Order of Operations & Arithmetic | Math Learning Guide

📋 Prerequisites

None — start here if you need to build basics

Arithmetic is the language of all mathematics. In competition math, you won't get points for showing arithmetic steps — but speed and accuracy here directly determine how much time you have for harder problems. Mastering order of operations and mental math shortcuts is one of the highest-ROI investments you can make.

📚 Key Concepts

Order of Operations (PEMDAS/BODMAS)

Every competition follows the same priority rules. Compute in this order:

Parentheses / Brackets — innermost first
Exponents / Orders — powers and roots
Multiplication & Division — equal priority, left to right
Addition & Subtraction — equal priority, left to right

Fraction bars act like parentheses: the entire numerator is grouped, and the entire denominator is grouped. For example, in $ rac{3+5}{2} $, you add before dividing.

Arithmetic Properties

These properties let you rearrange calculations to make them easier:

Commutative: $ a + b = b + a $ and $ a \times b = b \times a $
Associative: $ (a+b)+c = a+(b+c) $ and $ (a \times b) \times c = a \times (b \times c) $
Distributive: $ a(b + c) = ab + ac $ — the most useful for competition math

The distributive property works in reverse too: $ ab + ac = a(b+c) $. This is called factoring out or factoring by grouping, and it's the #1 mental math shortcut.

Negative Numbers

Key sign rules to memorize:

Adding a negative = subtracting: $ a + (-b) = a - b $
Subtracting a negative = adding: $ a - (-b) = a + b $
$Negative \times Negative = Positive:$ $ (-a) \times (-b) = ab $
$Negative \times Positive = Negative:$ $ (-a) \times b = -ab $

The most common mistake: distributing a negative sign. Remember that $ -(a - b) = -a + b $, not $ -a - b $. The negative applies to every term inside the parentheses.

Mental Math Shortcuts

Round and adjust: $ 99 \times 17 = (100 - 1) \times 17 = 1700 - 17 = 1683 $
Factor out common terms: $ 37 \times 53 + 37 \times 47 = 37 \times (53+47) = 3700 $
Pair to make round numbers: In $ 13 + 28 + 7 + 12 $, pair (13+7) and (28+12) = 20 + 40 = 60
Double & halve: $ 16 \times 35 = 8 \times 70 = 560 $ (halve one, double the other)

⚠️ Common Mistake: Don't do addition before subtraction just because "A" comes before "S" in PEMDAS. They have equal priority — always go left to right. Same with multiplication and division.

💡 Key Insight: Before computing anything, look for common factors or ways to rearrange. 90% of competition arithmetic problems are designed to be solved faster with the distributive property than with brute force.

✏️ Example Problems

📝 Example 1 (PEMDAS with nested parentheses)

Calculate: $ 3 + 4 \times 2 - (6 - 3)^2 $

Step 1: Parentheses first: $ 6 - 3 = 3 $

Now we have: $ 3 + 4 \times 2 - 3^2 $

Step 2: Exponent: $ 3^2 = 9 $

Now we have: $ 3 + 4 \times 2 - 9 $

Step 3: Multiplication: $ 4 \times 2 = 8 $

Now we have: $ 3 + 8 - 9 $

Step 4: Add and subtract left to right: $ 3 + 8 = 11 $, then $ 11 - 9 = 2 $

Answer: 2

📝 Example 2 (Distributive Property / Mental Math)

Calculate: $ 37 \times 99 + 37 $

Brute force way: $ 37 \times 99 = 3663 $, then $ 3663 + 37 = 3700 $. Slow and error-prone.

Smart way (distributive property in reverse):

Notice that $ 37 = 37 \times 1 $. So we have:

\[ 37 \times 99 + 37 \times 1 = 37 \times (99 + 1) = 37 \times 100 = 3700 \]

Factor out the common term 37, and the rest adds up to a clean 100. Done in your head.

Answer: 3700