Viral Math Problem - Explanation and Solution

Understanding the Order of Operations (PEMDAS/BODMAS)

When you encounter a mathematical expression, it can sometimes be unclear which operations to perform first. To avoid confusion, mathematicians follow a standard set of rules known as the order of operations. Two common acronyms used to remember these rules are PEMDAS and BODMAS.

What are PEMDAS and BODMAS?

PEMDAS stands for:

  • Parentheses
  • Exponents
  • Multiplication
  • Division
  • Addition
  • Subtraction

BODMAS (or sometimes BIDMAS) stands for:

  • Brackets
  • Orders (another term for exponents or powers)
  • Division
  • Multiplication
  • Addition
  • Subtraction

Both PEMDAS and BODMAS convey the same general hierarchy, just with slightly different words. The steps remain consistent regardless of which mnemonic you use.

The Hierarchy of Operations

  1. Parentheses/Brackets: Solve expressions within parentheses or brackets first.
  2. Exponents/Orders: Handle exponents, powers, and roots next.
  3. Multiplication and Division: Perform these operations as they appear from left to right.
  4. Addition and Subtraction: Perform these last, also from left to right.

Example:

Consider the expression 12 ÷ 3 × 2.

Multiplication and division have the same priority, so we handle them left to right:

  1. First, divide: 12 ÷ 3 = 4
  2. Then multiply: 4 × 2 = 8

The result is 8.

Ambiguity in Expressions

Sometimes, expressions are written in a way that can be confusing. For example, consider the expression:

8 ÷ 2(2+2)

If we first simplify (2+2), we get:

8 ÷ 2(4)

This can be read in two ways:

  1. Treat as left-to-right division and multiplication: 8 ÷ 2 = 4, then 4 × 4 = 16.
  2. Treat the denominator as the entire product 2(4): 8 ÷ (2×4) = 8 ÷ 8 = 1.

Because of this ambiguity, both 1 and 16 could be argued, depending on the chosen interpretation. Modern convention tends to favor the first interpretation, resulting in 16, but the best practice is to write the expression in a clearer way.

Clarifying Ambiguity

To remove any doubt about the intended order of operations, it’s best to use additional parentheses or fraction bars. For instance, if your intended meaning is:

If you mean the entire denominator to be 2(2+2):

Write it as:

8 / [2(2+2)]

Then it’s clear that you should first calculate (2+2) before dividing 8 by the result:

(2+2) = 4, so 2(4) = 8 and thus 8 / 8 = 1.

If you mean to divide 8 by 2 first, then multiply the result by (2+2):

Write it as:

(8/2) × (2+2)

Now it’s unambiguous that you do 8/2 = 4 first, and then 4 × 4 = 16.

By adding parentheses to show which operations belong together, you ensure anyone reading the expression will interpret it as intended.

Summary

The order of operations (PEMDAS/BODMAS) ensures consistency when solving math problems. To avoid confusion:

  • Use parentheses to clearly group related operations.
  • Follow the PEMDAS/BODMAS hierarchy: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (left to right), Addition and Subtraction (left to right).
  • Rewrite ambiguous expressions so that it’s always clear how they should be interpreted.