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Multiplication and division Multiplying multi-digit numbers Dividing multi-digit numbersHere you will learn about multiplying and dividing decimals, including how to multiply and divide decimals by whole numbers, multiply and divide decimals by decimals, and divide whole numbers by decimals.
Students will first learn about multiplying and dividing decimals as part of their work in number and operations in base 10 in 5th grade. They continue to build on this knowledge by learning the algorithm as a part of their work in the number system in 6th grade.
Multiplying and dividing decimals is when you multiply or divide with numbers that have decimals.
For example,
Multiplying decimal number by decimal number |
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Modeling with a Hundredths Grid |
\hspace{1.3cm} Factor \hspace{2.7cm} Factor \hspace{2.7cm} Product
The product is the overlapped shaded area. There are 15 shaded squares in the overlapped area, so the product is \bf{0.15} |
Using the Algorithm |
Use the algorithm for multi-digit multiplication to multiply decimal numbers. The product is \bf{0.15} |
Multiplying a whole number by decimal number |
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Modeling with a Hundredths Grid |
5 groups of 0.25 or 5 groups of 25 shaded squares (hundredths) The product is the sum of all the shaded squares, which is 125 hundredths. The product is \bf{1.25} |
Using the Algorithm |
Use the algorithm for multi-digit multiplication to multiply decimal numbers. The product is \bf{1.25} |
For example,
Dividing a decimal by a whole number |
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Modeling with a Hundredths Grid |
1.6 = 1.60 Divide into 5 equal groups. 32 hundredth squares in each group so the quotient is \bf{0.32} |
Using the Algorithm |
Divide the way you would whole numbers. |
Dividing a decimal by a decimal |
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Modeling with a Hundredths Grid |
How many groups of 0.5 are there in 1.1? You can see there are 2\cfrac{10}{50} \, groups. Since \cfrac{10}{50} \, = \cfrac{20}{100} \, the quotient is \bf{2.20} or \bf{2.2} |
Using the Algorithm |
● Check the divisor; if it’s a whole number, start dividing. |
Dividing a whole number by a decimal |
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Modeling with a Hundredths Grid |
How many groups of 0.2 \; (0.20) are there in 3? You can see there are 15 groups so thequotient is \bf{15.} |
Using the Algorithm |
● Check the divisor; if it’s a whole number, start dividing. |
How does this relate to 5th grade math and 6th grade math?
In order to multiply decimals using the standard algorithm.
In order to solve decimal multiplication word problems.
Use this worksheet to check your grade 5 to 6 students’ understanding of multiplying decimals. 15 questions with answers to identify areas of strength and support!
DOWNLOAD FREEUse this worksheet to check your grade 5 to 6 students’ understanding of multiplying decimals. 15 questions with answers to identify areas of strength and support!
DOWNLOAD FREEFind the product of 0.2 \times 0.8
Both factors have the same number of digits, so it does not matter which one is on top.
2Multiply the same as multi-digit whole numbers, regrouping when necessary.
3Count the number of digits after the decimal point for each factor.
In total, the factors have 2 decimal points.
4Put the same number of digits behind the decimal point for the product.
Move the decimal point two decimal places. The product will have two digits after the decimal point.
The product is 0.16.
On the hundredths grid, look at the overlapped shaded region. There are 16 hundredth squares or 0.16.
Find the product of 3 \times 1.03
Stack the number with the most digits on top.
1.03 has more digits than 3, so 1.03 will be on top.
Multiply the same as multi-digit whole numbers, regrouping when necessary.
Count the number of digits after the decimal point for each factor.
In total, the factors have two decimal places.
Put the same number of digits behind the decimal point for the product.
Move the decimal point two decimal places. The product will have two digits after the decimal point.
The product is 3.09.
Looking at the hundredths grids, you can see there are 3 groups of 1.03 that represent 309 shaded hundredth squares or 3.09.
Find the product of 5.7 \times 8.09.
Stack the number with the most digits on top.
8.09 has more digits than 5.7, so it will go on top.
Multiply the same as multi-digit whole numbers, regrouping when necessary.
Count the number of digits after the decimal point for each factor.
In total, the factors have 3 decimal points.
Put the same number of digits behind the decimal point for the product.
Move the decimal point three decimal places. The product will have three digits after the decimal point.
The product is 46.113.
Mark wants to paint a wall in his kitchen. It has a length of 12.5 feet and a width of 8.25 feet. How much paint, in square feet, does Mark need to paint the wall?
Create an equation to model the problem.
12.5 \times 8.25 = \; ?
Stack the number with the most digits on top.
Both factors have the same number of digits, so it does not matter which one is on top.
Multiply the same as multi-digit whole numbers, regrouping when necessary.
Count the number of digits after the decimal point for each factor.
In total, there are three decimal places after the decimal point.
Put the same number of digits behind the decimal point for the product.
Move the decimal point three decimal places. The product will have three digits after the decimal point.
The product is 103.125.
Label the product.
Mark will need 103.125 feet ^{2} of paint to paint the wall.
In order to divide decimals using the algorithm:
In order to solve real world problems using decimal division:
Solve the decimal division problem.
0.56 \div 7 = \; ?
If the divisor is a whole number, go to step 3. If not, decide the power of 10 to multiply it by that will make it a whole number.
The divisor is 7 which is a whole number → go to step 3.
If the dividend is a decimal number, line up the decimal point of the dividend with the decimal point of the quotient.
Do long division.
The answer makes sense because 56 hundredths on the grid is 7 groups of 8 hundredths.
Solve 24.36 \div 2.1 = \; ?
If the divisor is a whole number, go to step 3. If not, decide the power of 10 to multiply it by that will make it a whole number.
The divisor is 2.1, which is not a whole number. It needs to be multiplied by 10^1.
Multiply both the divisor and the dividend by the same power of 10 .
Multiply 2.1 \times 10 which is the same as 2.1 \times 10^1=21.
Multiply 24.36 \times 10 which is the same as 24.36 \times 10^1=243.6.
If the dividend is a decimal number, line up the decimal point of the dividend with the decimal point of the quotient.
Do long division.
Solve 9 \div 0.15 = \; ?
If the divisor is a whole number, go to step 3. If not, decide the power of 10 to multiply it by that will make it a whole number.
The divisor is 0.15 which is not a whole number. It needs to be multiplied by 10^2.
Multiply both the divisor and the dividend by the same power of 10 .
Multiply 0.15 \times 100 which is the same as 0.15 \times 10^2=15.
Multiply 9 \times 100 which is the same as 9 \times 10^2=900.
If the dividend is a decimal number, line up the decimal point of the dividend with the decimal point of the quotient.
The new equation is 900 \div 15, so there is no decimal in the dividend.
Do long division.
A recipe calls for 5.6 ounces of flour. How many complete recipes can be made with 265.44 ounces of flour?
Read the problem, identify keywords, and decide which number is the dividend and which number is the divisor.
Each recipe has 5.6 ounces and we need to find how many times 5.6 ounces goes into 265.44 ounces → 265.44 \div 5.6
If the divisor is a whole number, go to step 4. If not, decide the power of 10 to multiply it by that will make it a whole number.
The divisor is 5.6. It needs to be multiplied by 10^1.
Multiply both the divisor and the dividend by the same power of 10 .
Multiply 5.6 \times 10 which is the same as 5.6 \times 10^1=56.
Multiply 265.44 \times 10 which is the same as 265.44 \times 10^1=2,654.4.
If the dividend is a decimal number, line up the decimal point of the dividend with the decimal point of the quotient.
Do long division.
Label your answer appropriately.
The quotient is 47.4, which means that 47 complete recipes can be made with the flour.
1. Solve 0.7 \times 0.4 = \; ?
Both factors have the same number of digits, so it does not matter which one is on top.
In total, the factors have 2 decimal points.
Move the decimal point two decimal places. The product will have two digits after the decimal point.
The product is 0.28.
On the hundredths grid, look at the overlapped shaded region. There are 28 hundredth squares or 0.28.
2. Solve 17 \times 5.08 = \; ?
5.08 has more digits than 17, so it will go on top.
In total, the factors have 2 decimal points. The product will have two digits after the decimal point.
The product is 86.36.
3. Solve 9.1 \times 0.23 = \; ?
0.23 has more digits than 9.1, so it will go on top.
In total, the factors have 3 decimal points. The product will have three digits after the decimal point.
The product is 2.093.
4. A grocery store has 13.2 pounds of apples. They have 2.5 times more bananas than apples. How many pounds of bananas does the grocery store have?
0.33 pounds
33 pounds
3.3 pounds
33.33 pounds
There are 2.5 times more bananas → 13.2 \times 2.5
13.2 has more digits than 2.5, so it will go on top.
In total, the factors have 2 decimal points. The product will have two digits after the decimal point.
The product is 33. The grocery store has 33 pounds of bananas.
5. Solve 0.85 \div 5 = \; ?
The divisor is 5 which is a whole number, so we set up the long division. Line up the decimal place of the dividend with the decimal place of the quotient.
Use the algorithm of long division to divide.
The answer makes sense because 85 hundredths on the grid is 5 groups is 17 hundredths.
6. Solve 4.41 \div 1.5 = \; ?
The divisor is 1.5. Multiply the decimal divisor by 10 to make it a whole number. Also, multiply the dividend by 10.
1.5 \times 10=15
4.41 \times 10=44.1
Line up the decimal place of the dividend with the decimal place of the quotient.
Use the algorithm of long division to divide.
7. Solve 0.6 \div 20 = \; ?
The divisor is 20 which is a whole number, so we set up the long division.
Line up the decimal place of the dividend with the decimal place of the quotient.
Use the algorithm of long division to divide.
8. Twelve pizzas cost \$85.44. If each pizza costs the same, how much did one pizza cost?
\$85.44 has to be divided equally into 12 pizzas → 85.44 \div 12.
Line up the decimal place of the dividend with the decimal place of the quotient.
Use the algorithm of long division to divide.
The quotient is 7.12 which means one pizza costs \$7.12.
No, most whole number strategies will work with decimals, including partial products.
For example,
2.2 \times 3.4
Students will spend most of their time multiplying numbers with decimals in the tenths place, hundredths place and the thousandths place. However, once they learn the standard algorithm, students can multiply with any number of decimal positions.
Multiplying by the powers of ten increases the value of each digit in a number and therefore moves the decimal point. As a shortcut, you can just move the decimal point, but you must move it the same amount of times for both the divisor and the dividend to keep the equation equivalent.
Yes, the divisor can be smaller than the dividend. This will result in an answer that has no whole numbers, only decimals.
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