Multi-Step Whole Number Word Problems: Fifth Grade Mathematics

Solving multi-step whole number word problems requires dissecting the narrative, organizing hidden intermediate questions, and checking calculations against reasonable estimates.

The Strategy for Multi-Step Problems

Multi-step problems contain two or more mathematical operations embedded within a scenario. Success depends on identifying the hidden question that must be answered before solving the main question.

+--------------------------------------------------------------------------+
|                  MULTI-STEP PROBLEM SOLVING FRAMEWORK                    |
|                                                                          |
|   1. Read the entire story twice to understand the situation.            |
|   2. Circle the question being asked and state what units are needed.    |
|   3. Uncover the hidden question (the intermediate value required).       |
|   4. Write an equation with a letter (variable) for unknown values.       |
|   5. Solve each step sequentially.                                       |
|   6. Check reasonableness against an initial estimate.                   |
+--------------------------------------------------------------------------+

Visualizing with Tape Diagrams (Bar Models)

Tape diagrams illustrate part-whole and comparison relationships clearly, preventing you from applying the wrong operations.

Tape Diagram for Total Fundraiser Earnings:
  Ticket sales:  [  $450  ]
  Bake sale:     [  $320  ]
  Expenses:      [ -$180  ]

  Net Profit:    [  $450  ] + [  $320  ] - [  $180  ] = $590

Worked Multi-Step Example

Consider this scenario: A school garden club sold 45 decorative plant boxes for $18 each. They spent $140 on seeds and soil. With the remaining money, they purchased 8 garden tool sets of equal price. How much did each tool set cost?

Step 1: Find total revenue from plant boxes (Hidden Question 1):
        45 * 18 = 810 dollars.

Step 2: Subtract expenses to find remaining funds (Hidden Question 2):
        810 - 140 = 670 dollars.

Step 3: Divide remaining funds by number of tool sets (Main Question):
        670 / 8 = 83 R6.
        Expressed as money: 670.00 / 8 = $83.75 per set.

Writing a Single Comprehensive Expression

In fifth grade, you can represent an entire multi-step scenario with one numerical expression using grouping symbols: Total cost per tool set = [(45 * 18) - 140] / 8. This connects word problems directly to the order of operations.

Exercises and Solutions

Exercise 1: Three-Step Inventory Problem. A bookstore orders 16 boxes of hardcovers with 24 books in each box, and 25 boxes of paperbacks with 32 books in each box. Over the weekend, the store sells 420 books. How many books remain in the store's inventory?

Solution 1: Identify the hidden questions: first, find total hardcover books; second, find total paperback books; third, find combined inventory; fourth, subtract books sold. Step 1: 16 * 24 = 384 hardcovers. Step 2: 25 * 32 = 800 paperbacks. Step 3: Combine both totals: 384 + 800 = 1,184 books. Step 4: Subtract books sold: 1,184 - 420 = 764 books remain in the store's inventory.

Exercise 2: Comparison and Division Scenario. Marcus earned $480 mowing lawns over the summer. His sister Elena earned twice as much as Marcus. They decided to pool their earnings together, donate $150 to an animal shelter, and split the remaining money equally between the two of them. How much money did each person take home?

Solution 2: Step 1 (Hidden Question 1): Calculate Elena's earnings: 2 * 480 = $960. Step 2 (Hidden Question 2): Find their pooled earnings: 480 + 960 = $1,440. Step 3 (Hidden Question 3): Subtract the donation: 1,440 - 150 = $1,290 remaining. Step 4 (Main Question): Divide the remainder equally between the two siblings: 1,290 / 2 = $645. Each person took home $645.