1.5: Word Problems

Try Different Numbers

Word problems can be more difficult than non-word problems because they require reading comprehension in addition to mathematical comprehension. Sometimes simply having to do both at the same time can make things more difficult than expected. One useful strategy for solving word problems is to try a similar problem with "nicer" numbers so that the mathematical comprehension is easier. Then once the easier version is solved, the learner can analyze what they did and use that as a model for the original version. Consider the problem below.

Figure 2.3.1: A hot dog in a bun with mustard

Rosie ate \(2 \dfrac{3}{4}\) hot dogs. Edward ate \(1 \dfrac{1}{2}\) hot dogs. What part of all the hot dogs together did Rosie eat by herself?

Example \(\PageIndex{1}\)

Come up with an easier version of the hot dog problem. Then solve that problem.

Solution

Easier Version: Rosie ate five hot dogs. Edward ate three hot dogs. What part of all the hot dogs together did Rosie eat by herself.

Solution to Easier Version: Together, they ate 5+3 = 8 hot dogs. Rosie ate 5 out of 8 hot dogs, so she ate 5/8 of the hot dogs.

Now, mimic the same process for the easier version to solve the original hot dog problem that includes fractions.

Try Different words

Another useful strategy for solving word problems is to try rewriting the problem with less complicated sentences and simpler names to make the reading comprehension is easier. You may also want to get rid of some abbreviations (such as changing "mph" to "miles/hour", which is done below). Consider the problem below.

Figure 2.3.3: Train

Cara is riding in the South Train leaving Los Angeles and heading towards San Diego. Madeline is riding in the North Train leaving San Diego and heading towards Los Angeles. The South Train is traveling 84 mph while the North Train is traveling 92 mph. The distance between the two cities is 132 miles. Assuming the trains leave at the exact same time, how long will it take them to meet (crossing harmlessly on different rails)?

Example \(\PageIndex{1}\)

Rewrite the train problem to make it more straightforward.

Solution

Problem Rewrite: The South Train leaves point A towards point B. The North Train leaves point B towards point A. The South Train's speed is 84 miles/hour. The North Train's speed is 92 miles/hour. The two points are 132 miles apart. Assuming the trains leave at the exact same time, how long will it take them to meet?

Now we can take the information from that and draw a picture more easily. On a sheet of paper, draw two points, labeled A and B, and a line segment between them. Before drawing a point on that line segment to indicate where the trains meet, let's see if we can think ahead to make our picture more accurate. Will they meet closer to point A or point B? Another way of asking the question is, "Will the South Train have traveled more or less than the North Train?" Think about your answer to this question. Then read on.

Since the South Train's speed is less than the North Train's speed, the North Train will have traveled a longer distance in the amount of time it takes them to meet. So, this means the point where they meet should show a distance between that point and point B as greater than the distance between that point and point A. In other words, the meeting point should be closer to point A. Draw that meeting point.

Label the length of the line segment (from A to B) as representing 132 miles. We do not know how long it will take them to meet. That's the question we are trying to answer! So, let's give that time a name. How about \(t\) hours? How do speed, distance, and time relate to each other? You have learned that Distance = Rate x Time and that speed is a type of rate. But even if you don't remember that, you might notice that in this problem, the units of distance are miles, the units of speed are miles/hour, and the units of time are hours. So, using what we know about fraction multiplication, we can see that miles/hour x hours = miles/hour x hours/1 = miles. In other words, Speed x Time = Distance!

Since the South Train's distance traveled is the part of the line between point A and the meeting point, we can use its speed, 84 miles/hour, and the unknown elapsed time, \(t\) hours, to label that part of the figure. Do that. Now use the same idea to label the part of the line between the meeting point and point B. Your drawing with labels should look something like the diagram below (yours might have a different orientation, possibly representing north and south).

Figure \(\PageIndex{1}\): One possible diagram for the train problem (CC BY 4.0; Melissa Flora)

Now, use the diagram as your guide to set up an equation and solve for \(t\) to find how long after the trains leave they meet.

Try Diagrams Even Without a Story Context

People (especially children) tend to only want to draw pictures when solving problems if the problem has a visual component or some part of a narrative that is naturally visualized. But we need not limit ourselves this way! In the problem below, there is nothing that is naturally visual about it. But we can give the abstract concept of two numbers visual representations anyway.

One number is four times as large as another number, and their sum is 5285. What are the two numbers?

Example \(\PageIndex{1}\)

Draw a picture representing the number problem that two elements: one representing the larger number and one representing the smaller number. Make the drawing so that the relationship between the two numbers is reflected.

Solution

For this problem, we can use strips, squares, or anything else where we can represent the smaller number as one "thing" and the larger number as a "thing" or "things" 4 times larger. One example is below.

Figure \(\PageIndex{1}\): The two numbers (CC BY 4.0; Melissa Flora)

Since the sum of the two numbers is 5285, this means 5 squares (using the diagram in the example) represents 5285. Use this to determine how much one square represents. Then use that to determine the values of the smaller and larger numbers. 

Practice Problems

Try using one or more of the strategies described above to solve these practice problems. If none of them seem to help, what strategy or strategies do work? 

1. Grade 1: Some bicycles and tricycles are on the playground. There are seven seats and nineteen wheels. How many bicycles are there? How many tricycles are there? (Solve without using Algebra.)
2. Grade 2: Pam has three quarters. She wants to buy candy, which costs 54¢. How much change will she have after she buys the candy?
3. Grade 3: It takes a rocket 8 seconds to travel one mile. How long will it take the rocket to travel 7 miles?
4. Grade 4: Irwin decided to create designs using his cereal. In total, he created nine designs and used 63 pieces of cereal. How many cereal pieces were used in each design? Was there an equal number of pieces in each design?
5. Grade 5: Leo has 10 animals at his ranch, which are turtles, bunnies, and chickens. If there were 5 turtles, and 3 bunnies, what fraction of the animals are chickens?
6. Grade 6: Ellen wants to build a rectangular fenced area in her yard that is 10 meters by 8 meters. The fence will be made of glass panes between posts. The hardware store owner suggests installing a post every 2 meters. The posts cost $3.42 each. Ellen will also need to buy a glass pane to place between each pair of posts. The glass panes cost $50.84 per meter long. She also needs to buy one gate for $31.99. How much do all of the materials cost before tax?