Use Units to Solve Problems

Lesson 1 of 2: Units as a Problem-Solving Guide

In this lesson:

• Use units to decide what operation to perform
• Apply dimensional analysis to multi-step conversions

Unit Analysis Learning Objectives Today

By the end of this lesson, you should be able to:

1. Use units to understand a problem before solving it
2. Perform dimensional analysis to guide multi-step solving
3. Verify units combine correctly to produce the desired result
4. Chain conversion factors to transform between unit systems

Can Units Tell You What to Do?

You know the distance, the efficiency, and the gas price. Which numbers do you multiply? Which do you divide?

The Pool Problem: Let Units Decide

Pool: 1,500 gal; pump: 3 gal/min. How long to fill?

Units voted for division — before computing anything.

Units Tell You the Operation

Unit analysis: examine the units of each given quantity and the desired answer to determine what operation to perform.

• Given: gallons (pool size) and gallons/minute (fill rate)
• Desired: minutes
• Logic:

If the units work out, the operation is correct.

Painter Example: Reading Units Before Computing

Problem: A painter charges $25/hour, works 12 hours. Total cost?

• Charge: $25 per hour → rate in $/hr
• Time: 12 hours
• Desired: dollars

Unit check: ($/hr) × hr = $ ✓ → multiply

Compute: 25 × 12 = $300

Truck Delivery: Three Quantities, One Chain

40 boxes/trip × 25 lbs/box × 6 trips = ?

Unit chain:

Boxes cancel. Trips cancel. Pounds survive.

Compute: 40 × 25 × 6 = 6,000 lbs

Quick Check: Printer Setup Problem

A printer prints 8 pages/min. A document is 120 pages.

1. What unit should the answer have?
2. Write the unit setup — don't compute yet.

Which operation makes the units work out to minutes?

Error Detector: Spot the Mistake

A student set up this calculation:

Question: What went wrong — and how do you know?

The units make the error visible before you even check the number.

What If Units Don't Cancel Directly?

Units-first works when units already match.

But miles per hour → meters per second requires two conversions:

• miles ≠ meters
• hours ≠ seconds

You need a chain of conversion steps.

Dimensional analysis builds that chain systematically.

The Core Idea: Conversion Factors Equal 1

A conversion factor is a fraction equal to 1.

Since :

Multiplying by 1 changes only the unit — not the quantity.

Like 5 dollars = 500 cents: same money, different label.

Simple Conversion: 5 km → Meters

km cancels top and bottom — only m survives.

Two-Step Conversion: Hours → Seconds

Convert 3 hours to seconds:

• hr cancels; min cancels; seconds survive

Multi-Step: 45 mph → m/s

Goal: Convert 45 miles per hour to meters per second.

Cancellations:

• mi cancels (numerator × denominator)
• hr cancels (denominator × numerator)

Surviving units: m/s ✓

Your Turn: Build the Recipe Chain

A recipe needs 2 cups of flour. 1 cup ≈ 125 grams.

• What cancels? What survives?
• Compute the result.

Write the chain before computing.

Quick Check: Convert 72 km/h to m/s

Fill in the values, show cancellations, compute.

Write the full chain before advancing.

The Meta-Principle: Units Are Your Check

• Units work out → computation is almost certainly right
• Units don't work out → something is wrong

Set up the chain so units must work — then compute with confidence.

Worked Example: Factory Electricity Cost

3.5 kWh/widget × $0.12/kWh × 200 widgets/day

Compute: 3.5 × 0.12 × 200 = $84/day

Road-Trip Fuel Cost: Set Up the Chain

Your car gets 28 mpg. Trip: 350 miles. Gas: $3.50/gal.

• What cancels? What survives?
• Check units before computing.

Your Turn: Full Chain, No Prompts

A car gets 32 miles per gallon. A trip is 450 miles. Gas costs $3.60 per gallon.

Find the total fuel cost.

Set up the complete dimensional analysis chain — show all units, show all cancellations, then compute.

Two Common Unit Reasoning Errors

Error 1: Units stamped on after computing

"Got 4,500 — must be minutes." → Wrong operation.

Error 2: Thinking conversion changes the quantity

"5 km → 5,000 m — bigger!" → Same distance, different label.

Key Ideas from This Lesson

✓ Read units before computing — they choose the operation

✓ Conversion factors equal 1 — unit changes, quantity doesn't

✓ If units work out, the math is right

Units stamped on at the end don't catch errors

10 km + 500 m ≠ 510 — convert first

Coming Up in Lesson 2

You can now guide computation with unit analysis.

Lesson 2 applies the same reasoning in new contexts:

• Inside formulas — unit mismatches are invisible until you check
• On graph axes — scale choices change what you believe