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Multi-Step
Word Problems

Master the art of breaking down complex word problems into manageable steps using addition, subtraction, multiplication, and division.

ISEE Middle Level

What Is a Multi-Step Word Problem?

A multi-step word problem is a real-world scenario that requires two or more operations to reach the final answer. You cannot solve it in a single calculation: you must plan a sequence of steps.

Single-Step
"Sam has 8 apples and buys 5 more. How many?"
\( 8 + 5 = 13 \)
Multi-Step
"Sam has 8 apples, buys 5 more, then gives away 3. How many?"
\( (8 + 5) - 3 = 10 \)

The Four Operations Toolkit

Every multi-step word problem is built from combinations of these four operations. Recognizing keywords in the problem tells you which operation to use at each step.

\( + \) Addition
total, sum, combined, altogether, more than
\( - \) Subtraction
difference, left over, fewer, remain, less than
\( \times \) Multiplication
each, per, times, every, product, of
\( \div \) Division
split, share equally, per group, quotient, ratio

Why Do Multi-Step Problems Matter?

The ISEE tests your ability to think logically through a chain of reasoning. Multi-step problems appear frequently in the Quantitative Reasoning section because they measure both computation skill and strategic planning.

The R-U-P-S Framework

Use this four-step framework to tackle any multi-step word problem systematically.

R
Read the problem carefully. Identify what is given and what is being asked.
U
Underline key numbers and operation keywords (total, each, left over, etc.).
P
Plan your steps. Write a chain of expressions before calculating.
S
Solve each step in order, then check that your answer makes sense.

Visualizing the Step Chain

Step 1 Identify values Step 2 First operation Step 3 Second operation Answer Check it!

Order of Steps vs. Order of Operations

Students often confuse the order in which steps are described with mathematical order of operations. They are related but different.

Order of Steps Order of Operations
What it means The logical sequence described in the story PEMDAS rules for evaluating one expression
When to use Planning which calculations to do first Within a single step if it has mixed operations
Example "Buy 3 packs, then add loose ones" \( 3 \times 5 + 4 = 19 \)

How to Apply the Framework

1
Read the entire problem once without doing any math. Understand the story.
2
List known quantities and label each with its meaning (e.g., price per item, number of items).
3
Identify the final question. What exactly does the problem ask for?
4
Write the expression(s) step by step. Use parentheses to show grouping.
5
Compute each step. After the final answer, ask: "Does this make sense?"

Example 1: The Bakery Order

A bakery sells muffins for \( \$3 \) each. Maria buys \( 4 \) muffins and pays with a \( \$20 \) bill. How much change does she receive?

Step 1: Total cost
\( 4 \times \$3 = \$12 \)
Step 2: Change
\( \$20 - \$12 = \$8 \)
Answer: \( \$8 \)

Example 2: The Field Trip

A school has \( 126 \) students going on a field trip. Each bus holds \( 30 \) students. Bus tickets cost \( \$45 \) per bus. What is the total cost?

1
How many buses? \( 126 \div 30 = 4.2 \). Since you can't have a partial bus, round up: \( 5 \) buses.
2
Total cost: \( 5 \times \$45 = \$225 \).

Example 3: Saving for a Bike

A bike costs \( \$240 \). Jenna already saved \( \$80 \). She earns \( \$12 \) per hour babysitting. How many hours must she work to buy the bike?

1
Amount still needed: \( \$240 - \$80 = \$160 \).
2
Hours required: \( \$160 \div \$12 = 13.\overline{3} \).
3
Round up (she can't work a fraction of an hour for partial pay in this context): \( 14 \) hours.

Writing One Expression

Sometimes you can combine all steps into a single expression with parentheses. This helps you organize your thinking and avoids losing intermediate results.

Example 1 as a single expression
\( \$20 - (4 \times \$3) \)
Parentheses first Then subtract

Always Check Your Answer

After solving, run a quick reasonableness check. Ask yourself these three questions:

Right units?
If the question asks for dollars, is your answer in dollars? If hours, is it hours?
Right size?
Estimate: is the answer roughly what you'd expect? Too big or too small?
Right sign?
If asking "how much is left," the answer should be positive and less than the starting amount.

Common Mistakes to Avoid

✕ Skipping a Step
Jumping straight to the final operation without computing intermediate values. Always solve step by step.
✕ Wrong Operation
Multiplying when you should divide, or adding when you should subtract. Re-read the keywords carefully.
✕ Forgetting to Round
Getting a decimal answer like \( 4.2 \) buses and leaving it. Real-world context demands rounding up or down.

Strategy Summary

Strategy When to Use Example Cue
Multiply then subtract Buying items, making change "buys ... pays with"
Divide then round up Filling containers, seating "how many buses needed"
Subtract then divide Saving toward a goal "already has ... earns per"
Add then multiply Combining groups, scaling "combined ... times as many"
Chain of mixed ops Complex real-world scenarios Three or more quantities

Question 1 of 5

A store sells notebooks for \( \$4 \) each and pens for \( \$2 \) each. Liam buys \( 3 \) notebooks and \( 5 \) pens. How much does he spend in total?

Question 2 of 5

A farmer has \( 200 \) oranges. He packs them into boxes of \( 12 \). How many boxes does he need to pack all the oranges?

Question 3 of 5

Emma earns \( \$9 \) per hour. She works \( 6 \) hours on Saturday and \( 4 \) hours on Sunday. She then spends \( \$25 \) on a book. How much money does she have left?

Question 4 of 5

A rectangle has a length of \( 15 \) cm and a width of \( 8 \) cm. A square has the same perimeter as the rectangle. What is the side length of the square?

Question 5 of 5

A theater has \( 25 \) rows with \( 18 \) seats per row. For a show, \( 372 \) tickets were sold. How many seats are empty?