Solving PEMDAS Problems

PEMDAS Problems

PEMDAS problems are a common challenge for anyone using math – especially when you're learning PEMDAS in school for the first time, and without much real-life context. PEDMAS stands for Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction. It's a set of rules that tell you the order to follow when solving a math problem to get the right answer, and an answer which mathematicians can agree on internationally.  PEDMAS is the term in the US; BODMAS is the term used pretty much everywhere else.  I can't help wondering if this is going to end badly.

• Parentheses always come first. You must calculate everything inside the parentheses before moving on.
• Exponents (like squares, cubes, or square roots) are next.
• After that, it's the arithmetic functions: Divide, Multiply, Add, Subtract.  In BODMAS, Division always comes first.  For some strange reason I have not been able to fathom, in PEMDAS, you just do everything from left to right.  It's sure to cause confusion, and a quick look at social media will tell you that.  How can it not, if 3 *4 /3+2 is PEMDAS does 3*4 and then divide by three, and then add two?  If it's BODMAS then you do the division first, even though it's not on the left.  Let's leave that for a whole other discussion.

Why is PEMDAS Important?

Imagine you have this expression: 3 * 4 - 2

• Without PEMDAS, it's unclear whether you should multiply 3 by 4 first, or subtract 2 from 4.
• PEMDAS helps avoid confusion and makes sure everyone gets the same answer.  It's worth pointing out at this point that real mathematicians, who want to communicate unambiguously, always want to avoid confusion.  And by real mathematicians, I mean anybody except the fools who put stupid questions on social media just for the conflict.
  

Image from Hotpot

Let's Look at Some Examples:

1. (3 * 4) - 2

   • Parentheses first: 3 * 4 = 12
   • Subtraction: 12 - 2 = 10

2. 3 * (4 - 2)

   • Parentheses first: 4 - 2 = 2
   • Multiplication: 3 * 2 = 6

3. 4 * (6 - 2²)

   • Parentheses first:
     • Exponents: 2² = 4
     • Subtraction: 6 - 4 = 2
   • Multiplication: 4 * 2 = 8

Why Does PEMDAS Sometimes Cause Trouble?

• Tricky Online Problems: Some people intentionally write confusing problems (like 1 + 7 * 3) to see if others can solve them correctly.  There is a correct answer, but the ambiguity is a bigger problem than the math.
• Everyday Math: Even simple situations can involve PEDMAS. For example:
  • If each car has 4 wheels and is towing a caravan with 2 wheels, how many wheels are there in total for 4 cars?
    • Correct: 4 * (4 + 2) = 24 wheels 
    • Incorrect: (4 * 4) + 2 = 18 wheels.  Four wheels on four cars, and two for a caravan is not enough.

PEMDAS helps us avoid these kinds of mistakes and ensures we get the correct answer every time!

Image by Hotpot

Other online reading

The Math Equation Solver (will handle PEMDAS problems)

Other Calculator Fun and Games articles:

Snakes and Ladders (Collatz Conjecture)
Crafty Calculator Calculations (numerical anagrams with five digits)
More Multiplications (numerical anagrams, four digits)
Over and Out (reduce large numbers to zero as rapidly as possible)
Calculator Games: Front to Back
Calculator Games: Up, up and away with Ulam sequences
Calculator Games: The Kaprekar Constant