5th Grade Math
Multiplying and Dividing with Decimals
Unit 3
Multiplying and Dividing with Decimals
Patterns When Multiplying by Powers of 10s
We learned the patterns with power of 10 back in Unit 1, but this concept is important when multiplying decimals, so we will review it again!
"^" is the symbol for an exponent, so 10^4 means 10 raised to the power of 4.
36 x 10 = 36 x 10^1 = 360
36 x 10 x 10 = 36 x 10^2 = 3600
36 x 10 x 10 x 10 = 36 x 10^3 = 36,000
36 x 10 x 10 x 10 x 10 = 36 x 10^4 = 360,000
When you multiply a number by 10, it is just a shift in where the decimal point is. We noticed that every time we multiplied by 10, we placed a zero to the end of the number. That makes sense because each digit’s value became 10 times larger. To make a digit 10 times larger, we have to shift it one place value to the left.
When we multiplied 36 by 10, the 30 became 300. The 6 became 60 or the 36 became 360. So, we had to place a zero at the end to have the 3 tens represent 3 one-hundreds (instead of 3 tens) and the 6 ones represents 6 tens (instead of 6 ones).
We can also divide by powers of 10!
Here's an example: 350 ÷ 10^3 = 350 ÷ 1,000 = 0.350 = 0.35
Here's another example: 52.3 ÷ 101 = 5.23 The place value of the digits in 52.3 is decreased by one place.
Multiplying by negative powers of 10 is the same thing as dividing by positive powers of 10! If this is confusing, just ignore the bottom half of the image above for now :).
Ready to put it to practice? Find the pattern in placement of the decimal, when a decimal is divided by a power of 10. Hint: you may need to add zeroes in the quotient :)
Still confused? Watch this video to better understand how to multiply decimals!
Multiplying Decimals
Before we find the answer to multiplication problems with decimals, we can estimate the answer. Let's try it together!
Let's estimate 6 x 2.4
We can estimate that the answer is between 12 and 18 since 6 x 2 is 12 and 6 x 3 is 18 (and 2.4 is in between 2 and 3).
We can also give an estimate of a little less than 15 because we see the answer to be very close, but smaller than 6 x 2½. We can then think about 2½ groups of 6 as 12 (2 groups of 6) + 3(½ of a group of 6).
Multiplying decimals is the same thing as multiplying whole numbers!
Step 1: We place both numbers so that the longer factor is on the top and the shorter factor is on the bottom.
Step 2: We solve the multiplication problem as we normally would with whole numbers.
Step 3: We count the digits that come after the decimal points in all decimal numbers. We add up the number of decimal places in the first and second number. Then, we place the decimal point in the answer.
We can also solve multiplication problem with models. An example is given below!
A gumball costs $0.22. How much do 5 gumballs cost? Estimate the total, and then calculate. Was your estimate close
We estimate that the total cost will be a little more than a dollar. I know that 5 20’s equal 100 and we have 5 22’s. I have 10 whole columns shaded and 10 individual boxes shaded. The 10 columns equal 1 whole. The 10 individual boxes equal 10 hundredths or 1 tenth. My answer is $1.10. My estimate was a little more than a dollar, and my answer was $1.10. I was really close.
Dividing Decimals
Dividing decimals is almost the same as dividing whole numbers, except you use the position of the decimal point in the dividend to determine the decimal places in the result.
To divide decimal numbers:
If the divisor is not a whole number:
Move the decimal point in the divisor all the way to the right (to make it a whole number).
Move the decimal point in the dividend the same number of places.
Divide as usual. If the divisor doesn't go into the dividend evenly, add zeroes to the right of the last digit in the dividend and keep dividing until it comes out evenly or a repeating pattern shows up.
Position the decimal point in the result directly above the decimal point in the dividend.
These steps can be shown in the diagrams to the right and we solved an example problem (6.85 divided by .5)!
We can also solve division problems with models! An example is given below:
A relay race lasts 4.65 miles. The relay team has 3 runners. If each runner goes the same distance, how far does each team member run? Another way to write this problem is 4.65 ÷ 3. Make an estimate, find your actual answer, and then compare them.
This is an example response:
Our estimate is that each runner runs between 1 and 2 miles. If each runner went 2 miles, that would be a total of 6 miles which is too high (since 2 x 3 = 6). If each runner ran 1 mile, that would be 3 miles, which is too low (since 1 x 3 = 3).
We used the 5 grids above to represent the 4.65 miles. We are going to use all of the first 4 grids and 65 of the squares in the 5th grid. We have to divide the 4 whole grids and the 65 squares into 3 equal groups. We labeled each of the first 3 grids for each runner, so we know that each team member ran at least 1 mile.
We then have 1 whole grid and 65 squares to divide up. Each column represents one-tenth. If we give 5 columns to each runner, that means that each runner has run 1 whole mile and 5 tenths of a mile.
Now, we have 15 squares left to divide up. Each runner gets 5 of those squares. So, each runner ran 1 mile, 5 tenths and 5 hundredths of a mile. I can write that as 1.55 miles.
Our answer is 1.55 and our estimate was between 1 and 2 miles. We were pretty close! :-)
Still confused? Watch this video to better understand how to divide decimals!
Have some fun and get practice! Play this game to divide a decimal numbers! Hint: you can multiply the decimal by a power of 10 to convert it into a whole number first. After you have solved the question, then divide the answer by that same power of 10. For example, in 84 divided by 0.7, we can multiply the 0.7 by 10 to get 7. Then the problem becomes 84 divided by 7, which is equal to 12. Finally, we can divide 12 by 10 to get the answer, which is 1.2.