What is the 'place value shuffle'?

The 'place value shuffle' is a term for how the digits in a number shift to the left or right when you multiply or divide by powers of 10 (like 10, 100, 1000). Multiplying shifts digits to the left, making the number larger, while dividing shifts them to the right, making it smaller.

What is the difference between composing and decomposing decimals?

Decomposing a decimal means breaking it apart into the sum of its place values (e.g., 1.23 = 1 + 0.2 + 0.03). Composing is the opposite; it means putting the place values back together to form the original decimal.

How do you write a number from its expanded form like 500 + 4 + 0.8 + 0.02?

To compose a number from its expanded form, you add the values together, aligning them by their place value. For 500 + 4 + 0.8 + 0.02, the whole number part is 504. The decimal part is 0.8 + 0.02, which is 0.82. Combining them gives you 504.82.

Place Value Shuffle & Decomposing Decimals (شرح الجزء من ألف)

Learn 1: Place Value Shuffle

Multiplying by Powers of 10

When you multiply by 10, 100, or 1,000, the digits shift to the left, making the number larger.

If a whole number or a decimal is multiplied by 10, then each digit from this number shifts to the left one spot on the place-value chart and the value of each digit increases ten times.

Example: $714 \times 10$

Place value chart for multiplying 714 by 10 | Math Egypt - Mr. Ayman Hassan

Record 714 on the place-value chart.
Shift each digit to the left one spot to get the number "7,140".
Then 714 × 10 = 7,140.
The value of the whole number "714" increased when multiplying by 10.
The value of 7 increased from 700 to 7,000.
The value of 1 increased from 10 to 100.
The value of 4 increased from 4 to 40.

Another Example: $7.14 \times 100$

Place value chart for multiplying 7.14 by 100 | Math Egypt - Mr. Ayman Hassan

Note that:

When multiplying by 100, each digit shifts to the left two spots, then the value of each digit increases 100 times.
Then 7.14 × 100 = 714

Dividing by Powers of 10

When you divide by 10, 100, or 1,000, the digits shift to the right, making the number smaller.

If a whole number or a decimal is divided by 10, then each digit from this number shifts to the right one spot on the place-value chart and the value of each digit decreases ten times.

Example: $615 \div 10$

Place value chart for dividing 615 by 10 | Math Egypt - Mr. Ayman Hassan

Record 615 on the place-value chart.
Shift each digit to the right one spot to get the number 61.5
Then 615 ÷ 10 = 61.5
The value of the whole number "615" decreased when dividing by 10
The value of 6 decreased when dividing by 10 from 600 to 60
The value of 1 decreased when dividing by 10 from 10 to 1
The value of 5 decreased when dividing by 10 from 5 to 0.5

Another Example: $61.5 \div 100$

Place value chart for dividing 61.5 by 100 | Math Egypt - Mr. Ayman Hassan

Note that:

When dividing by 100, each digit shifts to the right two spots, then the value of each digit decreases 100 times.
Then 61.5 ÷ 100 = 0.615

Remark:

Dividing any number by 10 is the same as multiplying this number by $\frac{1}{10}$ . So, $362 \div 10 = 362 \times \frac{1}{10}$

Learn 2: Composing and Decomposing Decimals

Composing decimals means putting parts together to form a number.
Decomposing decimals means breaking a number apart into its place values.

You can decompose 843.572 in different ways:

Place value chart for decomposing 843.572 | Math Egypt - Mr. Ayman Hassan

1st way (Expanded Form): $843.572 = 800 + 40 + 3 + 0.5 + 0.07$ $+ 0.002$

2nd way (Whole & Decimal Part): $843.572 = 843 + 0.572$

3rd way (Mixed Form): $843.572 = 843 + 0.5 + 0.07 + 0.002$

Examples & Exercises

Example 1

Use the place-value charts to solve each problem.

a. $8.7 \times 10$

b. $1.35 \times 1,000$

c. $2.5 \div 10$

d. $6.2 \div 100$

Solution

Exercises on multiplying and dividing decimals by powers of 10 | Math Egypt - Mr. Ayman Hassan

Example 2

Record the number 504.82 in the place-value chart and decompose it in three different ways.

Place value chart for the number 504.82 | Math Egypt - Mr. Ayman Hassan

Solution

1st way (Expanded Form): $504.82 = 500 + 4 + 0.8 + 0.02$

2nd way: $504.82 = 500 + 4 + 0.82$

3rd way: $504.82 = 504 + 0.8 + 0.02$

Example 3

Decompose the following numerals using expanded form.

a. 640.078

b. Twenty-three and forty-two thousandths.

Solution

a. $640.078 = 600 + 40 + 0.07 + 0.008$

b. $23.042 = 20 + 3 + 0.04 + 0.002$

Example 4

Compose each of the following.

a. $4,000 + 80 + 7 + \frac{1}{10} + 0.002$

b. $420 + 0.2 + 0.07 + 0.009$

c. $5,900 + 0.3 + \frac{8}{1,000} + 70 + 2$

Solution

a. 4,087.102

b. 420.279

c. 5,972.308

Example 5

Complete each of the following.

a. 4.208 = ___ + 0.2 + 0.008

b. 70.106 = 70 + 0.1 + ___

c. 32.308 = ___ + 0.3 + 2 + 30

d. ___ = 120 + 0.204

e. ___ = 4 + 0.005 + 0.3

f. 34.012 = 34 + ___

Solution

a. 4

b. 0.006

c. 0.008

d. 120.204

e. 4.305

f. 0.012

Explore More Math Lessons

You're doing great! Keep learning by exploring these other lessons.

Previous: Decimals to the Thousandths

Go back and review the basics of decimals and place value up to the thousandths place.

Next: Rounding Decimals

Learn how to round decimals to the nearest tenth, hundredth, or thousandth.

Learn 1: Place Value Shuffle

Multiplying by Powers of 10

Another Example: 7.14×100

Dividing by Powers of 10

Another Example: 61.5÷100

Learn 2: Composing and Decomposing Decimals

Examples & Exercises

Example 1

Example 2

Example 3

Example 4

Example 5

Explore More Math Lessons

Previous: Decimals to the Thousandths

Next: Rounding Decimals

Another Example: $7.14 \times 100$

Another Example: $61.5 \div 100$