Order of Operations — Apply order of operations correctly. - SHSAT Math

SECTION 1

Why Do We Need Rules for Math?

Imagine you text a friend the expression 3 + 4 × 2. Your friend gets 11, but you got 14. Who is right? Without a shared set of rules, the same math problem can give different answers. That confusion is exactly why the order of operations was created — a universal agreement on which calculations to do first.

Mathematicians didn't always agree on these rules. Over hundreds of years, scholars in different countries slowly built the system we use today. Let's look at how it developed.

1500s • Early Algebra Takes Shape

European mathematicians began writing formulas with symbols like + and −. They needed a way to read these expressions consistently.

1600s • Parentheses Appear

Mathematicians started using parentheses to group parts of an expression. This let them show clearly which calculation should happen first.

1800s • Multiplication Before Addition

Textbooks began teaching that multiplication and division should be done before addition and subtraction. This became a widely accepted convention.

1900s • PEMDAS Goes Mainstream

Schools in the United States adopted the acronym PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction) to help students remember the correct order.

Today • A Global Standard

Calculators, computers, and every math class around the world follow the same order of operations. It is the shared language of mathematics.

So the big question this lesson answers is: In what order do you perform operations so that everyone gets the same correct answer?

SECTION 2

The Core Rules of PEMDAS

The order of operations tells you the exact sequence for simplifying any math expression. In the U.S., we use the acronym PEMDAS to remember the six steps. Each letter stands for a type of operation.

P — Parentheses

Always simplify what is inside parentheses (or brackets) first. Parentheses are like a VIP pass — they cut to the front of the line.

E — Exponents

Next, evaluate any exponents (powers). For example, 3² means 3 × 3, which equals 9.

MD — Multiply & Divide

Then handle multiplication and division from left to right. These two are equal in rank — just go left to right, like reading a sentence.

AS — Add & Subtract

Finally, do addition and subtraction from left to right. Like multiply and divide, these two share the same level of priority.

⚠

Common Mistake Alert

Many students think multiplication always comes before division, and addition always comes before subtraction. That is not true. Multiplication and division are done in order from left to right. The same goes for addition and subtraction.

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KEY TAKEAWAY

Think of PEMDAS like a four-level building. Parentheses live on the top floor and get served first. Exponents are on the third floor. Multiplication and division share the second floor. Addition and subtraction share the ground floor. You always start at the top and work your way down.

SECTION 3

Seeing the Order in Action

The diagram below shows the order of operations as a staircase. You start at the top step and work your way down. Each step must be completed before moving to the next one.

The PEMDAS staircase shows four priority levels. Start with Parentheses at the top. Notice that Multiplication/Division share one level, and Addition/Subtraction share another. Within each shared level, work left to right.

Notice the arrow on the left side of the diagram. It reminds you to always start at the top and move down. If an expression has no parentheses, skip that step and move to exponents. If there are no exponents, go straight to multiplication and division.

SECTION 4

The Rules Written as Steps

Let's write the order of operations as a clear set of steps. You can use this as a checklist every time you simplify an expression.

STEP 1 — PARENTHESES

Simplify inside ( ) or [ ] first

If parentheses are nested, like 3 × [2 + (5 − 1)], start with the innermost set of parentheses and work outward.

STEP 2 — EXPONENTS

Evaluate powers: aⁿ means a × a × … × a (n times)

For example, 2³ = 2 × 2 × 2 = 8. Always compute exponents before multiplication or addition.

STEP 3 — MULTIPLICATION & DIVISION

Perform × and ÷ from left → right

Multiplication does not automatically come before division. Whichever appears first as you read left to right, do it first.

STEP 4 — ADDITION & SUBTRACTION

Perform + and − from left → right

Same rule: addition does not beat subtraction. Just go left to right.

💡

SHSAT Tip

On the SHSAT, many wrong answer choices come from doing operations in the wrong order. If you see an answer that matches what you'd get by adding before multiplying, that is the trap answer. Always follow PEMDAS!

SECTION 5

Tracing Through an Expression Step by Step

The best way to learn the order of operations is to watch it in action. The diagram below traces the expression 5 + 3 × (4 − 2)² ÷ 6 step by step. Each colored box shows which operation is being performed at that stage.

Each row shows one operation being evaluated. The purple step handles parentheses, the pink step handles exponents, the yellow and orange steps handle multiplication then division left to right, and the cyan step finishes with addition. The final answer is 7.

Notice how Step 3 has two parts (3a and 3b). That is because multiplication and division share the same priority. We found multiplication first as we read left to right, so we did it first. Then we did the division.

SECTION 6

Worked Example

Let's solve a full SHSAT-style problem together. Take it slow and follow every step.

Simplify: 18 ÷ 3² + 4 × (7 − 5)

1
Step 1 — Parentheses

Look for parentheses. We see (7 − 5). Calculate inside: 7 − 5 = 2. The expression becomes:

18 ÷ 3² + 4 × 2
2
Step 2 — Exponents

Look for exponents. We see 3². Calculate: 3² = 3 × 3 = 9. The expression becomes:

18 ÷ 9 + 4 × 2
3
Step 3 — Multiplication & Division (left to right)

Scan left to right for × or ÷. The first one is 18 ÷ 9. Calculate: 18 ÷ 9 = 2. Now the expression is 2 + 4 × 2. Next, still scanning left to right, we find 4 × 2. Calculate: 4 × 2 = 8. The expression becomes:

2 + 8
4
Step 4 — Addition & Subtraction (left to right)

Finally, add: 2 + 8 = 10.

Final Answer: 10

💡

Check Your Work

A quick way to catch mistakes: if you had incorrectly added 18 ÷ 9 + 4 first (getting 18 ÷ 13 × 2), you would end up with a messy decimal. On the SHSAT, if your answer looks surprisingly ugly, re-check your order of operations.

SECTION 7

Common Mistakes vs. Correct Methods

Let's compare some common mistakes students make with the correct approaches. Knowing these traps will help you avoid them on test day.

Expression	Wrong Answer (Common Mistake)	Correct Answer (Using PEMDAS)
8 − 2 × 3	18 (subtracting before multiplying)	2 (multiply first: 2 × 3 = 6, then 8 − 6 = 2)
12 ÷ 4 × 3	1 (multiplying before dividing)	9 (left to right: 12 ÷ 4 = 3, then 3 × 3 = 9)
2 + 3²	25 (adding before the exponent)	11 (exponent first: 3² = 9, then 2 + 9 = 11)
10 − 6 + 2	2 (adding before subtracting)	6 (left to right: 10 − 6 = 4, then 4 + 2 = 6)

Common order-of-operations errors and how to fix them

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KEY TAKEAWAY

Think of the left-to-right rule like reading a book. You wouldn't jump to the last word of a sentence and then go backward. When you see operations at the same priority level (like × and ÷, or + and −), just read them in order from left to right.

SECTION 8

From PEMDAS to Algebraic Expressions

The order of operations isn't just about numbers. Once you start working with variables (letters like x and y that stand for unknown numbers), you'll use the same PEMDAS rules. The table below shows how the skills connect.

What You Know Now	What's Coming Next
Simplify 3 + 4 × 2 = 11	Simplify 3 + 4x when x = 2 → 3 + 4(2) = 11
Evaluate (5 + 1)² = 36	Evaluate (x + 1)² when x = 5 → 36
Left-to-right rule for × and ÷	Combining like terms: 3x + 2x = 5x follows grouping rules
Nested parentheses: 2 × [3 + (4 − 1)]	Distributive property: 2(3 + x) = 6 + 2x

How order of operations connects to algebra

Mastering PEMDAS now means you'll have a strong foundation for every algebra topic on the SHSAT. When you see a complicated algebraic expression, you'll already know the correct order to simplify it.

SECTION 9

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 carefully.

Question 1: CONCEPTUAL

In the expression 6 + 2 × 5, which operation should you perform first: addition or multiplication? Why?

Show answer

You should perform multiplication first. According to PEMDAS, multiplication has a higher priority than addition. So you compute 2 × 5 = 10 first, and then add 6 + 10 = 16. If you added first, you'd get 8 × 5 = 40, which is incorrect.

Question 2: BASIC CALCULATION

Simplify: 20 − 4 × 3 + 2

Show answer

First, do multiplication: 4 × 3 = 12. The expression becomes 20 − 12 + 2. Now do subtraction and addition from left to right: 20 − 12 = 8, then 8 + 2 = 10. The final answer is 10. A common mistake is to add 3 + 2 first, which would give the wrong answer of 5.

Question 3: INTERMEDIATE

Simplify: (8 + 2) × 3 − 4² ÷ 2

Show answer

Step 1 — Parentheses: 8 + 2 = 10. Step 2 — Exponents: 4² = 16. Step 3 — Multiply and divide left to right: 10 × 3 = 30, then 16 ÷ 2 = 8. Step 4 — Subtract: 30 − 8 = 22. The final answer is 22. Notice that 10 × 3 comes before 16 ÷ 2 because multiplication appeared first when reading left to right.

Question 4: APPLIED

A movie ticket costs $9. You buy 3 tickets and a large popcorn for $7. Your friend gives you $5 back. Write an expression for the total you spent and simplify it using order of operations.

Show answer

The expression is 3 × 9 + 7 − 5. Using PEMDAS: first multiply 3 × 9 = 27. Then add and subtract left to right: 27 + 7 = 34, and 34 − 5 = 29. You spent $29. This shows how order of operations applies to real life — you multiply the ticket price by the number of tickets first, then handle the addition and subtraction.

Question 5: CRITICAL THINKING

Place one pair of parentheses in the expression 2 + 3 × 4 − 1 so that the result equals 19. Where do they go, and why?

Show answer

Place the parentheses around (4 − 1), making the expression 2 + 3 × (4 − 1). Now simplify: parentheses first, 4 − 1 = 3. Then multiply: 3 × 3 = 9. Then add: 2 + 9 = 11. That gives 11, not 19. Try (2 + 3) × 4 − 1. Parentheses: 2 + 3 = 5. Multiply: 5 × 4 = 20. Subtract: 20 − 1 = 19. The parentheses go around 2 + 3. This problem shows that parentheses can completely change the value of an expression by overriding the normal priority of operations.

SUMMARY

Order of Operations — Quick Review

Summary

The order of operations is a set of rules that tells you the correct sequence for simplifying math expressions. Use the acronym PEMDAS to remember: Parentheses first, then Exponents, then Multiplication and Division from left to right, and finally Addition and Subtraction from left to right.

The most important detail to remember is the left-to-right rule: multiplication does NOT always come before division, and addition does NOT always come before subtraction. Operations at the same priority level are handled in order from left to right. Master these rules, and you will handle every SHSAT expression with confidence.