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Multi-Operation
Problems

Master the art of solving expressions that combine addition, subtraction, multiplication, and division in a single problem.

SSAT Upper Level: Quantitative

What Are Multi-Operation Problems?

A multi-operation problem is any expression or equation that requires you to perform two or more different arithmetic operations to find the answer. Instead of just adding or just multiplying, you must combine operations and apply them in the correct sequence.

Single-Operation
\( 15 + 27 = 42 \)
One step, one operation
Multi-Operation
\( 3 + 4 \times 5 - 2 = ? \)
Multiple steps, careful ordering

Why Order Matters

Consider the expression \( 3 + 4 \times 5 \). If you add first, you get \( 7 \times 5 = 35 \). If you multiply first, you get \( 3 + 20 = 23 \). Two different answers from the same expression: that is why mathematicians agreed on a universal set of rules.

Left to Right (Wrong)
\( 3 + 4 = 7 \)
\( 7 \times 5 = 35 \)
 ✕ 35 Multiply First (Correct)
\( 4 \times 5 = 20 \)
\( 3 + 20 = 23 \)
 ✓ 23

The Four Operations at a Glance

Addition
Combines quantities: \( a + b \)
Subtraction
Finds differences: \( a - b \)
Multiplication
Repeated addition: \( a \times b \)
Division
Splits into equal parts: \( a \div b \)

The Core Rule: PEMDAS

The Order of Operations tells you exactly which operation to perform first. The mnemonic PEMDAS encodes the priority levels:

P
Parentheses: evaluate expressions inside \( ( \; ) \) first
E
Exponents: compute powers and roots next
M/D
Multiplication & Division: left to right, equal priority
A/S
Addition & Subtraction: left to right, equal priority

PEMDAS Priority Hierarchy

( ) Parentheses 1st Exponents 2nd
\( \times \; \div \) (left to right)
 3rd
\( + \; - \) (left to right)
 4th HIGHEST LOWEST

Left-to-Right Tie-Breaking

Many students think multiplication always comes before division, or addition before subtraction. That is not correct. When operations share the same priority, you simply work from left to right.

Situation Common Mistake Correct Approach
\( 24 \div 6 \times 2 \) Multiply first: \( 24 \div 12 = 2 \) Left to right: \( 4 \times 2 = \) 8
\( 10 - 3 + 7 \) Add first: \( 10 - 10 = 0 \) Left to right: \( 7 + 7 = \) 14

How to Apply PEMDAS

1
Scan the expression. Identify all operations, parentheses, and exponents present.
2
Simplify parentheses first. Work from the innermost set outward if nested.
3
Evaluate exponents. Compute any powers or square roots.
4
Multiply and divide from left to right.
5
Add and subtract from left to right.
6
Double-check. Re-read the problem to confirm you answered what was asked.

Example 1: Basic Multi-Operation

Evaluate: \( \; 8 + 3 \times (10 - 4) \div 2 \)

Step 1: Parentheses
\( 10 - 4 = 6 \)
 Step 2: Multiply
\( 3 \times 6 = 18 \)
 Step 3: Divide
\( 18 \div 2 = 9 \)
 Step 4: Add
\( 8 + 9 = 17 \)
 Answer: 17

Example 2: Exponents and Fractions

Evaluate: \( \; \dfrac{6^2 - 4 \times 3}{12 \div 4 + 1} \)

1
Numerator: Exponent first: \( 6^2 = \) 36. Then multiply: \( 4 \times 3 = \) 12. Subtract: \( 36 - 12 = \) 24.
2
Denominator: Divide first: \( 12 \div 4 = \) 3. Then add: \( 3 + 1 = \) 4.
3
Final division: \( \dfrac{24}{4} = \) 6.

Example 3: Nested Parentheses

Evaluate: \( \; 2 \times [3 + (8 - 5)^2] - 10 \)

1
Innermost parentheses: \( 8 - 5 = \) 3
2
Exponent: \( 3^2 = \) 9
3
Brackets: \( 3 + 9 = \) 12
4
Multiply: \( 2 \times 12 = \) 24
5
Subtract: \( 24 - 10 = \) 14

Example 4: Word Problem Translation

A store sells notebooks for $3 each and pens for $1.50 each. Maria buys 4 notebooks and 6 pens, then uses a $5 coupon. What does she pay?

1
Translate: Total \( = 4 \times 3 + 6 \times 1.50 - 5 \)
2
Multiply: \( 4 \times 3 = \) 12 and \( 6 \times 1.50 = \) 9
3
Add then subtract: \( 12 + 9 - 5 = 21 - 5 = \) $16

The Distributive Property Shortcut

The distributive property lets you break apart a multiplication over addition or subtraction. This is a powerful shortcut for multi-operation problems:

\( a(b + c) = ab + ac \)
 Example:
\( 7 \times 98 \)
\( = 7(100 - 2) \)
\( = 700 - 14 \)
\( = 686 \)

Negative Numbers in Multi-Operation

Negative signs add an extra layer of complexity. Follow these rules carefully:

Negative × Positive
\( (-3) \times 4 = -12 \)
Result is negative
Negative × Negative
\( (-3) \times (-4) = 12 \)
Result is positive
Negative Squared
\( (-3)^2 = 9 \) but \( -3^2 = -9 \)
Parentheses matter!

Common Mistakes to Avoid

✕ Left-to-Right Only
Ignoring PEMDAS and just computing left to right. Always check for higher-priority operations first.
✕ M Before D Always
Thinking multiplication always comes before division. They are equal priority: process left to right.
✕ Dropping Negatives
Losing track of negative signs during multi-step work. Write every intermediate step to stay accurate.

Summary Cheat Sheet

Priority Operation Rule Example
1st Parentheses Innermost first \( (3+2) = 5 \)
2nd Exponents Compute powers \( 5^2 = 25 \)
3rd \( \times \; \div \) Left to right \( 6 \div 2 \times 3 = 9 \)
4th \( + \; - \) Left to right \( 10 - 3 + 7 = 14 \)

Question 1 of 5

What is the value of \( \; 5 + 3 \times 2^2 - 4 \)?

Question 2 of 5

Evaluate: \( \; 48 \div 8 \times 3 + 2 \)

Question 3 of 5

What is the value of \( \; \dfrac{(7 + 3)^2 - 20}{4 \times 5} \)?

Question 4 of 5

Evaluate: \( \; -2^3 + 4 \times (6 - 9) \)

Question 5 of 5

A bakery sells cupcakes for $2.50 each and muffins for $3 each. Sam buys 6 cupcakes and 4 muffins, then gets a 10% discount on the total. How much does he pay?