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Multi-Step
Operations

Master the art of combining addition, subtraction, multiplication, and division in complex, multi-step problems for the ISEE Upper Level.

ISEE Upper Level

What Are Multi-Step Operations?

A multi-step problem requires you to perform two or more arithmetic operations in sequence to arrive at the answer. Instead of one simple calculation, you chain together addition, subtraction, multiplication, and division.

Single-Step
\( 8 \times 5 = 40 \)
Multi-Step
\( 8 \times 5 + 12 \div 3 = ? \)

The Four Fundamental Operations

Addition (+)
Combines quantities together
Subtraction (−)
Finds the difference between quantities
Multiplication (×)
Repeated addition, scaling a value
Division (÷)
Splitting into equal groups

Why Order Matters

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
\( 3 + 4 = 7 \)
\( 7 \times 2 = 14 \)
\( 14 \) ✕
 ✓ Correct Order
\( 4 \times 2 = 8 \)
\( 3 + 8 = 11 \)
\( 11 \) ✓

The Core Rule: PEMDAS

The Order of Operations is the universal rule for evaluating any mathematical expression. Remember the acronym:

P
Parentheses: Evaluate everything inside grouping symbols first
E
Exponents: Calculate powers and roots next
M
Multiplication and Division: Left to right, equal priority
A
Addition and Subtraction: Left to right, equal priority

PEMDAS Priority Levels

Parentheses / Brackets Exponents / Roots Multiplication ← → Division Addition ← → Subtraction HIGHEST LOWEST Left → Right Left → Right Same-level operations are evaluated left to right

M/D vs A/S: Equal Priority Pairs

A common misconception is that multiplication always comes before division, or addition before subtraction. In reality, each pair shares equal priority and is resolved left to right.

Multiplication / Division Addition / Subtraction
Priority Higher (do first) Lower (do second)
Within the pair Left to right Left to right
Example \( 12 \div 3 \times 2 = 8 \) \( 10 - 4 + 2 = 8 \)
Common error \( 12 \div 6 = 2 \) ✕ \( 10 - 6 = 4 \) ✕

How to Solve Any Multi-Step Problem

1
Read carefully. Identify all operations and grouping symbols in the expression.
2
Parentheses first. Simplify everything inside parentheses or brackets.
3
Exponents next. Evaluate any powers or roots.
4
Multiply and divide from left to right.
5
Add and subtract from left to right.
6
Verify. Re-check each step for arithmetic errors.

Example 1: Basic Multi-Step

Evaluate: \( \;15 - 3 \times 4 + 8 \div 2 \)

1
Identify operations: subtraction, multiplication, addition, division
2
Multiplication: \( 3 \times 4 = \) 12
3
Division: \( 8 \div 2 = \) 4
4
Substitute: \( 15 - \) 12 \( + \) 4
5
Left to right: \( 15 - 12 = 3 \), then \( 3 + 4 = \) 7

Example 2: Parentheses Change Everything

Evaluate: \( \;(15 - 3) \times (4 + 8) \div 2 \)

1
First parentheses: \( 15 - 3 = \) 12
2
Second parentheses: \( 4 + 8 = \) 12
3
Multiply left to right: 12 \( \times \) 12 \( = \) 144
4
Divide: 144 \( \div 2 = \) 72

Example 3: ISEE-Style Word Problem

A store sells notebooks for $4 each and pens for $2 each. Maria buys 5 notebooks and 8 pens, then uses a $10 coupon. What does she pay?

1
Translate to math: \( 5 \times 4 + 8 \times 2 - 10 \)
2
Multiply: \( 5 \times 4 = \) 20 and \( 8 \times 2 = \) 16
3
Add and subtract left to right: 20 \( + \) 16 \( - 10 = 36 - 10 = \) 26

Example 4: Nested Complexity

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

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 \( - 4 = \) 20

Common Mistakes to Avoid

✕ Left-to-Right Only
Performing all operations from left to right without respecting PEMDAS priorities leads to wrong answers.
✕ Multiply Before Divide
Assuming multiplication always comes before division. They share equal priority: work left to right.
✕ Ignoring Parentheses
Skipping or misreading grouping symbols. Parentheses override all other priority rules.
✕ Sign Errors
Losing track of negative signs during subtraction or when multiplying by negatives.

Quick Reference Summary

Step Operation Rule
1st Parentheses / Brackets Innermost first
2nd Exponents / Roots Left to right
3rd Multiplication and Division Left to right (equal rank)
4th Addition and Subtraction Left to right (equal rank)

ISEE Strategy Tips

Estimate First
Round numbers to get a ballpark answer, then eliminate obviously wrong choices.
Show Your Work
Write each step on scratch paper. One arithmetic slip can change the final answer.
Check with Plugging
Plug your answer back into the original context to verify it makes logical sense.

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

What is the value of \( 20 - 2^3 + 4 \times 3 \)?

Question 4 of 5

A baker makes 3 batches of cookies with 24 cookies per batch. She gives away half of all the cookies and then eats 4. How many cookies remain?

Question 5 of 5

Evaluate: \( 5 \times [10 - (2 + 3)] + 6 \div 2 \)