Multi-Step Problem Solving — Solve a multi-step problem...

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Where Did Multi-Step Problems Come From?

People have been solving multi-step problems for thousands of years. Whenever someone needed to figure out how much grain to store, how many workers to hire, or how far a ship could travel, they had to think through more than one calculation. Multi-step problem solving means breaking a big question into smaller pieces and working through them in order.

1800 BCE

Ancient Babylon

Babylonian scribes wrote clay tablets with word problems about land, grain, and wages. Many required two or three steps to solve.

300 BCE

Euclid's Elements

Greek mathematicians showed that complex results could be proved step by step, building each new fact on a previous one.

820 CE

Al-Khwarizmi's Algebra

The Persian scholar al-Khwarizmi wrote the first textbook on algebra (the word itself comes from Arabic). He taught people to use equations to solve real-world problems.

1972

The SHSAT is Introduced

New York City begins using the Specialized High Schools Admissions Test. Multi-step word problems become a key part of the math section because they test reasoning, not just memorization.

The big idea has not changed: real life rarely gives you a one-step question. The SHSAT tests whether you can read a situation, decide which operations or equations to use, and carry them out in the right order. That is exactly the skill you will build in this lesson.

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Core Principles of Multi-Step Problem Solving

Before you touch any numbers, you need a game plan. Every multi-step problem follows the same basic principles. Learn these, and you will have a framework (a step-by-step guide) you can use on any problem.

Read & Identify

Read the entire problem once. Then read it again and underline the given information and the question being asked.

Plan Your Steps

Decide which operations (addition, subtraction, multiplication, division) or which equation you need. Figure out the order before you calculate.

Assign Variables

If a quantity is unknown, give it a letter like x. Write an equation (a math sentence) that connects what you know to what you need to find.

Solve Step by Step

Work through your plan one step at a time. Write each step clearly so you can check your work later.

Check & Interpret

Plug your answer back into the original problem. Does it make sense? Does it actually answer the question that was asked?

✦ KEY TAKEAWAY

Think of a multi-step problem like following a recipe. You would not toss all your ingredients into a bowl at once — you follow the steps in order. If a step asks for "half the batter," you need to know the whole amount first. Each step feeds into the next one, and skipping ahead leads to a mess.

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Seeing the Steps — A Visual Map

The diagram below shows how a multi-step word problem flows from reading the question all the way to checking your answer. Notice how each stage feeds into the next, just like a chain where every link matters.

Follow the numbered boxes from left to right. Each step uses the result of the one before it. The example at the bottom shows a quick preview of the equation method you will learn next.

The flowchart highlights something important: you do not start calculating right away. Steps 1 through 3 are all about understanding the problem. Only then do you write an equation and solve. This habit saves time because you avoid careless mistakes that come from jumping in too fast.

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The Math Behind Multi-Step Problems

There are two main math tools you will use on the SHSAT: writing and solving equations and proportional reasoning. Let's look at the key formulas you need.

Tool 1 — Setting Up Equations

BASIC EQUATION SETUP

Unknown quantity = x, then translate words into math

Common translations: "more than" → add, "less than" → subtract, "times as many" → multiply, "split equally" → divide, "is / was / equals" → = sign.

SOLVING A LINEAR EQUATION

ax + b = c → ax = c − b → x = (c − b) ÷ a

Here a is the number multiplied by x, b is any number added or subtracted, and c is the total. You get x alone by undoing operations in reverse order.

Tool 2 — Proportional Reasoning

PROPORTION

a / b = c / d → a × d = b × c (cross-multiply)

A proportion (two equal ratios) lets you find a missing value when three of the four numbers are known. Cross-multiplying turns the proportion into a simple equation.

UNIT RATE

Unit rate = Total quantity ÷ Number of units

A unit rate tells you how much per one item. For example, if 5 notebooks cost $15, the unit rate is $15 ÷ 5 = $3 per notebook.

💡 SHSAT Tip

Many SHSAT problems mix both tools. You might use a proportion to find a rate and then plug that rate into an equation. Stay flexible!

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Common SHSAT Multi-Step Problem Types

Not every multi-step problem looks the same. The SHSAT uses several common setups. The diagram below sorts them into categories so you can recognize each type quickly on test day.

The tree shows three main categories. Equation problems ask you to find an unknown using algebra. Proportion problems give you a rate or ratio and ask you to scale it. Mixed problems combine both tools.

Keyword table to help you decide which math tool fits the problem.

Clue Words	Likely Tool	Example Phrase
"total," "combined," "altogether"	Equation (addition)	"Altogether they scored 54 points."
"more than," "fewer than"	Equation (add/subtract)	"Jake has 8 more cards than Lila."
"for every," "per," "ratio"	Proportion / Unit Rate	"She types 45 words per minute."
"times as many," "twice"	Equation (multiplication)	"The bus holds 3 times as many people."
"at this rate," "how long"	Proportion (rate × time)	"At this rate, how many miles in 5 hours?"

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Worked Example — Step by Step

Let's work through a full SHSAT-style problem together. Follow each step carefully.

📝 PROBLEM

A store sells notebooks for $4 each and pens for $2 each. Maya bought a total of 15 items and spent exactly $42. How many notebooks did Maya buy?

Solution

Step 1 — Identify the Knowns and UnknownKnowns: notebooks cost $4, pens cost $2, total items = 15, total cost = $42. Unknown: how many notebooks Maya bought.

Step 2 — Assign a VariableLet n = number of notebooks. Since there are 15 items total, the number of pens is 15 − n.

Step 3 — Write an Equation for CostCost of notebooks + cost of pens = total cost. That gives us: 4n + 2(15 − n) = 42.

Step 4 — Distribute and Simplify4n + 30 − 2n = 42. Combine like terms: 2n + 30 = 42.

2n + 30 = 42

Step 5 — Solve for nSubtract 30 from both sides: 2n = 12. Divide both sides by 2: n = 6.

n = 6 notebooks

Step 6 — Check Your Answer6 notebooks at $4 = $24. Pens: 15 − 6 = 9 pens at $2 = $18. Total: $24 + $18 = $42 ✓. Items: 6 + 9 = 15 ✓. Both conditions match!

✔ WHY THE CHECK MATTERS

The check step is like proofreading an essay. You might feel done, but one quick read-through can catch a silly mistake and save you a point on the SHSAT.

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Equations vs. Proportions — When to Use Which

Sometimes students are unsure whether to set up an equation or a proportion. The table below compares the two approaches so you can make a smart choice quickly.

Side-by-side comparison of the equation and proportion approaches.

Feature	Equation Approach	Proportion Approach
Best for	Problems with a fixed total, a difference, or combined quantities	Problems with a rate, ratio, or scaling pattern
Typical clue words	"total," "combined," "how many more"	"per," "for every," "at this rate"
Setup	Translate words into ax + b = c	Set two ratios equal: a/b = c/d
Solve method	Isolate x by undoing operations	Cross-multiply, then solve the resulting equation
Common mistake	Forgetting to distribute or combine like terms	Setting up the ratio upside down

✦ KEY TAKEAWAY

Think of equations as a balance scale — whatever you do to one side, you must do to the other. Think of proportions as a copy machine that enlarges or shrinks everything by the same factor. Pick the tool that matches the story the problem is telling you.

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Connecting to Harder Concepts

The skills you learn here do not stop at the SHSAT. In high school math, multi-step reasoning grows into systems of equations, linear modeling, and even calculus. But the core habit — translate, set up, solve, check — stays the same.

How today's skills connect to future math courses.

What You Learn Now	What It Becomes Later
One equation with one unknown (ax + b = c)	Systems of two equations with two unknowns (Algebra 1)
Proportions and unit rates	Slope of a line and direct variation (Algebra 1 & 2)
Translating words into math	Creating functions and models from real data (Pre-Calculus)
Checking your answer by plugging it back in	Verifying solutions to equations and inequalities (all higher math)

So every time you practice a multi-step problem, you are not just preparing for one test. You are building a foundation that will help you in every math class to come.

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Practice Problems

Try these five problems on your own. They get harder as you go. After each one, check the answer and read the explanation.

PROBLEM 1 — CONCEPTUAL

A word problem says: "Carlos has twice as many stickers as Dina. Together they have 36 stickers." If you let d = Dina's stickers, which equation correctly represents the situation? (A) d + d = 36 (B) 2d + d = 36 (C) d + 2 = 36 (D) 2d − d = 36

PROBLEM 2 — BASIC CALCULATION

A car travels 180 miles in 3 hours. At this same rate, how many miles will it travel in 7 hours?

PROBLEM 3 — INTERMEDIATE

Aisha and Ben together have $78. Aisha has $14 more than Ben. How much money does each person have?

PROBLEM 4 — APPLIED

A printer can print 24 pages in 4 minutes. Maria needs to print a 156-page report. She has already printed 36 pages. How many more minutes will it take to finish printing the rest of the report?

PROBLEM 5 — CRITICAL THINKING

At a school fair, tickets cost $3 for students and $5 for adults. A total of 200 tickets were sold, and the total revenue was $720. How many student tickets and how many adult tickets were sold? (Hint: you will need two pieces of information to set up your equation.)

SUMMARY

Pulling It All Together

Multi-step problem solving is the skill of breaking a complex question into manageable pieces. On the SHSAT, you will see problems that require two, three, or even four steps. The process always starts with reading carefully and identifying what you know and what you need to find. Next, you assign a variable and write an equation or set up a proportion. Then you solve step by step and check your answer by plugging it back in.

Use equations when the problem involves totals, differences, or combined quantities. Use proportions when you see rates, ratios, or scaling. Look for clue words like "per," "for every," "total," and "more than" to guide your choice. Practice these strategies regularly, and you will feel confident tackling any multi-step problem on test day!