Powers of Ten and Decimal Products

What happens to $2.5$ when you multiply it by $10$? Think of $2.5$ as $2$ ones and $5$ tenths. Make each part $10$ times as large.

• $2$ ones times $10$ is $20$ ones, which is $2$ tens.
• $5$ tenths times $10$ is $50$ tenths, which is $5$ ones.

So $2.5 \times 10 = 25$. Each digit has moved one place to the left.

Dividing by $10$ does the opposite. It makes each part one tenth as large, so each digit moves one place to the right. The $2$ ones become $2$ tenths, and the $5$ tenths become $5$ hundredths. So $2.5 \div 10 = 0.25$.

Multiplying or dividing by $100$ makes this move twice, because $100 = 10 \times 10$. Multiplying or dividing by $1{,}000$ makes it three times.

You do not need a chart every time. Count the zeros in the power of ten: $10$ has one, $100$ has two and $1{,}000$ has three. That is how many places each digit moves. Digits move left when you multiply and right when you divide.

Sometimes the move leaves an empty place between the digits and the decimal point. Fill it with a placeholder zero, a $0$ that holds the other digits in their places. In $2.5 \times 100 = 250$, a zero fills the ones place. In $0.6 \div 100 = 0.006$, zeros fill the tenths and hundredths places.

It can look as if the decimal point jumps instead: to the right when you multiply and to the left when you divide. That picture gives the same answer. Really, the point stays between the ones and the tenths while the digits move.

A quick size check catches many slips. $10$, $100$ and $1{,}000$ are all greater than $1$. So when you start with a positive number, multiplying by one of them gives a larger number, and dividing by one of them gives a smaller number. Zero is different. It stays zero: $0 \times 100 = 0$ and $0 \div 100 = 0$.

Keep those conditions attached to the rule. Division does not always shrink a number. Dividing by a positive number below $1$ can make a positive number larger. You will work with divisions like that in the next unit.

Now multiply a decimal by a whole number. The numbers you multiply are called factors. The answer is called the product.

$4 \times 0.3$ means $4$ groups of $0.3$. You could add them: $0.3 + 0.3 + 0.3 + 0.3 = 1.2$. Here is a faster way to think. $0.3$ is $3$ tenths, so $4$ groups of $3$ tenths make $12$ tenths. That is the whole-number fact $4 \times 3 = 12$, counted in tenths. And $12$ tenths is $12 \div 10 = 1.2$.

The same plan works when the decimal ends in the hundredths or thousandths place. Read the decimal as a whole number of those units. Multiply the whole numbers. Then divide by $100$ or $1{,}000$ to turn the product back into a decimal.

What does it mean to multiply by a decimal such as $0.2$? $0.2$ is $2$ tenths. Taking $2$ tenths of a number means cutting it into $10$ equal parts and keeping $2$ of them.

Try it with $0.3 \times 0.2$. Cut $0.3$ into $10$ equal parts: $0.3 \div 10 = 0.03$. Keep $2$ of the parts: $2 \times 0.03 = 0.06$. So $0.3 \times 0.2 = 0.06$. Look at the units. A tenth of a tenth is a hundredth, so tenths times tenths gives hundredths.

There is a quicker route to the same answer. Leave out the decimal points for a moment and find $3 \times 2 = 6$. Now, $3$ is $10$ times $0.3$, and $2$ is $10$ times $0.2$. Making one factor $10$ times as large makes the product $10$ times as large. You saw that with $4 \times 3 = 12$ and $4 \times 0.3 = 1.2$. Here you did it twice, so $6$ is $10 \times 10 = 100$ times too large. Divide by $100$ to undo it.

Here is the idea to keep. Multiplying or dividing by $10$, $100$ or $1{,}000$ moves every digit one, two or three places. Multiplying decimals is whole-number multiplication followed by one of those moves: multiply the whole numbers, count the decimal places in both factors, and move the digits that many places to the right. Then check the size. When you start with a positive number, a factor greater than $1$ gives a larger product, a factor of $1$ leaves it unchanged, and a factor between $0$ and $1$ gives a smaller one.

Next, you will use the same digit moves to divide decimals by whole numbers, and then to divide by decimals.