books       could you stack in that box? 8 × 1.5 cm + 3 cm = 15 cm 15 × 3 cm + 1.5 cm = 46.5 cm (15 cm − 3 cm) ÷ 1.5 cm = 8 (15 cm − 1.5 cm) ÷ 3 cm = 4.5 15 cm + 3 cm + 1.5 cm = 19.5 cm (15 cm ÷ 3 cm) + 1.5 cm = 6.5 (15 cm ÷ 1.5) + 3 cm = 13

15. Write a single expression (number sentence with several operations) to match this problem. Solve.

How much is left from 5 meters of material
after you cut off four 0.6 meter pieces?

16. Joe has 0.85 kg of meat. How many 0.3 kg servings can he get from that?

Also, “convert” this problem into grams, remembering that 1 kg has 1,000 grams.

17. Divide and place the answers in the cross-number puzzle.

 Across:       a. 1 ÷ 0.04      b. 0.018 ÷ 9      c. 0.044 ÷ 0.004      d. 5 ÷ 10       e. 0.9 ÷ 0.09 Down: a. 0.9 ÷ 0.06b. 0.09 ÷ 3c. 8.4 ÷ 0.7d. 1 ÷ 100 e. 0.32 ÷ 8

18. Figure out the pattern and continue it for at least two more problems.

 a.  0.025 ÷ 0.005 =       0.25 ÷  0.05  =         2.5 ÷   0.5   = b.  1000 ÷   20  =       100  ÷    2   =        10   ÷  0.2  = c.  4,200 ÷  40  =       420   ÷   4   =        42   ÷  0.4  =

 Based on what you observed in the previous exercise, change    the decimal division 0.987 ÷ 0.021 into a WHOLE NUMBER    division problem with the same answer, and solve.

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.