Dimensional Consistency Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Dimensional Consistency.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The principle that every term added or equated in a valid equation must share the same physical dimensions or units.

You can't add meters to seconds — dimensionally inconsistent equations don't make physical sense.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Units must balance across both sides of an equation — just as numbers must. A dimensionally inconsistent formula is guaranteed wrong.

Common stuck point: Even when working symbolically, track what each variable represents.

Sense of Study hint: Write the units next to every number and variable, then verify that both sides of the equation have matching units.

Worked Examples

Example 1

easy
Is the equation v = d + t (velocity = distance + time) dimensionally consistent?

Solution

  1. 1
    Step 1: [v] = \text{m/s}, [d] = \text{m}, [t] = \text{s}.
  2. 2
    Step 2: \text{m} + \text{s} — you can't add meters and seconds!
  3. 3
    Step 3: Not dimensionally consistent. The equation must be wrong.

Answer

No, dimensionally inconsistent.
Dimensional consistency requires all terms being added or set equal to have the same units. You can only add quantities of the same dimension — this is a fundamental check for equation validity.

Example 2

medium
Check: E = mc^2 where E is energy (kg·m²/s²), m is mass (kg), c is speed (m/s).

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Is A = l + w a valid formula for area?

Example 2

medium
In the equation x^2 + 3x = 10 (where x is in meters), are all terms dimensionally consistent?
← Back to Dimensional Consistency Formula Explained Practice Mode

Background Knowledge

These ideas may be useful before you work through the harder examples.

equations