Multiply and divide numbers with decimals by powers of 10
Multiplying or dividing by powers of $10$10 ($10$10, $100$100, $1000$1000, and so on) is much easier than multiplying by other numbers, because our number system is base $10$10. Let's have a look at what this means.
Consider the number $253$253. We can write this as:
$253=2\times100+5\times10+3\times1$253=2×100+5×10+3×1
When we multiply $253$253 by $100$100, we can write it like this:
| $\left(2\times100+5\times10+3\times1\right)\times100$(2×100+5×10+3×1)×100 | $=$= | $2\times10000+5\times1000+3\times100$2×10000+5×1000+3×100 |
| $=$= | $25300$25300 |
Notice that when we multiplied by $100$100 the place value of the "$5$5" changed from tens to thousands, since $10\times100=1000$10×100=1000. In fact, all of the digits moved up by two place values.
Multiplying by $10$10 increases the place of each digit by one place.
Dividing by $10$10 decreases the place of each digit by one place.
Multiplying or dividing by $100=10\times10$100=10×10 moves the values by two places, multiplying or dividing by $1000=10\times10\times10$1000=10×10×10 moves the values by three places, and so on.
You may have to add or take away zeros, and you may have to add or take away the decimal point, after you have finished moving the values. This depends on which columns the values end up in.
Worked example
Example 1
Evaluate $5.62\times1000$5.62×1000.
Think: Since we are multiplying, the digits will all move up in place value. Since there are three zeros in $1000$1000, each digit will move up by three place values.
Do: We can evaluate the multiplication by moving each digit up by three place values.
So $5.62\times1000=5620$5.62×1000=5620.
Reflect: Notice that we added a zero in the Ones place, to properly represent the new places that our values ended up in. We also removed the decimal point, since we didn't need it after the multiplication.
Example 2
Evaluate $7.43\div100$7.43÷100.
Think: Since we are dividing, the digits will all move down in place value. Since there are two zeros in $100$100, each digit will move down by two place values.
Do: We can evaluate the division by moving each digit down by two place values.
So $7.43\div100=0.0743$7.43÷100=0.0743.
Reflect: We moved the digits up in place value for multiplication and down for division. The number of zeroes in the conversion unit told us how many place values to move by. We ended up adding in some extra zeros to keep track of the places for each of our values.
Practice questions
Question 2
What is $0.65\times10$0.65×10? Write your answer in decimal form.
Question 3
What is $8.4\div100$8.4÷100? Write your answer in decimal form.