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Scaling &
Unit Rates

Master proportional reasoning, convert between rates, and apply scale factors to solve real-world problems on the ISEE.

ISEE Upper Level

What Is a Rate?

A rate compares two quantities measured in different units. It tells you how much of one thing corresponds to a given amount of another.

Speed
\(\frac{120 \text{ miles}}{2 \text{ hours}}\)
Price
\(\frac{\$9.00}{3 \text{ pounds}}\)
Density
\(\frac{500 \text{ people}}{4 \text{ sq mi}}\)

The Unit Rate

A unit rate is a rate with a denominator of 1. It answers: "How much per one unit?"

Original Rate
\(\frac{120 \text{ mi}}{2 \text{ hr}}\)
 two hours
\(\div 2\)
 Unit Rate
\(\frac{60 \text{ mi}}{1 \text{ hr}}\)
 one hour

Why Unit Rates Matter

Unit rates make comparison and prediction easy. On the ISEE, you'll use them to compare deals, compute distances, and solve proportion problems quickly.

1
Compare fairly: Which store is cheaper per ounce?
2
Predict quickly: At \(60\) mph, how far in \(3.5\) hours?
3
Set up proportions: Unit rates become the constant of proportionality \(k\) in \(y = kx\).

What Is a Scale Factor?

A scale factor is the ratio that describes how every length in one figure relates to the corresponding length in another figure.

Original
\(4 \text{ cm}\)
\(2.5 \text{ cm}\)
\(\times 3\)
 Scaled Copy
\(12 \text{ cm}\)
\(7.5 \text{ cm}\)
 Scale factor:
\(k = 3\)

The Core Formulas

Unit Rate
$$\text{Unit Rate} = \frac{\text{Total Quantity}}{\text{Number of Units}}$$
Scale Factor
$$k = \frac{\text{New Length}}{\text{Original Length}}$$

Proportional Relationships

When two quantities are proportional, their graph is a straight line through the origin. The unit rate equals the slope.

Hours Miles 0 1 2 3 0 60 120 180 240
\(+1 \text{ hr}\)
\(+60 \text{ mi}\)

Rate vs. Scale Factor

\(\frac{\text{new}}{\text{original}}\)
Unit Rate Scale Factor
Definition Quantity per one unit Ratio of corresponding lengths
Units Has units (mi/hr, $/lb) Dimensionless (pure number)
Formula \(\frac{\text{total}}{\text{count}}\)
Key Question "How much per one?" "How many times bigger/smaller?"
Example \(\$3\) per gallon Map scale \(1 : 50{,}000\)

Step-by-Step Strategy

1
Identify the two quantities and their units (e.g., miles and hours, dollars and pounds).
2
Write the rate as a fraction: \(\frac{\text{quantity}_1}{\text{quantity}_2}\).
3
Simplify to a unit rate by dividing numerator and denominator by the denominator value.
4
Multiply or set up a proportion to find the unknown: \(\text{unit rate} \times n = \text{answer}\).
5
Check units: does your answer have the right label? If not, re-examine.

Example 1: Best Deal

Problem: Brand A sells 12 oz of juice for \(\$3.60\). Brand B sells 16 oz for \(\$4.48\). Which is cheaper per ounce?

Brand A
\(\frac{\$3.60}{12 \text{ oz}} = \$0.30\) per oz
Brand B
\(\frac{\$4.48}{16 \text{ oz}} = \$0.28\) per oz

Example 2: Map Scales

Problem: On a map, 2 cm represents 50 km. Two cities are 7 cm apart on the map. What is the actual distance?

Map Distance
\(2 \text{ cm} = 50 \text{ km}\)
\(7 \text{ cm} = ?\)
1
Unit rate: \(\frac{50 \text{ km}}{2 \text{ cm}} = 25\) km per cm.
2
Actual distance: \(25 \times 7 = \) 175 km.

Example 3: Scaling and Area

Problem: A rectangle is 4 cm by 6 cm. If the scale factor is \(k = 2.5\), what is the area of the scaled rectangle?

\(4 \times 6\)
\(6 \text{ cm}\)
\(4 \text{ cm}\)
\(\times 2.5\)
\(10 \times 15\)
\(15 \text{ cm}\)
\(10 \text{ cm}\)
\(\text{Area} = 24 \text{ cm}^2\)
\(\text{Area} = 150 \text{ cm}^2\)

Scaling Area and Volume

When a figure is scaled by factor \(k\), different measures scale differently. This is a critical ISEE trap.

Length
Scales by \(k\)
Area
Scales by \(k^2\)
Volume
Scales by \(k^3\)

Converting Between Rates

Problem: A printer prints 15 pages per minute. How many pages per hour?

1
Start with the unit rate: \(15\) pages/min.
2
Conversion factor: \(1\) hour \(= 60\) minutes.
3
Multiply: \(\frac{15 \text{ pages}}{1 \text{ min}} \times \frac{60 \text{ min}}{1 \text{ hr}} = 900\) pages/hr.

Common Mistakes

✕ Flipping the Fraction
Writing \(\frac{\text{hours}}{\text{miles}}\) instead of \(\frac{\text{miles}}{\text{hours}}\). Always check: which unit do you want per one of?
✕ Forgetting to Square
Using \(k\) for area instead of \(k^2\). If lengths double, area quadruples: not doubles.
✕ Misreading Map Scales
Treating "\(1 \text{ cm} : 50 \text{ km}\)" as "\(1 \text{ cm} = 50 \text{ cm}\)." Keep track of which units are real and which are on the map.

Quick Reference Summary

Numerator/denominator order\(k < 1\) means shrinkSquare the factorCube the factorUnits must cancel
Concept Formula Watch Out For
Unit Rate \(\frac{\text{total}}{\text{units}}\)
Scale Factor \(k = \frac{\text{new}}{\text{original}}\)
Scaled Area \(A_{\text{new}} = k^2 \cdot A_{\text{old}}\)
Scaled Volume \(V_{\text{new}} = k^3 \cdot V_{\text{old}}\)
Rate Conversion Multiply by \(\frac{\text{new unit}}{\text{old unit}}\)

Question 1 of 5

A car travels 255 miles using 8.5 gallons of gasoline. What is its fuel efficiency in miles per gallon?

Question 2 of 5

On a blueprint, 3 inches represents 12 feet. If a room is 8 inches long on the blueprint, what is its actual length?

Question 3 of 5

A square has side length \(5 \text{ cm}\). If it is scaled by a factor of \(k = 3\), what is the area of the new square?

Question 4 of 5

Store X sells 5 lb of rice for \(\$8.50\). Store Y sells 8 lb for \(\$12.80\). Which store offers the lower unit price?

Question 5 of 5

A model building has a scale of \(1 : 200\). The model is \(15 \text{ cm}\) tall. If the real building's floor area is \(800 \text{ m}^2\), what is the model's floor area?