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MATH 114 · Module 2 of 14

Problem Solving & Units

Most real-world math comes with units attached. This module shows you how to track units, convert between them, and use them to guide your problem solving.

Start learning Course overview

Learn the Concepts

1 · Working with units

A number alone rarely tells the whole story — 5 means something very different as 5 dollars, 5 miles, or 5 hours. The unit is part of the answer.

  • "Per" means division: miles per hour is miles ÷ hours.
  • "Of" often means multiplication: 0.25 of 80 is 0.25 × 80.
  • Square / cubic units come from multiplying lengths: ft × ft = ft² (area), ft × ft × ft = ft³ (volume).

Learn the Concepts

2 · Unit (dimensional) analysis

Treat units like numbers you can multiply and cancel. A conversion factor is a fraction equal to 1, such as (5280 ft) / (1 mi). Multiplying by it doesn't change the amount — only the units it's written in. Line up the factors so the units you don't want cancel and the units you do want remain.

A dependable four-step strategy for any problem:

  1. Understand — what is given, and what are you asked to find (in what units)?
  2. Plan — choose conversion factors or a formula that connects them.
  3. Solve — carry out the arithmetic, keeping units attached.
  4. Check — do the units and the size of the answer make sense?

Worked Examples

See it done, step by step

Try each first, then reveal the solution.

Example 1 — Convert 60 miles per hour to feet per second

Chain conversion factors so "miles" and "hours" cancel, leaving feet over seconds.

60 mi/hr × (5280 ft / 1 mi) × (1 hr / 3600 s)

= (60 × 5280 ÷ 3600) ft/s = 88 ft/s.

Example 2 — A car travels 360 miles on 12 gallons. What is its gas mileage?

Gas mileage is measured in miles per gallon, so divide miles by gallons.

360 mi ÷ 12 gal = 30 miles per gallon.

Example 3 — How many square feet of carpet for a 12 ft by 10 ft room?

Area of a rectangle is length × width, and ft × ft = ft².

12 ft × 10 ft = 120 square feet.

Watch & Review

Resources for this module

Slides

Lecture slides

The Module 2 slides on units and problem solving.

Open slides

Check Yourself

Quick self-check

Answer, then reveal.

1. Convert 3 feet to inches.

3 ft × (12 in / 1 ft) = 36 inches.

2. Apples cost $1.50 per pound. What do 4 pounds cost?

4 lb × ($1.50 / 1 lb) = $6.00.

3. A trip of 150 miles at 50 miles per hour takes how long?

150 mi ÷ (50 mi/hr) = 3 hours.

Ready to turn it in?

When the concepts feel solid, complete the Module 2 homework and quiz in Canvas, where the work is graded and due dates are posted.

Submit in Canvas