Word problems for comparing

The following diagrams show the three types of comparison word problems: Difference Unknown, Unknown Big Quantity, Unknown Small Quantity. Scroll down the page for examples and solutions.

Types of Comparison Word Problems

There are three main types of comparison word problems.

Difference Unknown
Connie has 15 red marbles and 28 blue marbles. How many more blue marbles than red marbles does Connie have?
This is a subtraction problem.
28 — 15 = 13

Unknown Big Quantity
Connie has 15 red marbles and some blue marbles. She has 13 more blue marbles than red ones. How many blue marbles does Connie have?
This is an addition problem.
15 + 13 = 28

Unknown Small Quantity
Connie has 28 blue marbles. She has 13 more blue marbles than red ones. How many red marbles does Connie have?
This is a subtraction problem.
28 -13 = 15

How to solve comparison word problems using Bar Models or Tape Diagrams?

This video explains how to use bar modeling in Singapore math to solve word problems that deal with comparing. This technique of using model drawings to solve word problems is recommended by the Common Core mathematics standards.

Example:
Adam has 11 fewer lollipops than Hope. If Adam has 16 lollipops, how may lollipops does Hope have?

• Show Video Lesson

Bar Model (Comparison)
This video employs a visual way to solve world problems using bar modeling. This type of word problem uses the comparison model. Because the part is missing, this is a subtraction problem.

Example:
Cayla did 88 sit-ups in the morning. Nekira did 32 sit-ups at night. How many more sit-ups did Cayla do than Nekira?

• Show Video Lesson

How to solve comparison word problems using Comparison Bars?

This is another strategy that we can use for story problems that involves comparison.

Examples:

1. Bob has 14 crackers. Steve ate five fewer than Bob. How many did Steve eat?
2. David has 5 marbles. Anna has 8 marbles. How many more does Anna have than David?
3. Ellie has 9 goldfish. Laney has 5 more goldfish than Ellie. How many goldfish does Laney have?
4. Mark earned $428 doing yard work. Troy earned $186. How much less did Troy earn?
5. Billy has 679 gumballs. He has 278 more gumballs than Lee. How many gumballs does Lee have?
6. Jake has 219 bottlecaps. Then, he found some more. Now he has 347 bottlecaps. How many bottlecaps did he find?

• Show Video Lesson

Comparison Bars

Examples:

1. Claire has 8 marbles. Sasha has 15 marbles. How many more marbles does Sasha have than Claire?
2. Bill read 5 books. Beth read 2 more books than Bill. How many books did Beth read?
3. Beth read 8 books. Bill read 3 fewer books than Beth. How many books did Bill read?

Example:
The Nature Center has a collection of snakes. The redbelly snake in the collection is 9 inches long. The eastern ribbon snake is 21 inches long. How much longer is the eastern ribbon snake than the redbelly snake?

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Comparison word problems in elementary schools

Students may start solving comparison word problems as early as first
grade. Then, these problems get more and more challenging in 4th or 5th
grade.  There are two types of comparison in math: additive and
multiplicative.

Next we show some examples commonly used in elementary schools.

Additive comparison

In additive comparison, the problem may have the following expressions where x can be a whole number.

• How much more?  
• How much less?
• x more
• x less

Example #1

Darline has 6 candies. Sylvia has 5 more. How many candies does Sylvia have? Here is the model for this problem.

Sylvia has 6 + 5 candies or 11 candies.

Example #2

John has 15 books. Peter has 6 less. How many books does Peter have? Here is the model for this problem.

Peter has 15 — 6 or 9 books

Multiplicative comparison

In multiplicative comparison, the problem may have the following expressions where x can be a whole number.

• x times as many
• x times more … than

Example #3

Noemy has 3 times as many dolls as Tatiana. Tatiana has 2 dolls. How many dolls does Noemy have?

Step 1

Draw a model for this situation.

Step 2

Use the model to write an equation. Let n be the number of dolls Noemy has.

n = 3 x 2

Step 3 

Solve the equation. n = 3. so Noemy has 6 dolls.

A challenging comparing word problem in elementary school.

Example #4

Kirk has 3 times as many baseball cards as Nancy. Together, they have 20 baseball cards. How many cards does Kirk have?

Step 1

Draw a box with the letter n in it to show that Nancy has an unknown number of cards. Kirk has 3 times as many cards as Nancy, so draw three identical boxes to represent Kirk’s cards.

Step 2

Use a model to write an equation. There are 4 equal boxes and each box has n cards.

To get the total numbers of cards, you need to do 4 x n. This amount must equal to 20 cards. So 4 x n = 20

Step 3

Solve the equation to find n.

4 times what number is equal to 20? Since 4 x 5 = 20, n = 5.

Nancy has 5 baseball cards.

Step 4

Find how many cards Kirk has. Kirk has 3 times as many as Nancy or 3 x 5.

Kirk has 15 cards.

Example #5

Kirk has 3 times as many baseball cards as Nancy. Together, they have 20 baseball cards. How many cards does Kirk have?

Compare word problems can be some of the most difficult addition and subtraction problems for students to solve. In this type of problem, students are asked to compare two groups of objects and determine which group has more/less/the same. This can be a difficult concept for students to grasp, but with some careful instruction and practice, they can master it!

Do your students struggle with word problems?  What about those really hard ones: compare word problems?  This is the most difficult addition and subtraction word problem by problem type.

Here are some tips for teaching students to solve compare word problems:

Use Real-World Problems

Teach students how to compare objects by using real-world examples. In our second-grade classroom, we measure our hands and feet and then spend quite a bit of time comparing those measurements.

We attach mathematical language to the activity and help students see how the words more than and less then relate to the measurements. We also show students how those terms can be changed around in the question and answer and the importance of really reading the question.

Model Compare Word Problems

One of the best ways to teach students to solve compare word problems is to model the process for them. Show them how you would approach the problem and think aloud as you work through it. This will help them see that there is a method to solving these types of problems.

Write out a few examples on the board and have students work through them as a class. Then, have students work in small groups or pairs to solve additional problems. Finally, allow students time to work independently on a few problems. By the end of the lesson, students should have had ample opportunity to see and solve compare word problems.

Use Manipulatives

Encourage students to use manipulatives. Many students find it helpful to use manipulatives (e.g., counters, linking cubes, etc.) when solving compare word problems. Manipulatives can help solidify the concept for some students and make the problem-solving process less abstract. 

Compare Word Problems – Facebook Live

I have a great Facebook Live where I show you the differences between the Compare problems types.  You can watch the Facebook Live below or visit this link to see it on Facebook.

Some other Word Problem Resources

• Join Word Problems – Facebook Live
• Separate Word Problems – Facebook Live
• Addition & Subtraction Word Problems by Problem Type – Product Overview
• 5-Tips: How to Teach Students to Solve Word Problems
• The Problem with Using Keywords to Solve Word Problems

Free Word Problems Sample

Do you want a free sample of the word problems I use in my classroom?  Click the link or the image below. FREE Sample of Word Problems by Problem Type

Word Problems Teaching Resources for Your Classroom

Are you teaching word problems and do you need more resources to meet the needs of all of your learners? Check out these word problem resources for your classroom.

More Ideas for Teaching Word Problems

Are you looking for more ideas for teaching word problems to your elementary students? Check out these blog posts.

Problem 1 :

Anna’s recipe for lemonade calls for 2 cups of lemonade concentrate and 3 cups of water. Bailey’s recipe calls for 3 cups of lemonade concentrate and 5 cups of water. Whose recipe makes stronger lemonade ?  How do you know ?

Solution :

Anna’s recipe :

2 cups of lemonade and 3 cups of water.

So, the ratio is 2 : 3.

Let us write equivalent ratios to the ratio 2 : 3.

Bailey’s recipe :

3 cups of lemonade and 5 cups of water.

So, the ratio is 3 : 5.

Let us write equivalent ratios to the ratio 3 : 5.

Find two columns, one in each table, in which the amount of water is the same. Circle those two columns.

From the circled columns, we get two ratios. 

They are, 

10 : 15 and 9 : 15

In these two ratios, the second quantity (water) is same. 

So, we have to compare the first quantity (Lemonade).

The first quantity (10) in the first ratio is more than the first quantity (9) in the second ratio. 

When the amount of water is same (15) in both recipes, Anna’s recipe has more quantity of lemonade concentrate.

Therefore, Anna’s recipe has stronger lemonade. 

Problem 2 :

There are two alloys A and B, both are made up of gold and copper. The ratio between gold and copper in each alloy is given below.

Alloy A (G : C)—-> 2 1/3  :  3  1/3

Alloy B (G : C) —-> 3.6 : 4.8

In which alloy do we have more gold ?

Solution :

To know the alloy in which we have more gold, we have to compare the given two ratios.

To compare two ratios, both the terms of the ratio must be integers. 

Let us convert the terms of the first ratio into integers. 

2 1/3  :  3  1/3  =  (7/3) : (10/3)

2 1/3  :  3  1/3  =  7 : 10 ——> multiplied by 3 

Let us write equivalent ratios to the ratio 7 : 10

Let us convert the terms of the second ratio into integers. 

3.6 : 4.8  =  36 : 48 ——> multiplied by 10 

3.6 : 4.8  =  3 : 4 ——> divided by 12 

Let us write equivalent ratios to the ratio 3 : 4

Find two columns, one in each table, in which the the second term is same. Circle those two columns.

From the circled columns, we get two ratios. 

They are, 

14 : 20 and 15 : 20

In these two ratios, the second quantity (copper) is same. 

So, we have to compare the first quantity (gold).

The first quantity (15) in the second ratio is more than the first quantity (14) in the first ratio. 

When the quantity of copper is same (20) in both the alloys, Alloy A has more quantity of gold. 

Therefore, we have more gold in alloy A. 

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