What pattern describes how the decimal point moves when you multiply a decimal by any power of ten

If you multiply a number by 100%, you are simply multiplying by 1. You can multiply by 100% to change a decimal number or fraction to a percent. (0.25)(100%)=25%. You can also divide by 100% to change a percent to a decimal number. 65%/100%= 65/100 or 0.65.

This is a complete lesson with a video & exercises showing, first of all, the common shortcut for multiplying & dividing decimals by powers of ten: you move the decimal point as many steps as there are zeros in the number 10, 100, 1000 etc.

Then, I also show where this shortcut originates, using place value charts. In reality, the decimal point moving is sort of an illusion, and instead, the digits of the number move within the place value chart. This explanation can really help students to understand the reason behind the "trick" of moving the decimal point.

The lesson below explains the shortcut in more detail, plus has different kinds of exercises, word problems, and even a fun riddle for students.

Remember? When you multiply whole numbers by 10, 100, 1000, and so on (powers of ten), you can simply “tag” as many zeros on the product as there are in the factor 10, 100, 1000 etc.

There is a similar shortcut for multiplying decimal numbers by numbers such as 10, 100, and 1000:

Move the decimal point to the right as many places as there are zeros in the factor.

10 × 0	.	4	9	= 04.9 = 4.9

Move the decimal point one step to the right (10 has one zero).

100 × 2	.	6	5	= 265. = 265

Move the decimal point two steps to the right (100 has two zeros). The number 265. is 265 (as shown above).

1000 × 0	.	3	7	0		= 3	7	0	. = 370

1000 means we move the point three steps. Write a zero at the end of 0.37 so that the decimal point can “jump over to” that place.

1. Multiply.

a. 10 × 0.04 = ________

b. 100 × 0.04 = ________

c. 1000 × 0.04 = ________

d. 10 × 0.56 = ________

e. 100 × 0.56 = ________

f. 1000 × 0.56 = ________

g. 10 × 0.048 = ________

h. 100 × 0.048 = ________

i. 1000 × 0.048 = _______

Another helpful shortcut! Since 100 × 2 = 200, obviously the answer to 100 × 2.105 will be a little more than 200. Hence, you can just write the digits

2105 and put the decimal point so that the answer is 200-something: 210.5.

2. Let's practice some more.

a. 100 × 5.439 = ________

b. 100 × 4.03 = ________

c. 1000 × 3.06 = ________

d. 100 × 30.54 = ________

e. 30.73 × 10 = ________

f. 93.103 × 100 = _______

105 × 0	.	1	2	0		0		0		= 12000. = 12,000

105 = 100,000 has five zeros. Again, write additional zeros so that the decimal point can “jump over to” those places.

3. Now let's practice using powers of ten.

a. 102 × 0.007 = _____________

103 × 2.01 = _____________

105 × 4.1 = ______________

b. 105 × 41.59 = _____________

3.06 × 104 = ______________

0.046 × 106 = _____________

The shortcut for division by 10, 100 and 1000 (powers of ten) is similar. Can you guess it?

Move the decimal point to the ( left / right ) for as many places (steps) as there are _________________________ in the factor 10, 100, or 1000.

Move the decimal point two steps to the ____________. You need to write zeros in front of the number.

0 0 0 5 6.	÷ 104 = 0.0056

Move the decimal point four steps to the ____________. You need to write zeros in front of the number.

4. Divide.

a. 0.4 ÷ 10 = ________

0.4 ÷ 100 = ________

4.4 ÷ 100 = ________

b. 15.4 ÷ 100 = ________

21.03 ÷ 10 = ________

0.39 ÷ 10 = ________

c. 5.6 ÷ 10 = ________

34.9 ÷ 100 = ________

230 ÷ 1000 = ________

5. Now let's practice using powers of ten.

a. 0.7 ÷ 102 = _____________

45.3 ÷ 103 = _____________

568 ÷ 105 = _____________

b. 2.1 ÷ 104 = _____________

4,500 ÷ 106 = _____________

9.13 ÷ 103 = _____________

Why does this SHORTCUT work?

When 0.01 (a hundredth) is multiplied by ten, we get ten hundredths, which is equal to one tenth. Or, 10 × 0.01 = 0.1.

The entire number moved one “slot” to the left on the place value chart. This looks like moving the decimal point in the number to the right.

A hundred times two tenths is like multiplying each tenth by 10, and by 10 again. Ten times two-tenths gives us two, and ten times that gives us 20.

Again, it is like moving the number over two “slots” to the left in the place value chart, or moving a decimal point in 0.2, two steps to the right.

When 3.915 is multiplied by 100, we get 391.5. Each part of the number (3, 9 tenths, 1 hundredth, 5 thousandths) is multiplied by 100, so each one of those moves two “slots” in the place value chart. This is identical to thinking that the decimal point moves two steps to the right.

The similar shortcut for division works because division is the opposite operation of multiplication—it “undoes” multiplication. If we move the decimal point to the right when multiplying by 10, 100, 1000 and so on, then it is quite natural that the rule for division would work the “opposite” way.

Fractions vs. division. If we move the decimal point to solve 6 ÷ 100, we get:

0 0 6	.	0	÷ 100 = 0.060 = 0.06

Let’s write 6 ÷ 100 using the fraction line: it is 6/100 or 6 hundredths, which is written 0.06 as a decimal. Therefore, in this case you do not need the “shortcut,” but you can just think of fractions and decimals. These kinds of “connections” make mathematics so neat!

6. Divide. Think of fractions to decimals, or use the shortcut. Compare the problems in each box!

7. A 10-lb sack of nuts costs $72.
How much does one pound cost?

8. Find the price of 100 ping-pong balls if one ping pong ball costs $0.89.

Thinking more about fractions and decimals

If we divide any whole number by 1,000, the answer will have thousandths or three decimal digits. This makes it easy to divide whole numbers by 1,000: simply copy the dividend as your answer (without the commas), and then make it have three decimal digits:

Examples:

41,300

1000

= 41.300 = 41.3

Notice in the last two cases, we can simplify the results: 41.300 to 41.3 and 8.000 to 8.

9. Divide whole numbers by 1000. Simplify the final answer by dropping any ending decimal zeros.

Similarly:

If you divide any whole number by 10, copy the dividend and make it have one decimal digit.
If you divide any whole number by 100, copy the dividend and make it have two decimal digits.

Examples:

82,000

10

= 8200.0 = 8,200

10. Divide whole numbers by 10 and 100.

11. Find one-tenth of...

a. $8

b. $25.50

c. $126

12. Find one-hundredth of...

a. $78

b. $4

c. $390

13. A pair of shoes that cost $29 was discounted by 3/10 of its price. What is the new price?(Hint: First find 1/10 of the price.)

14. Find the discounted price:

a. A bike that costs $126 is discounted by 2/10 of its price.

b. A $45 cell phone is discounted by 5/100 of its price.
(Hint: First find 1/100 of the price.)

15. One-hundredth of a certain number is 0.03. What is the number?

16. Which vacuum cleaner ends up being cheaper? Model A, with the initial price $86.90, is discounted by 3/10 of its price.

Model B costs $75 now, but you will get a discount of 1/4 of its price.

An important tip

In the problem ____ × 3.09 = 309, the number 3 becomes 300, so obviously
the missing factor is 100. You do not even have to consider the decimal point!

The same works with division, too. In the problem 7,209 ÷ _____ = 7.209, the missing divisor
is one thousand, because the value of the digit 7 was first 7000, and then it became 7.

Of course, in some problems it will be easier to think in terms of “moving the decimal point.”