043. Multiplying and Dividing Decimals by Powers of Ten

Learning Intentions

• To understand that a power of ten is a number like $10$, $100$, $1000$ etc.
• multiply a decimal by a power of ten
• divide a decimal by a power of ten

Pre-requisite Summary

• Understand the meaning of the decimal point as separating whole-number parts from fractional parts
• Know the place value of digits to the left and right of the decimal point
• Recognise powers of ten such as $10,100,1000$ as ${10}^1,{10}^2,{10}^3$
• Understand that multiplying or dividing by $10$ changes place value
• Be able to compare the size of decimals before and after a calculation
• Know that placeholder zeros may be needed to show place value correctly

Worked Examples

Worked Example 1

a) Explain what a power of ten is.
b) State three examples of powers of ten.
c) Explain how multiplying by a power of ten affects place value.

Worked Example 2

Multiply each decimal by a power of ten:
a) $4.7\times 10$
b) $0.36\times 100$
c) $5.082\times 1000$

Worked Example 3

Multiply each decimal by a power of ten:
a) $3.45\times 100$
b) $0.908\times 10$
c) $12.06\times 1000$

Worked Example 4

Divide each decimal by a power of ten:
a) $8.4÷10$
b) $53.7÷100$
c) $624.5÷1000$

Worked Example 5

Divide each decimal by a power of ten:
a) $47.2÷10$
b) $6.09÷100$
c) $305.8÷1000$

Worked Example 6

a) Compare $4.26$, $4.26\times 100$, and $4.26÷100$.
b) Explain why one result is greater and one result is smaller than the starting number.
c) State the role of the decimal point in each calculation.

Problems

Problem 1

a) Explain what a power of ten is.
b) State three examples of powers of ten.
c) Explain how dividing by a power of ten affects place value.

Problem 2

Multiply each decimal by a power of ten:
a) $5.8\times 10$
b) $0.47\times 100$
c) $6.031\times 1000$

Problem 3

Multiply each decimal by a power of ten:
a) $2.64\times 100$
b) $0.709\times 10$
c) $14.08\times 1000$

Problem 4

Divide each decimal by a power of ten:
a) $9.6÷10$
b) $48.3÷100$
c) $731.2÷1000$

Problem 5

Divide each decimal by a power of ten:
a) $56.4÷10$
b) $7.08÷100$
c) $402.9÷1000$

Problem 6

a) Compare $3.75$, $3.75\times 100$, and $3.75÷100$.
b) Explain why one result is greater and one result is smaller than the starting number.
c) State the role of the decimal point in each calculation.

Exercises

Understanding and Fluency

1. State whether each number is a power of ten:
   a) $10$
   b) $100$
   c) $200$

2. Write the next three powers of ten after:
   a) $10$
   b) $100$
   c) $1000$

3. Multiply each decimal by $10$:
   a) $4.3$
   b) $0.8$
   c) $12.06$

4. Multiply each decimal by $100$:
   a) $3.47$
   b) $0.09$
   c) $15.2$

5. Multiply each decimal by $1000$:
   a) $2.506$
   b) $0.304$
   c) $7.08$

6. Divide each decimal by $10$:
   a) $6.5$
   b) $48.2$
   c) $903.1$

7. Divide each decimal by $100$:
   a) $7.4$
   b) $56.09$
   c) $120.5$

8. Divide each decimal by $1000$:
   a) $8.36$
   b) $420.7$
   c) $9056.2$

9. Complete the table:
   a) $4.26,4.26\times 10,4.26\times 100$
   b) $5.08,5.08÷10,5.08÷100$

10. Mixed practice:
    a) $0.37\times 100$
    b) $46.2÷100$
    c) $3.005\times 1000$
    d) $781.4÷10$

Reasoning

11. Explain why $3.6\times 100=360$.

12. A student says $4.28\times 10=4.280$. Explain the mistake.

13. Explain why dividing $52.7$ by $100$ gives a smaller number than $52.7$.

14. A student says $8.4÷10=84$. Explain why this is incorrect.

Problem-solving

15. A bottle holds $1.25$ L of juice. Express this amount in decilitres by multiplying by $10$.

16. A ribbon is $2.48$ m long. Express this length in centimetres by multiplying by $100$.

17. A container has $375.6$ mL of water. Express this amount in litres by dividing by $1000$.

18. A runner records a distance of $4.275$ km. Express this in metres by multiplying by $1000$.

Potential Misunderstandings

• Students may think multiplying by a power of ten means “adding zeros” without considering decimal place value
• Students may move digits rather than understanding that the value of each digit changes place
• Students may confuse multiplying by $10$ with dividing by $10$
• Students may think the decimal point moves on its own rather than the digits changing place value around it
• Students may forget to use placeholder zeros when needed, for example in $0.4\times 100=40$
• Students may believe that dividing by $100$ makes a number larger because more digits appear after the decimal point
• Students may write answers without checking whether the result should be larger or smaller than the original number
• Students may not recognise that powers of ten are specifically $10,100,1000,\dots$ and not just any multiple of $10$

Next: 044. Multiplying Decimals