N.2 Multi-step word problems with multiplication and division
What are multi-step word problems?
Multi-step word problems are math stories that require two or more operations to find the final answer. In fourth grade, these problems often combine multiplication and division with addition or subtraction. Solving them means you must read carefully, decide the correct order of operations, and sometimes find a hidden piece of information before you can answer the final question.
- A simple problem: 3 × 4 = 12 (one step).
- A multi-step problem: A bakery bakes 24 muffins in one batch. They make 5 batches in the morning and sell 38 muffins. How many muffins are left? You must first multiply 24 × 5, then subtract 38.
Always read the problem twice. The first time, find what the question is asking. The second time, look for the numbers and clues that tell you which operations to use.
Understanding multiplication and division in word problems
Multiplication is used when you have equal groups and need to find the total. Look for keywords such as each, per, in all, total, or altogether. Division is used when you know the total and the number of equal groups, and you need to find how many are in each group, or when you need to find how many groups there are. Keywords include each, per, split, share, equal groups, and divided among.
- Multiplication clue: "There are 8 boxes with 12 pencils in each box." → 8 × 12 = 96 pencils total.
- Division clue: "96 pencils are shared equally among 8 students." → 96 ÷ 8 = 12 pencils per student.
- Division clue: "96 pencils are put into bags with 12 pencils in each bag." → 96 ÷ 12 = 8 bags.
In multi-step problems, one operation often gives you a number you need for the next operation. Underline or write down the answer from step one before moving to step two.
Using the four-step plan to solve
A reliable method for solving multi-step word problems is the four-step plan: Understand, Plan, Solve, Check. This strategy helps you organize your thinking and avoid careless mistakes.
A school ordered 15 cases of juice boxes. Each case contains 24 juice boxes. The school gave 120 juice boxes to the kindergarten classes. How many juice boxes remain for the other grades?
- Understand: What do I know? 15 cases, 24 per case, 120 given away. What do I need to find? How many juice boxes are left.
- Plan: First, find the total number of juice boxes (multiplication). Then, subtract the boxes given away (subtraction).
- Solve: Step 1: 15 × 24 = 360 juice boxes total. Step 2: 360 − 120 = 240 juice boxes remaining.
- Check: Does 240 make sense? 240 is less than the total. Add 240 + 120 = 360, which matches the total. The answer is correct.
Do not skip the "check" step. Even a small error in multiplication or division can change your final answer. Use inverse operations to check your work.
Finding hidden questions
Many multi-step word problems contain a hidden question—a piece of information you must figure out before you can answer the final question. Recognizing the hidden question is key to setting up the correct operations.
- Miguel scored 142 points in a video game. His sister scored 3 times as many points. How many more points did Miguel’s sister score than Miguel?
Hidden question: How many points did Miguel’s sister score? (142 × 3 = 426). Then, subtract to find the difference: 426 − 142 = 284 more points. - A farmer has 1,248 eggs. He packs them into cartons that hold 12 eggs each. He sells 84 cartons. How many cartons does he have left?
Hidden question: How many cartons can he fill with 1,248 eggs? (1,248 ÷ 12 = 104 cartons total). Then subtract the sold cartons: 104 − 84 = 20 cartons left.
Ask yourself: “What number do I need to find before I can answer the main question?” If the problem says “twice as many,” “in each,” or “split equally,” those are clues that a hidden calculation is needed.
Multiplication and division with larger numbers
In fourth grade, you will multiply up to 4-digit numbers by 1-digit numbers and multiply two 2-digit numbers. You will also divide up to 4-digit dividends by 1-digit divisors. Multi-step word problems may require these skills in combination.
A theater has 48 rows of seats. Each row has 32 seats. For a special performance, the theater manager wants to set up 1,200 chairs on the main floor in addition to the regular seats. How many total seats will be available for the performance?
- Step 1: Find the regular seats: 48 × 32.
- 48 × 30 = 1,440; 48 × 2 = 96; 1,440 + 96 = 1,536 regular seats.
- Step 2: Add the additional chairs: 1,536 + 1,200 = 2,736 total seats.
When multiplying larger numbers, use the standard algorithm or break apart numbers by place value. Show your work neatly so you can check each step. Remember to line up your place values correctly when adding.
Interpreting remainders in division
Sometimes division problems leave a remainder. In multi-step word problems, you must decide what to do with the remainder. You might need to ignore it, round up to the next whole number, or use it as a fraction or part of the answer.
- Ignore the remainder: “A bus holds 36 students. How many buses will be completely full when there are 150 students?” 150 ÷ 36 = 4 remainder 6. The answer is 4 full buses. (The remainder is not enough for another full bus.)
- Round up the remainder: “How many buses are needed to transport 150 students if each bus holds 36?” 150 ÷ 36 = 4 remainder 6. Since 6 students still need a bus, you need 5 buses total.
- Use the remainder as part of the answer: “A baker has 150 cookies. She puts 36 cookies in each box. How many cookies are left over?” 150 ÷ 36 = 4 remainder 6. The remainder of 6 is the answer.
Read the final question carefully. If it asks “how many groups” and the remainder cannot form a complete group, you may need to round up or ignore the remainder. If it asks “how many are left over,” the remainder is your answer.
Using estimation to check reasonableness
Before solving a multi-step problem, make an estimate to know about what size your answer should be. After solving, compare your exact answer to your estimate. If they are very different, you may have made a mistake.
A factory produces 2,385 toy cars each day. The factory operates for 6 days. The toys are packed into boxes that hold 24 cars each. How many full boxes are packed?
- Estimate: 2,385 rounds to 2,400. 2,400 × 6 = 14,400 total cars. 14,400 ÷ 24 = 600 boxes.
- Exact solve: Step 1: 2,385 × 6 = 14,310 total cars. Step 2: 14,310 ÷ 24 = 596 remainder 6. So, 596 full boxes.
- Check: 596 is close to the estimate of 600, so the answer is reasonable.
Estimation is a powerful check. If your exact answer is far from your estimate, review your multiplication and division calculations. Common errors include forgetting to regroup or misplacing a digit.
Common mistakes and how to avoid them
Even strong math students make errors on multi-step word problems. Knowing the most common mistakes can help you avoid them.
- Performing operations in the wrong order: Always follow the order implied by the problem. If the problem says, “add the cost of three items, then divide by the number of people,” do the addition first, then the division.
- Forgetting to complete all steps: If you stop after the first operation, you will get an incomplete answer. Look for clues like “how many in all” or “how many remain” to know if you need another step.
- Misreading the question: Sometimes students solve for the wrong thing. For example, if the problem asks for “how many more,” you must subtract. If it asks for “total,” you must add or multiply.
- Not labeling the answer: Always include the unit (boxes, miles, dollars, students) so your answer is clear.
Slow down and read each problem word by word. Underline the question and circle the numbers. Write each step clearly, and always label your final answer with the correct unit.
Practice strategy for multi-step problems
To become confident with multi-step word problems, use a consistent practice routine. This builds accuracy and speed over time.
- Read the problem aloud: Hearing the words helps you notice details you might miss when reading silently.
- Restate the question in your own words: “What am I trying to find?”
- Identify the given information: List all numbers and what they represent.
- Decide the order of operations: Write down the steps you will take before you start calculating.
- Solve one step at a time: Write each intermediate answer clearly.
- Check your answer: Use estimation or inverse operations to verify.
Practice with real-world scenarios.For example, multi-step problems appear in shopping, cooking, sports statistics, travel planning, etc. Connecting math to everyday life makes it easier to understand when to multiply and when to divide.
Common Core alignment: CCSS.MATH.CONTENT.4.OA.A.3 – Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.
Notes for teachers
This lesson is aligned with CCSS.MATH.CONTENT.4.OA.A.3. All content is 100% free, use it for whole-class instruction, independent study and practice, or homework.