Multi-Step Word Problems | Fifth Grade
Complete Notes & Formulas
1. Write Numerical Expressions for Word Problems
Definition: Convert word problems into numerical expressions (without solving them) by identifying numbers, operations, and the order in which operations should be performed.
📝 Steps to Write Expressions:
- Step 1: Read the problem carefully
- Step 2: Identify all numbers
- Step 3: Find key words for operations
- Step 4: Determine the order of operations
- Step 5: Write the expression using parentheses if needed
✏️ Example:
Problem: Sarah bought 3 packs of pencils with 12 pencils in each pack. She gave away 8 pencils. Write an expression for how many pencils she has left.
Solution:
• Total pencils = 3 packs × 12 pencils each
• Then subtract 8 pencils given away
Expression: (3 × 12) - 8
2. Use Numerical Expressions to Solve Multi-Step Word Problems
Definition: Write the expression AND evaluate it to find the answer using order of operations (PEMDAS).
📐 Process:
Write Expression → Evaluate Using PEMDAS → Answer with Units
✏️ Example:
Problem: A bakery made 240 cookies. They packed them in boxes of 12. They sold 15 boxes. How many boxes are left?
Solution:
Step 1: Write expression: (240 ÷ 12) - 15
Step 2: Evaluate parentheses: 240 ÷ 12 = 20
Step 3: Subtract: 20 - 15 = 5
Answer: 5 boxes are left
3. Represent Multi-Step Problems Using Equations
Definition: Write an equation (with an equal sign) to represent the relationship between the known and unknown quantities in a problem.
🔑 Equation vs Expression:
| Type | Has Equal Sign? | Example |
|---|---|---|
| Expression | NO | (5 × 3) + 2 |
| Equation | YES | (5 × 3) + 2 = 17 |
✏️ Example:
Problem: Tom has $50. He buys 3 books for $12 each. How much money does he have left?
Solution:
Let m = money left
Equation: 50 - (3 × 12) = m
50 - 36 = m
m = 14
Answer: Tom has $14 left
4. Use Equations with Unknown Numbers to Solve Multi-Step Word Problems
Definition: Use a variable (letter) to represent the unknown quantity, write an equation, and solve to find the value of the variable.
📝 Steps to Solve:
- Step 1: Identify what you need to find (the unknown)
- Step 2: Let a variable represent the unknown (x, n, etc.)
- Step 3: Write an equation showing the relationship
- Step 4: Solve the equation
- Step 5: Check your answer
✏️ Example:
Problem: Maria multiplied a number by 4 and then added 15. The result was 47. What was the original number?
Solution:
Let n = the original number
Equation: (n × 4) + 15 = 47
4n + 15 = 47
4n = 47 - 15
4n = 32
n = 32 ÷ 4
n = 8
Answer: The original number was 8
5. Multi-Step Word Problems
Definition: Problems that require two or more operations to solve. They often involve real-world situations with multiple pieces of information.
🔑 Problem-Solving Strategy:
- Read - Read the problem carefully
- Understand - What is being asked?
- Plan - What operations are needed?
- Solve - Do the calculations step by step
- Check - Does the answer make sense?
✏️ Example:
Problem: A school library has 1,250 books. They bought 345 new books and donated 180 old books. How many books does the library have now?
Solution:
Step 1: Start with original: 1,250 books
Step 2: Add new books: 1,250 + 345 = 1,595
Step 3: Subtract donated: 1,595 - 180 = 1,415
Answer: 1,415 books
6. Multi-Step Word Problems Involving Remainders
Definition: Problems where division results in a remainder, and you must interpret what to do with the remainder based on the context.
📐 Three Ways to Interpret Remainders:
1. Round Up:
When you need enough for everyone
Example: 47 students, 6 per van → Need 8 vans (not 7 R5)
2. Drop the Remainder:
When partial amounts can't be used
Example: 50 apples, 8 per bag → Can fill 6 bags (2 left over)
3. Remainder is the Answer:
When the leftover is what's asked
Example: 50 apples, 8 per bag → How many left? Answer: 2
✏️ Example:
Problem: A factory makes 1,000 toys. They pack 48 toys in each box. How many boxes can they completely fill? How many toys are left over?
Solution:
1,000 ÷ 48 = 20 R40
Answer: 20 boxes can be filled; 40 toys left over
7. Multi-Step Word Problems: Identify Reasonable Answers
Definition: Determine if an answer makes sense in the context of the problem by estimating or using logical reasoning.
🔑 Strategies to Check Reasonableness:
- Estimate first - Round numbers to get approximate answer
- Check units - Does the answer have correct units?
- Compare to known values - Is it too big or too small?
- Think about real life - Does it make sense?
- Work backwards - Plug answer back into problem
✏️ Example:
Problem: A bicycle costs $185. Jake saved $45 each month for 4 months. Does he have enough to buy the bike?
Solution:
Estimate: $45 ≈ $50, so 50 × 4 = $200
Exact: $45 × 4 = $180
Compare: $180 < $185
Answer: No, he needs $5 more
8. Multi-Step Word Problems: Multiplicative Comparison
Definition: Problems where one quantity is described as "times as many" as another quantity. Use multiplication or division to compare.
🔑 Key Phrases:
| Phrase | Meaning | Operation |
|---|---|---|
| "3 times as many" | Multiply by 3 | × 3 |
| "5 times as much" | Multiply by 5 | × 5 |
| "How many times" | Divide to compare | ÷ |
✏️ Examples:
Example 1: Lisa has 8 stickers. Maya has 4 times as many stickers as Lisa. Then Maya gives away 12 stickers. How many stickers does Maya have now?
Solution:
Step 1: Maya's stickers = 8 × 4 = 32
Step 2: After giving away: 32 - 12 = 20
Answer: Maya has 20 stickers
Example 2: A restaurant served 120 customers on Monday and 480 customers on Friday. How many times as many customers did they serve on Friday?
Solution:
480 ÷ 120 = 4
Answer: 4 times as many
Multi-Step Problem-Solving Guide
📝 5-Step Problem-Solving Process:
1
Read Carefully
2
Identify Info
3
Write Expression
4
Solve Step-by-Step
5
Check Answer
| Concept | Key Points |
|---|---|
| Expression | Numbers + operations (NO equal sign) |
| Equation | Expression = value (HAS equal sign) |
| Variable | Letter representing unknown number |
| Remainder | What's left after division - interpret based on context |
| Multiplicative Comparison | "Times as many/much" - use multiplication |
🔑 Key Tips for Success:
- Always read the problem twice before starting
- Underline or highlight important numbers and key words
- Draw pictures or diagrams if helpful
- Use parentheses to show which operations come first
- Include units in your final answer
- Check: Does your answer make sense?
📚 Fifth Grade Multi-Step Word Problems - Complete Study Guide
Master these concepts for math excellence! ✨