Place Value Shuffle

Learn 1: Place value shuffle

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.

For Example:

714 \times 10

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

Another Example:

7.14 \times 100

Place value chart showing 7.14 times 100

Note that

When multiplying by [100] each digit shifts to the left two spots, then the value of each digit increases 100 times.

Then $7.14 \times 100 = 714$

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.

For Example:

615 \div 10

Place value chart showing 615 divided by 10

Record 615 on the place-value chart.
Shift each digit to the right one spot to get the number 61.5
Then $615 \div 10 = 61.5$
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 showing 61.5 divided by 100

Note that

When dividing by [100] each digit shifts to the right two spots, then the value of each digit decreases 100 times.

Then $61.5 \div 100 = 0.615$

Remark

Dividing any number by 10 is the same as multiplying this number by $\frac{1}{10}$ .

So, $362 \div 10 = 362 \times \frac{1}{10}$

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

a. $8.7 \times 10 = 87$

✓The value of 8 increases from 8 to 80.
✓The value of 7 increases from 0.7 to 7.

b. $1.35 \times 1,000 = 1,350$

✓The value of 1 increases from 1 to 1,000.
✓The value of 3 increases from 0.3 to 300.
✓The value of 5 increases from 0.05 to 50.

c. $2.5 \div 10 = 0.25$

✓The value of 2 decreases from 2 to 0.2.
✓The value of 5 decreases from 0.5 to 0.05.

d. $6.2 \div 100 = 0.062$

✓The value of 6 decreases from 6 to 0.06.
✓The value of 2 decreases from 0.2 to 0.002.

Learn 2: Composing and decomposing decimals

Composing decimals means [put together]
Decomposing decimals means [broken apart]
You can decompose 843.572 in different ways using place-value chart :

Unit form

8 hundreds + 4 tens + 3 ones + 5 tenths + 7 hundredths + 2 thousandths

Expanded form

$800 + 40 + 3 + 0.5 + 0.07 + 0.002$

Another expanded form

$(8 \times 100) + (4 \times 10) + (3 \times 1) + (5 \times \frac{1}{10}) + (7 \times \frac{1}{100}) + (2 \times \frac{1}{1000})$

▸ 1^st way (expanded form): $843.572 = 800 + 40 + 3 + 0.5 + 0.07 + 0.002$

▸ 2^nd way: $843.572 = 843 + 0.572$

▸ 3^rd way: $843.572 = 843 + 0.5 + 0.07 + 0.002$

There are many answers that equal 843.572 when composed.

Example 2

Record the number 504.82 in the place-value chart and decompose this number in expanded form then decompose it in two other ways.

Solution

• 1^st way [expanded form]: $504.82 = 500 + 4 + 0.8 + 0.02$
• 2^nd way: $504.82 = 500 + 4 + 0.82$
• 3^rd 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

Lessons 2 & 3