Solve multi-step word problems using the four operations. - ISEE Lower Level Quantitative Reasoning

SECTION 1

Why Do We Solve Word Problems?

People have been solving math problems in real life for thousands of years! Long before calculators existed, word problems helped people figure out everyday things. Think about it — when you buy snacks at a store, you are solving a word problem!

Math has always been about solving real-life puzzles. Let's look at how word problems have been important throughout history.

1650 BC • Ancient Egypt

Egyptians wrote math problems on paper made from plants called papyrus. They used word problems to divide bread and grain fairly.

300 BC • Ancient Greece

Greek mathematicians solved word problems about distances, areas, and trade. They loved puzzles that required multiple steps!

800 AD • Medieval Math

Scholars in the Middle East created algebra. They wrote problems as stories and solved them step by step — just like we do today!

Today • The ISEE and Beyond

Tests like the ISEE use word problems to see how well you can think through real situations. These skills help you every single day!

Here is the big question: When a problem needs more than one step to solve, how do you figure out what to do first? That is exactly what this lesson will teach you!

SECTION 2

The Four Building Blocks

Every multi-step word problem uses a mix of the four operations (that means addition, subtraction, multiplication, and division). Think of them as your four superpowers. Each one does a different job.

Addition (+)

Use addition when you are putting things together or finding a total. Clue words: "in all," "total," "combined," "altogether."

Subtraction (−)

Use subtraction when you are taking away, finding a difference, or seeing how much is left. Clue words: "left," "fewer," "difference," "remain."

Multiplication (×)

Use multiplication when you have equal groups. Clue words: "each," "every," "per," "times."

Division (÷)

Use division when you are splitting into equal groups or sharing. Clue words: "split," "shared equally," "divided," "each gets."

✦

KEY TAKEAWAY

Think of a word problem like building with LEGO bricks. Each step is one brick. You snap them together one at a time until your answer is complete. Don't try to do everything at once — solve one step, then use that answer in the next step.

On the ISEE, you will see word problems that need two or even three steps. The trick is to read carefully, find the clue words, and figure out which operations to use in which order. You've got this!

SECTION 3

The Step-by-Step Strategy

Here is a simple plan you can follow every time you see a multi-step word problem. We call it the R.U.N.S. strategy. The diagram below shows you how it works!

The R.U.N.S. strategy helps you stay organized. Read, Underline, write a Number sentence, then Solve and check. Notice how the example at the bottom uses two operations: multiplication first, then subtraction.

Look at the example in the diagram. Sara's problem needed two steps. First, you found the total with multiplication. Then you subtracted. If you tried to do it all at once, you might get confused. Breaking it into steps makes it easy!

SECTION 4

How the Operations Work Together

Multi-step problems often combine two or three operations. Here are the most common patterns you will see on the ISEE. Knowing these patterns helps you work faster!

PATTERN 1: MULTIPLY THEN ADD

Total = (group size × number of groups) + extra

Example: You buy 4 packs of 6 stickers, then your friend gives you 3 more. Total = (4 × 6) + 3 = 24 + 3 = 27 stickers.

PATTERN 2: MULTIPLY THEN SUBTRACT

Remaining = (group size × number of groups) − amount removed

Example: You have 5 bags of 10 marbles, and you lose 8. Remaining = (5 × 10) − 8 = 50 − 8 = 42 marbles.

PATTERN 3: ADD THEN DIVIDE

Each person's share = (amount A + amount B) ÷ number of people

Example: You have 12 cookies and bake 8 more, then split them equally among 4 friends. Each gets (12 + 8) ÷ 4 = 20 ÷ 4 = 5 cookies.

PATTERN 4: SUBTRACT THEN DIVIDE

Each share = (starting amount − amount used) ÷ number of groups

Example: You have 30 cards, give away 6, then split the rest equally among 3 friends. Each gets (30 − 6) ÷ 3 = 24 ÷ 3 = 8 cards.

💡

ISEE Tip!

On the ISEE, there is no penalty for guessing. If you are stuck, eliminate answer choices that are way too big or way too small, then pick from what is left. You can often cross out 1 or 2 choices quickly by estimating!

SECTION 5

Clue Words That Tell You Which Operation to Use

Word problems are like little mysteries. The clue words hidden inside the sentences tell you exactly which operation to use. Learning to spot these clues is one of the biggest keys to doing well on the ISEE!

Use this guide like a detective's cheat sheet! When you see clue words like "each" and "total" in the same problem, that tells you to multiply first, then maybe add or subtract.

⚠

Be Careful!

Some clue words can mean different things depending on the sentence. For example, "how many more" can mean addition OR subtraction. Always read the whole problem before deciding which operation to use.

SECTION 6

Worked Example: Step by Step

Let's solve a real ISEE-style problem together. Follow along and notice how we use the R.U.N.S. strategy!

ℹ

Problem

Ms. Chen bought 6 bags of apples. Each bag had 8 apples. She used 15 apples to make pies and gave 9 apples to her neighbor. How many apples does Ms. Chen have left?

Solution Using R.U.N.S.

1
Step 1 — Read and Underline

Read the whole problem. Underline the important numbers: 6 bags, 8 apples each, used 15, gave 9. The question asks: how many are left? The word "left" is a subtraction clue.
2
Step 2 — Find the Total (Multiply)

"6 bags with 8 apples each" means equal groups. That's multiplication! Multiply: 6 × 8 = 48. Ms. Chen started with 48 apples.

6 × 8 = 48 apples
3
Step 3 — Subtract the Apples Used for Pies

She "used" 15 apples. "Used" means they are gone, so subtract: 48 − 15 = 33 apples remain after making pies.

48 − 15 = 33 apples
4
Step 4 — Subtract the Apples Given Away

She "gave" 9 apples to her neighbor. More are leaving, so subtract again: 33 − 9 = 24 apples.

33 − 9 = 24 apples
5
Step 5 — Check Your Answer

Does 24 make sense? She started with 48 and gave away 15 + 9 = 24 total. 48 − 24 = 24. ✓ It checks out!

Answer: 24 apples

See how we used three steps? First multiplication, then two subtractions. Each step was simple on its own. That's the power of breaking it down!

SECTION 7

Common Mistakes and How to Avoid Them

Even smart students make mistakes on multi-step problems. Here are the most common traps — and how to dodge them!

Common Mistake	Why It Happens	How to Fix It
Stopping too early (only doing one step)	You solve the first part and think you are done	Re-read the question at the end. Does your answer match what was asked?
Using the wrong operation	You add when you should subtract, or multiply when you should divide	Circle clue words. Ask: Am I putting together or taking away? Equal groups or sharing?
Doing steps in the wrong order	You rush and mix up what comes first	Write each step on a new line. Number them 1, 2, 3.
Forgetting a number from the problem	You miss a number that was mentioned in the problem	Underline every number in the problem before you start solving.

The four most common mistakes on multi-step ISEE word problems

✦

KEY TAKEAWAY

Think of a multi-step problem like following a recipe. If you skip a step or mix them up, the cookies won't taste right! Always re-read the question after you find your answer to make sure you answered what was actually asked.

SECTION 8

Building Toward Harder Math

The skills you are learning right now are the same skills you will use in harder math later! Let's see how multi-step word problems connect to what comes next.

What You Learn Now	What It Becomes Later
Finding a total, then subtracting	Solving equations with two operations (algebra!)
Picking the right operation from clue words	Translating word problems into algebraic expressions
Breaking big problems into small steps	Solving multi-step equations and proofs
Checking if your answer makes sense	Verifying solutions and estimating reasonableness

How today's skills build toward advanced math

Every time you solve a multi-step word problem, you are training your brain to think like a mathematician. The ISEE tests this because schools want to see that you can think through problems logically — not just memorize facts. Keep practicing and you will be ready for anything!

SECTION 9

Practice Problems

Time to put your skills to the test! These problems go from easier to harder, just like the real ISEE. Remember to use the R.U.N.S. strategy. Read carefully, underline clue words, set up your number sentences, and solve step by step. You've got this!

Question 1: CONCEPTUAL

Emma has 14 stickers. Her friend gives her 8 more stickers. Then Emma gives 5 stickers to her sister. How many stickers does Emma have now?

A. 9
B. 17
C. 22
D. 27

Show answer (B)

The correct answer is B (17). Step 1: Add the stickers her friend gave her: 14 + 8 = 22. Step 2: Subtract the stickers she gave to her sister: 22 − 5 = 17. Choice A (9) comes from subtracting both numbers from 14 without adding first. Choice C (22) is the result of only the first step — you stopped too early! Choice D (27) comes from adding all three numbers, but she gave stickers away.

Question 2: BASIC CALCULATION

A pet store has 4 tanks. Each tank has 7 goldfish. The store sells 10 goldfish. How many goldfish are left?

A. 1
B. 11
C. 18
D. 28

Show answer (C)

The correct answer is C (18). Step 1: Find the total goldfish by multiplying: 4 × 7 = 28 goldfish. Step 2: Subtract the ones sold: 28 − 10 = 18 goldfish left. Choice A (1) comes from subtracting 7 − 4 − 10, which does not make sense. Choice B (11) comes from adding 4 + 7 = 11, but you should multiply because each tank has 7 fish. Choice D (28) is the total before selling — you forgot to subtract!

Question 3: INTERMEDIATE

Ben earns $5 each day walking dogs. He works for 6 days. Then he spends $12 on a book and $8 on a toy. How much money does Ben have left?

A. $10
B. $18
C. $20
D. $30

Show answer (A)

The correct answer is A ($10). Step 1: Find how much Ben earns: 5 × 6 = $30. Step 2: Find how much he spends total: $12 + $8 = $20. Step 3: Subtract spending from earnings: $30 − $20 = $10. Choice B ($18) comes from only subtracting the book cost: $30 − $12 = $18. Choice C ($20) is just the total spending. Choice D ($30) is his total earnings before spending anything.

Question 4: APPLIED

A bakery made 96 muffins in the morning. They sold 40 muffins before lunch. After lunch, they split the remaining muffins equally into 8 boxes. How many muffins are in each box?

A. 5
B. 7
C. 12
D. 56

Show answer (B)

The correct answer is B (7). Step 1: Find remaining muffins after selling: 96 − 40 = 56 muffins. Step 2: Divide equally into 8 boxes: 56 ÷ 8 = 7 muffins per box. Choice A (5) comes from dividing 40 ÷ 8 = 5, using the wrong number. Choice C (12) comes from dividing the original total: 96 ÷ 8 = 12, but you forgot to subtract the sold muffins first. Choice D (56) is the remaining muffins before dividing into boxes — you only did the first step.

Question 5: CRITICAL THINKING

Mia has $50. She buys 3 notebooks that cost $4 each and 2 packs of markers that cost $7 each. She wants to buy a poster that costs $15. Does she have enough money left for the poster, and if so, how much money will she have after buying everything?

A. $9
B. $13
C. $24
D. $35

Show answer (A)

The correct answer is A ($9). Step 1: Find the cost of 3 notebooks: 3 × $4 = $12. Step 2: Find the cost of 2 packs of markers: 2 × $7 = $14. Step 3: Find the total spent on notebooks and markers: $12 + $14 = $26. Step 4: Subtract from $50: $50 − $26 = $24 remaining. Step 5: Subtract the cost of the poster: $24 − $15 = $9. Mia has enough money for the poster and will have $9 left after buying everything.

SUMMARY

Putting It All Together

Summary

You now know how to solve multi-step word problems using the four operations: addition, subtraction, multiplication, and division. Remember the R.U.N.S. strategy — Read, Underline, write a Number sentence, and Solve & check. Look for clue words like "each," "left," "total," and "shared equally" to pick the right operation for each step.

The biggest tip for the ISEE: break every problem into small, simple steps. Solve one step at a time and use the answer from each step in the next one. Always re-read the question to make sure you answered what was asked. And remember — on the ISEE there is no penalty for guessing, so never leave a question blank. Eliminate wrong answers, take your best guess, and move on. You are ready!