Multiplying and dividing decimals

Here 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.

What is multiplying and dividing decimals?

Multiplying and dividing decimals is when you multiply or divide with numbers that have decimals.

For example,

Multiplying decimal number by decimal number
Modeling with a Hundredths Grid
\hspace{2.3cm} 0.3 \times 0.5 = \, ? \hspace{1.3cm} Factor \hspace{2.7cm} Factor \hspace{2.7cm} Product \hspace{1.1cm} 0.3 = 0.30 \hspace{2.2cm} 0.5 = 0.50 \hspace{2.7cm} 0.15 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
Modeling with a Hundredths Grid
\hspace{2.3cm} 5 \times 0.25 = \, ? 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
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. But first: ● Check to make sure the divisor is a whole number. ● Line up decimal and place it where the answer will go. ● Add 0’ s as place holders as necessary. ● Do long division. Quotient \bf{= 0.32}

Dividing a decimal by a decimal
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. ● If the divisor isn’t a whole number, decide what power of 10 to multiply it by that will make it a whole number. ● Multiply the dividend by the same power of ten you multiplied the divisor by. ● Line up decimal point of dividend with quotient, add necessary 0’ s as place holders. ● Do long division. Quotient \bf{= 2.2}

Dividing a whole number by a decimal
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. ● If the divisor isn’t a whole number, decide what power of 10 to multiply it by that will make it a whole number. ● Multiply the dividend by the same power of ten you multiplied the divisor by. ● Line up decimal point of dividend with quotient, add necessary 0’ s as place holders. ● Do long division. Quotient \bf{=15}

What is multiplying and dividing decimals?

Common Core State Standards

How does this relate to 5th grade math and 6th grade math?

• Grade 5: Number & Operations Base Ten (5.NBT.B.7)
  Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

• Grade 6: The Number System (6.NS.B.3)
  Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.

How to multiply decimals

In order to multiply decimals using the standard algorithm.

1. Stack the number with the most digits on top.
2. Multiply the same as multi-digit whole numbers, regrouping when necessary.
3. Count the number of digits after the decimal point for each factor.
4. Put the same number of digits behind the decimal point for the product.

In order to solve decimal multiplication word problems.

1. Create an equation to model the problem.
2. Stack the number with the most digits on top.
3. Multiply the same as multi-digit whole numbers, regrouping when necessary.
4. Count the number of digits after the decimal point for each factor.
5. Put the same number of digits behind the decimal point for the product.
6. Label the product.

Multiplying decimals examples

Example 1: multiplying a decimal number by a decimal number

Find the product of ​​ 0.2 \times 0.8

1. 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.

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.

Example 2: multiplying a decimal number by a whole number

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.

Example 3: multiplying a decimal by a decimal

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.

Example 4: word problem multiplying decimal by a decimal

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?

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.

How to divide decimals

In order to divide decimals using the algorithm:

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.
2. Multiply both the divisor and the dividend by the same power of 10 .
3. If the dividend is a decimal number, line up the decimal point of the dividend with the decimal point of the quotient.
4. Do long division.

In order to solve real world problems using decimal division:

1. Read the problem, identify keywords, and decide which number is the dividend and which number is the divisor.
2. If the divisor is a whole number, go to step 4. If not, decide the power of 10 that will make the divisor a whole number.
3. Multiply both the divisor and the dividend by the same power of 10 .
4. If the dividend is a decimal number, line up the decimal point of the dividend with the decimal point of the quotient.
5. Do long division.
6. Label your answer appropriately.

Dividing decimals examples

Example 5: dividing a decimal by a whole number

Solve the decimal division problem.

0.56 \div 7 = \; ?

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.

Example 6: dividing a decimal by a decimal

Solve 24.36 \div 2.1 = \; ?

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.

Example 7: dividing a whole number by a decimal

Solve 9 \div 0.15 = \; ?

Do long division.

Example 8: Decimal division word problem

A recipe calls for 5.6 ounces of flour. How many complete recipes can be made with 265.44 ounces of flour?

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.

Teaching tips for multiplying and dividing decimals

• Students who are just learning how to multiply and divide with decimals should use visual models, such as hundredths grids, to solve. This helps them see the equal groups and consider the size of the products and quotients.

• Give students opportunities to use different strategies to solve these decimal operations, such as partial products or the relationship between multiplication and division. It is important that students have a deep understanding of multiplying and dividing decimals before being introduced to the algorithm. In time, provide opportunities for them to make connections between their strategies and the standard algorithm. This helps students understand why the steps work for all decimal numbers.

• Encourage students to use mental math to estimate what the product or quotient should be. This will help them catch simple mistakes, like a wrong calculation or placing the decimal point in the wrong position.

• Do not rely on only practice worksheets for these skills. Furthermore, when using practice worksheets, value understanding over speed. Decimal calculations take time to do by hand. Do not expect students to answer rapidly, but instead encourage them to take their time to estimate, solve and understand what they are doing.

Easy mistakes to make

• Putting the decimal point in the wrong position in the product
  The total number of decimal positions in the numbers being multiplied is how many the product should have.
  For example,
  5.6 \times 0.7
  There are 2 decimal positions being multiplied, so the product should also have 2 decimal positions.
  
  

  

• Forgetting to add back the decimal after multiplying
  The algorithm for multiplying decimals is the same as whole numbers, except there is an additional step. The decimal needs to be added back to the product. Forgetting to do so, leads to the wrong final answer.
  For example,
  5.6 \times 0.7
  There are 2 decimal positions being multiplied, so the product should also have 2 decimal positions.
  
  

  

• Lining up the decimal points of the products before multiplying
  The decimals do not need to be lined up when using the standard algorithm to multiply, like they do when adding and subtracting. Doing so is not incorrect, but can create extra work and more room for errors.
  For example,
  2 \times 1.23
  Without lining up the decimals: Lining up the decimals:
  
  

  

• Switching the dividend and the divisor when using long division
  The dividend goes on the inside and the divisor goes on the outside.
  For example,
  0.58 \div 2
  
  

  

• Not moving the decimal point of the dividend and divisor the same amount
  In order to keep the equations equivalent, the decimal needs to move the same amount for both the dividend and divisor. This is the same multiplying the dividend and divisor by the same power of 10.
  For example,
  2 \div 0.5
  
  

  

• Forgetting to align the decimal point of the dividend with the quotient
  For example,
  0.8 \div 2
  
  

  

• Thinking that the larger number is always the dividend
  Smaller numbers can be, and often are, divided by larger numbers.
  For example,
  A gardener has 0.3L of water for his 3 plants. If the gardener will water the plants equally, how much water does each plant get?
  
  

  

• Not being mindful of place value by forgetting to divide zeros in the dividend
  All digits in the dividend need to be divided when using the standard algorithm, including zeros. Otherwise the place value positions of the digits shifts, changing the value.
  For example,
  0.15 \div 9 = 15 \div 900
  
  

  

Related lessons on decimals

• Decimals
• Adding decimals
• Decimal places
• Decimal number line
• Decimal place value
• Multiplying decimals
• Dividing decimals
• Adding and subtracting decimals
• Subtracting decimals
• Comparing decimals

The next lessons are

• Percent
• Converting fractions, decimals, and percentages
• Exponents
• Algebraic expression
• Math equations

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