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Multi-Operation
Problems

Master the art of solving multi-step problems that combine addition, subtraction, multiplication, and division: the key to SSAT Quantitative success.

SSAT Middle Level: Quantitative

What Are Multi-Operation Problems?

A multi-operation problem requires you to use two or more arithmetic operations: addition, subtraction, multiplication, or division: in a single problem. You must decide which operations to use and in what order to perform them.

+ & −
Combine or find differences between quantities
× & ÷
Scale quantities up or split them into groups
( )
Parentheses group operations that happen first

Why Does Order Matter?

Consider the expression \( 3 + 4 \times 2 \). If you add first, you get \( 7 \times 2 = 14 \). If you multiply first, you get \( 3 + 8 = 11 \). The correct answer is 11 because multiplication comes before addition.

✕ Wrong Order
\( 3 + 4 \times 2 \)
 Add first:
\( 7 \times 2 = 14 \)
 ✕ Incorrect ✓ Correct Order
\( 3 + 4 \times 2 \)
 Multiply first:
\( 3 + 8 = 11 \)
 ✓ Correct

Where You'll Use This

Multi-operation problems appear everywhere: calculating a shopping total with tax, splitting a restaurant bill, or figuring out how much paint you need for a room. On the SSAT, nearly half of all quantitative questions require more than one operation.

Shopping
Buy 3 items at $5 each, use a $4 coupon: \( 3 \times 5 - 4 \)
Cooking
Triple a recipe, then halve it for leftovers: \( 3 \times r \div 2 \)
Sports
Average points over 4 games: \( (a+b+c+d) \div 4 \)

The Order of Operations

P Parentheses ( ) first E Exponents powers & roots M Multiplication D Division left to right A Addition S Subtraction left to right

Parentheses Change Everything

Parentheses override the default order. Whatever is inside must be solved first, regardless of the operation.

No parentheses
\( 8 - 2 \times 3 \)
\( = 8 - 6 \)
\( = 2 \)
 With parentheses
\( (8 - 2) \times 3 \)
\( = 6 \times 3 \)
\( = 18 \)

Priority vs. Left-to-Right

Students often confuse two ideas. Priority decides which operation type goes first. Left-to-right decides the order within same-priority operations.

Priority Rule Left-to-Right Rule
When to use Different operation types Same-level operations
Example \( + \) vs. \( \times \): multiply first \( \times \) then \( \div \): go left to right
Common error Adding before multiplying Dividing before a multiplication to its left

Step-by-Step Strategy

1
Read the whole problem. Identify every operation and any parentheses or exponents.
2
Solve parentheses first. Simplify everything inside ( ) before moving on.
3
Evaluate exponents. Compute any powers or square roots.
4
Multiply and divide left to right. Scan the expression and handle each \( \times \) or \( \div \) in order.
5
Add and subtract left to right. Finish with the remaining operations.
6
Check your answer. Re-read the problem to make sure you answered what was asked.

Example 1: Basic

Evaluate: \( 12 + 6 \div 3 - 1 \)

\( 12 + 6 \div 3 - 1 \)
 Divide first
\( 12 + \color{#fbbf24}\mathbf{2} \color{#e2e8f0} - 1 \)
 Add left to right
\( \color{#4ade80}\mathbf{14} \color{#e2e8f0} - 1 = \color{#4ade80}\mathbf{13} \)
Answer: \(13\)

Example 2: With Parentheses

Evaluate: \( 5 \times (8 - 3) + 4^2 \)

\( 5 \times (8 - 3) + 4^2 \)
 Parentheses first
\( 5 \times \color{#818cf8}\mathbf{5} \color{#e2e8f0} + 4^2 \)
 Exponent next
\( 5 \times 5 + \color{#a78bfa}\mathbf{16} \)
 Multiply
\( \color{#fbbf24}\mathbf{25} \color{#e2e8f0} + 16 = \color{#4ade80}\mathbf{41} \)
Answer: \(41\)

Example 3: Word Problem

Maria buys 4 notebooks at $3 each and 2 pens at $1.50 each. She pays with a $20 bill. How much change does she get?

1
Translate to math: \( 20 - (4 \times 3 + 2 \times 1.50) \)
2
Parentheses: Multiply inside: \( 4 \times 3 = \) \(12\) and \( 2 \times 1.50 = \) \(3\)
3
Add inside parentheses: \( 12 + 3 = 15 \)
4
Subtract: \( 20 - 15 = \) \($5.00\)

Example 4: Nested Parentheses

Evaluate: \( 2 \times [3 + (10 - 4)] \)

1
Innermost first: \( (10 - 4) = \) \(6\)
2
Outer brackets: \( [3 + 6] = \) \(9\)
3
Multiply: \( 2 \times 9 = \) \(18\)

Translating Words to Operations

Word/Phrase Operation Example
sum, total, combined, more than \( + \) "5 more than 3" → \( 3 + 5 \)
difference, less than, fewer \( - \) "8 less than 12" → \( 12 - 8 \)
product, times, each, per \( \times \) "4 groups of 6" → \( 4 \times 6 \)
quotient, divided by, split \( \div \) "20 split among 5" → \( 20 \div 5 \)

Common Mistakes

✕ Left-to-Right for Everything
Solving \( 2 + 3 \times 4 \) as \( 5 \times 4 = 20 \) instead of \( 2 + 12 = 14 \). Always check for multiplication/division first.
✕ Skipping Parentheses
Ignoring ( ) and jumping to multiplication. Parentheses are the highest priority: solve what's inside them before anything else.
✕ Multiply Before Divide Always
Thinking \( \times \) comes before \( \div \). They're equal: use left-to-right. \( 12 \div 3 \times 2 = 4 \times 2 = 8 \), not \( 12 \div 6 = 2 \).

Summary Checklist

Step Action Mnemonic
1 Solve inside parentheses P
2 Evaluate exponents E
3 Multiply and divide left to right M / D
4 Add and subtract left to right A / S
5 Check your answer against the question ✓

Question 1 of 5

What is the value of \( 6 + 2 \times 5 - 3 \)?

Question 2 of 5

Evaluate: \( (9 + 3) \div 4 \times 2 \)

Question 3 of 5

A store sells pencils for $0.75 each and erasers for $0.50 each. Tom buys 6 pencils and 4 erasers. What is the total cost?

Question 4 of 5

What is the value of \( 3^2 + 4 \times (7 - 5) \)?

Question 5 of 5

Evaluate: \( 48 \div (2 + 6) - 1 \times 3 \)