Two-Step Word Problems Using the Four Operations - Third Grade Mathematics

Real-life mathematical challenges rarely come labeled with a single neat arithmetic sign; instead, they present complex situations that combine addition, subtraction, multiplication, and division in different combinations. Solving two-step word problems is the crowning achievement of third grade mathematics, demonstrating that you can analyze a story, discover hidden questions, and apply the correct operations in the right order. In this chapter, you will master the strategic tools needed to conquer any two-step problem with four operations.

The Secret to Two-Step Problems: The Hidden Question

In every two-step problem, there is an unstated, intermediate question that must be solved before you can answer the main question.

Story Problem:
"Maya bought 3 packs of stickers with 8 stickers in each pack.
She gave 5 stickers to her brother.
How many stickers does Maya have left?"

Hidden Question:
How many stickers did Maya have in total BEFORE giving any away?
Calculation 1: 3 x 8 = 24 stickers.

Main Question:
How many stickers does Maya have now?
Calculation 2: 24 - 5 = 19 stickers left!

If you jump straight to subtracting 5 without first multiplying 3 x 8, you cannot solve the problem. Finding the hidden question is your master key!

The Four Operation Combinations

Two-step problems can pair operations in many exciting combinations:

+-----------------------+-----------------------------------+-------------------+
| Operation Pairing     | Example Situation                 | Number Sentence   |
+-----------------------+-----------------------------------+-------------------+
| Multiply then Add     | Bought 4 boxes of 6, found 3 more | (4 x 6) + 3 = 27  |
| Multiply then Subtract| Bought 4 boxes of 6, gave 5 away  | (4 x 6) - 5 = 19  |
| Add then Divide       | Earned 14 and 16 dollars, split 3 | (14 + 16) ÷ 3 = 10|
| Subtract then Divide  | Had 30 cookies, ate 6, shared by 4| (30 - 6) ÷ 4 = 6  |
| Divide then Add       | Split 20 by 4, bought 7 more      | (20 ÷ 4) + 7 = 12 |
+-----------------------+-----------------------------------+-------------------+

The Four-Phase Problem-Solving Process

To keep your thinking structured and confident, follow these four phases:

Phase 1: Understand

Read the entire problem carefully. Ask yourself:
- What is happening in the story?

- What numbers am I given?

- What is the question asking me to find?


Phase 2: Plan

  • What do I need to figure out first? (Identify the hidden question).
  • Which operation does the hidden question require?
  • Which operation does the second step require?
  • Write a plan using words or a tape diagram!

Phase 3: Solve

Perform Step 1 carefully. Then take that result and use it in Step 2.

Phase 4: Check

Does your answer make sense in the context of the story? If you started with 3 packs of stickers and gave some away, an answer of 19 makes sense (less than 24, but more than 0). If you had calculated 120, a quick sanity check would catch the mistake immediately!

Worked Examples

Let us dissect two classic multi-operation word problems.

Example 1: Add Then Divide

Story: Liam had 18 blue marbles and 14 green marbles. He put all his marbles together and divided them equally into 4 pouches. How many marbles went into each pouch?

Step 1 (Hidden Question): How many marbles does Liam have in total? 18 + 14 = 32 marbles in all.

Step 2 (Main Question): How many marbles in each pouch? 32 ÷ 4 = 8 marbles in each pouch.

Complete Equation: (18 + 14) ÷ 4 = 32 ÷ 4 = 8 marbles.

Example 2: Multiply Then Subtract

Story: A baker baked 5 trays of muffins with 6 muffins on each tray. At breakfast, customers purchased 18 muffins. How many muffins were left in the bakery?

Step 1 (Hidden Question): How many muffins were baked altogether? 5 x 6 = 30 muffins.

Step 2 (Main Question): How many muffins remain? 30 - 18 = 12 muffins remaining.

Complete Equation: (5 x 6) - 18 = 30 - 18 = 12 muffins.

Chapter Practice Exercises

  1. Identify the hidden question and solve: a. Sophia had 4 bags with 5 apples in each bag. She used 8 apples to bake an apple pie. How many apples does Sophia have left? b. Noah earned 15 dollars on Saturday and 25 dollars on Sunday. He spent all the money equally on 5 new books. How much did each book cost? c. A farmer had 36 eggs. He kept 6 eggs for himself and packed the rest equally into cartons of 6. How many cartons did he fill? d. A toy store had 3 boxes of action figures with 8 figures in each box. The store received a delivery of 12 more figures. How many figures does the store have now?
  2. Write a two-step word problem that matches the equation: (4 x 6) - 10 = 14.
  3. Write a two-step word problem that matches the equation: (20 + 15) ÷ 5 = 7.
  4. Marcus has 40 dollars. He buys 3 comic books that cost 6 dollars each. How much change does Marcus receive?
  5. A school bus had 24 students. At the first stop, 8 students got off. At the second stop, the remaining students split equally into 2 groups to walk into two different buildings. How many students were in each group?

Solutions and Step-by-Step Answers

  1. Solutions: a. Hidden question: How many apples in total? 4 x 5 = 20 apples. Main step: 20 - 8 = 12 apples left. b. Hidden question: How much total money earned? 15 + 25 = 40 dollars. Main step: 40 ÷ 5 = 8 dollars per book. c. Hidden question: How many eggs to pack? 36 - 6 = 30 eggs. Main step: 30 ÷ 6 = 5 cartons filled. d. Hidden question: How many figures originally? 3 x 8 = 24 figures. Main step: 24 + 12 = 36 action figures in all.
  2. Sample word problem for (4 x 6) - 10 = 14: "A florist has 4 vases with 6 roses in each vase. She sells 10 roses to a customer. How many roses does the florist have left? (24 - 10 = 14 roses)."
  3. Sample word problem for (20 + 15) ÷ 5 = 7: "Zoe collected 20 sea glass pieces on Friday and 15 pieces on Saturday. She divided all the pieces equally among 5 craft jars. How many pieces did she put in each jar? (35 ÷ 5 = 7 pieces)."
  4. Marcus's comic purchase:
  5. Step 1: Cost of 3 comics = 3 x 6 = 18 dollars.
  6. Step 2: Change remaining = 40 - 18 = 22 dollars.
  7. School bus students:
  8. Step 1: Students remaining after first stop = 24 - 8 = 16 students.
  9. Step 2: Split into 2 groups = 16 ÷ 2 = 8 students in each group.