Choosing Direction

Lesson 3 of 3: Context Drives the Unit Rate

In this lesson:

• Decide which direction to compute from a given ratio
• Work with like-unit ratios (scale factors)
• Apply to area and mixed-unit contexts

Learning Objectives

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

1. Identify the two quantities and determine which direction yields the desired unit rate
2. Set up the complex fraction representing that unit rate
3. Compute and simplify the result
4. Interpret the unit rate in context with correct units
5. Apply unit rate reasoning to lengths, areas, and like-unit quantities

One Ratio — Two Unit Rates

A pipe discharges gallon in minute.

Both are valid unit rates. The question determines which one you need.

Read the Answer Unit First

The routine:

1. Read the question — identify what you want per what
2. Write the answer unit as a fraction: "miles per hour" → miles/hour
3. Put the A-quantity in the numerator, B-quantity in the denominator
4. Set up the complex fraction to match, then compute

"Gallons per minute" → gallons on top, minutes on bottom.
"Minutes per gallon" → minutes on top, gallons on bottom.

Two Directions: The Printing Press

Same ratio — same context — two different unit rates.

Worked Example: Printing Press (Both Directions)

A press uses pint of ink for of a page.

Pints per page (ink cost for one full page):

Pages per pint (pages from one pint of ink):

Quick Check: Choosing the Right Rate

A manager needs to order ink for a 500-page edition.

Which rate do you multiply by 500 to get pints of ink?

Think: what are the units of "500 pages × ___" that gives pints?

Like Units: When Units Cancel

A scale model: every ft of model = ft actual.

Scale = actual feet per model foot:

Interpretation: Every 1 model foot corresponds to 2.5 actual feet.

Like Units: The Scale Factor Diagram

• Both quantities measured in the same unit → units cancel in the result
• The result is a scale factor — a pure multiplier, no unit label
• Applies to: maps, architectural plans, similar figures

Worked Example: Area Context

A floor tile covers sq ft and costs .

Cost per square foot:

Units: dollars ÷ square feet → dollars per square foot ✓

Your Turn: Choose Your Direction

For each problem, identify the answer unit before computing.

1. A hiker covers mile in hour. Speed in miles per hour?

2. A map: inch represents mile. Scale in inches per mile?

3. Carpet: sq yd costs . Cost per square yard?

Write the answer unit as a fraction first. Then build and compute.

Answers

1. Miles per hour: mph

2. Inches per mile: inch per mile

3. Dollars per sq yd: per sq yd

Key Takeaways

✓ One ratio → two valid unit rates (each is the reciprocal of the other)
✓ Read the answer unit first — it determines which quantity goes in the numerator
✓ Like-unit ratios → dimensionless scale factor (units cancel in the answer)

Watch out: wrong direction gives the reciprocal of the correct answer

• "Per page" → pages in the denominator; "per pint" → pints in the denominator
• If your answer seems like it should be bigger (or smaller), check your direction

Watch out: always write units in labeled answers

• is incomplete; gal/min is correct
• Exception: like-unit answers are dimensionless (e.g., 2.5 for a scale factor — no label)

Well Done — You've Completed 7.RP.A.1

What you can now do:

• Set up any fractional unit rate — same question, same setup, fraction ÷ fraction
• Simplify complex fractions — reciprocal, cancel, multiply, simplify
• Choose the right direction — read the answer unit, build the fraction to match

Coming up in 7.RP.A.2:
The unit rate you computed is also called the constant of proportionality in .

When comes from a ratio of fractions — which it often does — you will use exactly the procedure from Lessons 1 and 2 to find it.