Chapter 5 · Numerical expressions
Represent and Solve Multi-Step Word Problems
Turn each step of a story problem into an equation, combine the steps into one expression, and check that your answer and its unit fit the question.
Some story problems hide a question inside them. Before you can answer what is asked, you have to find a number the story never gives you.
A multi-step word problem is a story problem that takes two or more steps to answer. Each step is one operation: adding, subtracting, multiplying or dividing.
In this lesson you will write each step with numbers and symbols. Then you will put all the steps on one line, and you will make sure your answer fits the question.
Read this story.
Mia buys 4 packs of stickers. Each pack has 12 stickers. She gives 9 stickers to her brother. How many stickers does Mia have left?
You cannot take away the 9 yet. First you need the hidden number: how many stickers Mia bought. So this story has two steps.
An equation is a number sentence with an equals sign. It says that the two sides are the same amount. You can write one equation for each step. The table shows each step in words and in symbols.
| Step | What the story says | Equation | What the result means |
|---|---|---|---|
1 | Mia buys 4 packs with 12 stickers in each pack. | Mia buys 48 stickers. | |
2 | She gives 9 of her 48 stickers to her brother. | Mia has 39 stickers left. |
Look at the two equations. The 48 you find in step 1 is the number you start with in step 2. That link lets you tell the whole story on one line.
You already know that a numerical expression is made of numbers and operation signs such as and . It can also have parentheses, the curved marks ( ) that group part of it. It has no equals sign.
In step 2, write in place of the 48. You can do this because they are the same amount: both are the stickers Mia bought. Now you have one expression for the whole story: .
To evaluate an expression is to work out the one number it is equal to. That number is its value. You already know the order of operations for doing this:
- First, work inside parentheses.
- Next, multiply and divide from left to right.
- Last, add and subtract from left to right.
The expression has no parentheses, so you multiply first: . Then you subtract: . That is the same order as the story.
Write the value after an equals sign, and you have one equation for the whole problem.
Worked example
A step that has to come first
Ms. Lee fills gift bags. She puts 4 pencils and 6 erasers in each bag. She fills 3 bags. How many items does she use in all?
Find the hidden number first: how many items go in one bag. The pencils and the erasers go into the bag together, so add them.
Each of the 3 bags holds 10 items. The bags are equal groups, so multiply 10 by 3.
Write one expression for the whole story. Put 4 + 6 in place of the 10, because they are the same amount. The story adds first. Without parentheses, the order of operations would multiply 6 by 3 first. So put parentheses around 4 + 6 to make the adding happen first.
Evaluate the expression. Work inside the parentheses first, then multiply. The value matches the two step equations.
Answer
Ms. Lee uses items in all.
Key concept
One expression can tell the whole story
Each step of a multi-step problem is one operation. You can write an equation for each step, or you can put all the steps into one numerical expression. When both match the story, they give the same value.
The expression must do the steps in the order the story needs. The order of operations decides which step is done first. So when the story needs a step done earlier than the order of operations would do it, put that step in parentheses.
In a word problem, a number by itself is not a full answer. Your answer needs its unit. The unit is the word that tells what the number counts or measures, such as stickers, pages or minutes.
Then check that you answered the question that was asked. Read the question one more time and ask yourself:
- Does my number count the thing the question asks about?
- Did I do every step, or did I stop at a number I found along the way?
Think about the sticker story. The number 48 is true: Mia did buy 48 stickers. But the question asks how many stickers she has left. The answer to that question is 39 stickers.
Worked example
Three steps: do not stop too soon
A baker makes 5 trays of muffins. Each tray holds 12 muffins. She sells 28 muffins. Then she packs the rest equally into 4 boxes. How many muffins go in each box?
Find how many muffins she bakes. There are 5 equal trays of 12 muffins, so multiply.
Find how many muffins are left. Selling 28 takes them away from the 60, so subtract.
Pack the 32 muffins that are left into 4 equal boxes. Sharing equally means dividing.
Write one expression and evaluate it. You must find the muffins left before you share them, so put that part in parentheses. Without the parentheses, the division would come before the subtraction and would divide only the 28. Inside the parentheses, multiply first and then subtract. Then divide.
Check the question. It asks how many muffins go in each box. The 60 counts muffins baked and the 32 counts muffins left. You found both along the way, so neither one is the answer. The 8 counts the muffins in one box.
Answer
Each box holds muffins.
Worked example
Write the expression straight from the story
A notebook has 200 pages. Ravi fills 25 pages each week for 4 weeks. Then he fills 20 pages each week for 3 weeks. How many pages are still blank?
Plan before you write. To find the blank pages, start with all 200 pages and take away the pages Ravi fills. He fills pages in two parts: 4 weeks at 25 pages a week, and 3 weeks at 20 pages a week.
Write the pages he fills. Each part is equal groups of pages, so each part is a multiplication. He fills both parts, so join them by adding.
Take all the filled pages away from the 200 pages. Put the filled pages in parentheses so that they become one amount you find first. Without the parentheses, you would subtract only the first part and then add the second part.
Evaluate. Inside the parentheses, do both multiplications and then add. Then subtract from 200.
Check the question. It asks for blank pages. The 160 counts pages Ravi filled, so it is not the answer. The 40 counts blank pages.
Answer
The notebook still has blank pages.
Remember: a multi-step problem is a chain of steps, and each step is one operation. You can write one equation for each step, or one expression for the whole story. Put a step in parentheses when the story needs it done earlier than the order of operations would do it. Finish with the unit, and read the question again to be sure your number answers it.
Next, you will solve multi-step problems where one amount is a number of times as many as another. You will also use a letter for the unknown number in an equation.