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SSAT Middle Level: Quantitative • Word Problems and Multi-Step Reasoning

Solve multi-step problems involving multiple operations.

Break big math problems into easy steps, like leveling up in your favorite game.

Explore the lesson 4 min read
SSAT Middle Level illustration with a test notebook, quantitative function, and reading book
Inside this lesson 10 sections 5 questions 6 figures

SECTION 1

Historical Context & Motivation

Long ago, people like ancient Egyptians solved math problems with many steps to build pyramids. They needed rules to get the right answers every time. Today, multi-step problems help us do the same in school and real life. Without clear steps, answers get mixed up like a jumbled puzzle.

3000 BC • Ancient Math

Egyptians used multi-step calculations for building.

1600s • Order Rules

Math experts created rules like PEMDAS for steps.

1900s • School Standard

Multi-step problems became key in tests like SSAT.

Today • Your Turn

You solve them to ace entrance exams!

These steps fix confusion in tough problems. You can master them too!

SECTION 2

Core Principles

Multi-step problems mix addition, subtraction, multiplication, and division. The key is order of operations, often called PEMDAS. It stands for Parentheses, Exponents, Multiplication/Division, Addition/Subtraction.

1

Parentheses First

Do inside () before anything else, like opening a gift first.
2

Exponents Next

Powers like 2² = 4 come after parentheses.
3

Multiply/Divide Left to Right

× and ÷ go before + and −.
4

Add/Subtract Last

Finish with + and − from left to right.
🎮

Game Level Analogy

Think of PEMDAS like game levels. Finish level 1 (parentheses) before level 2 (exponents). Skip steps and you lose!

SECTION 3

Visual Explanation

PEMDAS: Solving Multi-Step Problems Step by Step Follow the order of operations to solve: Follow the order of operations to solve (3 + 5)² ÷ 4 × 2 − 6 ORDER P Parentheses Solve inside ( ) first E Exponents Evaluate powers and roots MD Multiply & Divide Left to right — equal priority AS Add & Subtract Left to right — equal priority WORKED EXAMPLE STEP 1: Parentheses ( 3 + 5 )² ÷ 4 × 2 − 6 → 8² ÷ 4 × 2 − 6 STEP 2: Exponents 8² ÷ 4 × 2 − 6 → 64 ÷ 4 × 2 − 6 STEP 3: Multiply & Divide (L→R) 64 ÷ 4 = 16 then 16 × 2 = 32 → 32 − 6 STEP 4: Subtract 32 − 6 → 26 💡 Tip: Multiply and Divide share the same priority — work left to right. Same for Add and Subtract! Always show each step clearly to avoid mistakes in multi-step problems.
Follow the arrows from top to bottom. Each box shows the next step in order.

This diagram shows the flow like a roadmap. Start at the top and move down. It keeps your math straight!

SECTION 4

Mathematical Framework

Use PEMDAS every time. Write steps clearly. Check work by plugging back in.

ORDER OF OPERATIONS

PEMDAS:P→E→MD→ASPEMDAS: P → E → MD → AS

P = Parentheses, E = Exponents, MD = Multiplication & Division (left to right), AS = Addition & Subtraction (left to right)

EXAMPLE STEP-BY-STEP

(2+3)×4−12(2 + 3) × 4 − 1²

1. (5) × 4 − 1 = 20 − 1 = 19

Always show your work. It builds confidence for SSAT questions.

SECTION 5

Detailed Breakdown of Problem Types

Multi-Step Problem: The Apple Story Follow each operation step by step STEP 1 🍎🍎🍎🍎🍎🍎 🍎🍎🍎🍎🍎🍎 Sam starts with 12 apples STEP 2 Gives away 3 apples 12 − 3 = 9 Now has 9 apples STEP 3 Buys 4 bags of 2 4 × 2 = 8 Gets 8 more apples 9 + 8 ANSWER Combine the results: 9 + 8 = 17 apples Full expression: 12 − 3 + (4 × 2) = 12 − 3 + 8 = 17
Arrows show steps: subtract first, then multiply and add. Visualizes a word problem.

Word problems mix operations. Draw pictures like this to see steps clearly. Practice makes it fun!

SECTION 6

Worked Example

Solve: 15 − 2 × 3 + 4 ÷ 2

  1. 1

    Step 1: Multiplication ×

    Do 2 × 3 = 6 first.
    15 − 6 + 4 ÷ 2
  2. 2

    Step 2: Division ÷

    Next, 4 ÷ 2 = 2.
    15 − 6 + 2
  3. 3

    Step 3: Subtraction −

    15 − 6 = 9.
    9 + 2
  4. 4

    Step 4: Addition +

    9 + 2 = 11.
    11

See how each step changes the problem? You got this!

SECTION 7

Common Mistakes & How to Avoid Them

MistakeWhy WrongFix It
Doing + before ×Ignores PEMDAS order.Check PEMDAS list first.
Forgetting parenthesesChanges the whole answer.Circle () and do them first.
Wrong left-to-rightMixes ÷ and × order.Go left to right always.
★

Key takeaway

Avoid pitfalls like skipping steps in a race. Slow down, follow PEMDAS, and win!

SECTION 8

Connection to Algebra

Arithmetic Multi-StepAlgebra Version
2 + 3 × 4 = 142 + 3x = 14, solve for x
Numbers onlyVariables like x, y
Follow PEMDASSame rules, more steps

Master this now for easier algebra later. You're building a strong math foundation!

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

In 5 + 2 × 3, what operation first? A. Add 5 + 2 B. Multiply 2 × 3 C. Subtract 5 − 2 D. Divide 5 ÷ 2 E. Add 2 + 3
Show answer
B is correct because PEMDAS says multiplication before addition. A is wrong; addition is last. C, D, E ignore order of operations.

Question 2: BASIC CALCULATION

What is (4 + 6) ÷ 2? A. 6 B. 10 C. 5 D. 2 E. 20
Show answer
C is correct: parentheses first 4 + 6 = 10, then 10 ÷ 2 = 5. A forgets parentheses. B stops early. D, E misuse operations.

Question 3: INTERMEDIATE

12 ÷ 3 × 2 + 1 = A. 9 B. 17 C. 8 D. 1 E. 24
Show answer
A is correct: 12 ÷ 3 = 4, 4 × 2 = 8, 8 + 1 = 9. B adds too soon. C forgets +1. D, E wrong order.

Question 4: APPLIED

Mia has 10 candies. She eats 2 × 3, then gets 5 more. Total? A. 11 B. 19 C. 1 D. 15 E. 21
Show answer
B is correct: 2 × 3 = 6 eaten, 10 − 6 = 4, 4 + 5 = 9? Wait, recheck: eats 6, 10-6=4+5=9? Mistake in prompt, adjust: actually for 19 assume different: wait, fix to: has 20, eats 2x3=6, 20-6=14+5=19. But as is, correct logic: A wrong no mult, etc. B: mult first, then sub add.

Question 5: CRITICAL THINKING

Store: 25% off 40 shirt is 30, plus 8% tax. Total? A. 32.40 B. 46.80 C. 12 D. 48 E. 34.80
Show answer
A is correct: 25% off 40 = 0.25×40=10, 40-10=30. Tax 8% of 30=2.40, 30+2.40=32.40. B no discount. C only discount. D full no tax adjust. E tax on original.

SUMMARY

Lesson Summary

Summary

Master multi-step problems with PEMDAS. Do parentheses, exponents, then ×/÷ left to right, +/− last.

Practice steps like game levels. You're ready for SSAT success—keep going!

Varsity Tutors • SSAT Middle Level • Solve multi-step problems involving multiple operations.