Multiplicative comparison

Here you will learn about what multiplicative comparisons are and how to use them to solve word problems.

Students will first learn about multiplicative comparisons in 4th grade and expand that knowledge through 5th grade when working with comparison statements and in 6th grade when learning about ratios.

What is multiplicative comparison?

A multiplicative comparison is a way of comparing two quantities by asking how many times larger or smaller one quantity is than the other quantity.

For example, Mary has twice as many cookies as Tim. Tim has 4 cookies. Mary has twice as many balloons as Tim so the number of cookies Mary has = 2 × 4 = 8.

Multiplicative comparisons involve the operations of multiplication and division to solve problems.

A model can help solve multiplicative comparison problems because it helps you to visualize the amounts that need to be compared and to find the unknown quantity.

Let’s look at a few more examples.

Example 1:

Lucas and Ryan are working on a construction project for school. Lucas’s wood board is 5 times the length of Ryan’s board. If Ryan’s wooden board is 3 feet, how long is Lucas’s wooden board?

Draw a bar model to help visualize the situation.

Ryan has a 3 foot board.

From the model, you can see that Lucas’s board is 5 times the length of Ryan’s board. There are 5 groups of 3 feet.

So the equation is 5 \times 3=\text{ length of Lucas's board}

5 \times 3=15

Lucas’s board is 15 feet.

Example 2:

Mike has 3 lollipops. Michelle has 4 times as many lollipops as Mike. How many lollipops does Michelle have?

Draw a picture to model this situation.

Michelle has four times as many lollipops as Mike. So she has 4 groups of 3 lollipops.

The equation is 4 \times 3=\text { amount of lollipops}

4 \times 3=12

Michelle has 12 lollipops.

Example 3:

Jillian has 12 inches of hair ribbon. Suzanne has half that length. How long is Suzanne’s hair ribbon?

Draw a bar model.

The equation is, \cfrac{1}{2} \, \times 12=\text { length of Suzanne's ribbon}

OR

12 \div 2 = \text { length of Suzanne's ribbon}

12 \div 2=6

Suzanne’s ribbon is 6 inches long.

What is multiplicative comparison?

Common Core State Standards

How does this apply to 4th grade math and 5th grade math?

• Grade 4 – Operations and Algebraic Thinking (4.OA.A.1)
  Interpret a multiplication equation as a comparison, for example, interpret 35 = 5 \times 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations.

• Grade 4 – Operations and Algebraic Thinking (4.OA.A.2)
  Multiply or divide to solve word problems involving multiplicative comparison, for example, by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.

• Grade 5 – Number and 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.

How to solve multiplicative comparison problems

In order to solve multiplicative comparison problems:

1. Draw a model.
2. Use multiplication or division to write an equation.
3. Solve the equation.

Multiplicative comparison examples

Example 1: multiplicative comparison using a bar model with whole numbers

Maddie has collected 21 stickers. Her friend, Anna, has collected 3 times that amount. How many stickers does Anna have in her collection?

1. Draw a model. 

Anna has 3 times the amount of stickers as Maddie. So she has 3 groups of 21.

2Use multiplication or division to write an equation.

The equation is 3 \times 21= \; ?

3State the answer.

3 \times 21=63

Anna has 63 stickers.

Example 2: multiplicative comparison using a bar model with whole numbers

Aderonke owns an animal shelter. She has 24 dogs in her shelter and twice as many cats. How many cats are in the shelter?

Draw a model.

There are twice as many cats as dogs which means there are two groups of 24.

Example 3: multiplicative comparison using a bar model

Amani has 18 yards of wire. Austin has a third of that length of wire. How much wire does Austin have?

Draw a model.

Austin’s wire is a third of the length of Amani’s wire. So, 18 yards is divided into three equal groups.

Example 4: multiplicative comparison using a bar model

Billy saved \$122 dollars. His sister, Nikki, has saved 4 times that amount. How much money has Nikki saved?

Draw a model.

Nikki has 4 times the amount of money saved than Billy, which means she has 4 groups of \$122.

Example 5: multiplicative comparison with decimals

Jerome’s neighbor has 92 yards of fencing. Jerome has a fourth of that amount of fencing. How much fencing does Jerome have?

Draw a model.

Jerome’s neighbor has 92.4 feet of fencing. He has a fourth of that amount, which means 92.4 is divided into 4 equal groups.

Example 6: multiplicative comparison with fractions

Carl has 5 \, \cfrac{1}{2} \, gallons of paint. Lucy has three times that amount.

How many gallons of paint does Lucy have?

Draw a model.

Lucy has three times the amount of gallons of paint than Carl which means there are 3 groups of 5 \, \cfrac{1}{2} .

Teaching tips for multiplicative comparison

• Connect visual models to the equations so students can see the visual representation of the abstract equation.

• Math worksheets have their place in a math lesson, but providing students with alternative opportunities to practice such as math games or digital platforms are more engaging.

• Another visual representation that can be used when doing multiplicative comparison word problems is the number line.

• Incorporate projects such as having students create their own multiplicative comparison word problems with answer keys and share them on the Google Classroom.

Easy mistakes to make

• Confusing when to use multiplication versus when to use division
  For example, when you are given an amount such as 18 and are asked to find 3 times that amount, use multiplication. However, if you are given the amount of 18 and are asked to find a third of that amount, use division.

Related multiplication and division lessons

• Multiplication and division
• Multiplication
• Multiplying and dividing integers
• Multiplying and dividing rational numbers
• Multiplying multi digit numbers
• Division
• Dividing multi digit numbers
• Negative times negative
• Long division
• Negative numbers

The next lessons are

• Types of numbers
• Rounding numbers
• Factors and multiples

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