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Unit Rate
Comparisons

Learn how to find unit rates and use them to compare quantities: which deal is better, who runs faster, and what costs less per unit.

SSAT Upper Level: Quantitative

What Is a Rate?

A rate is a ratio that compares two quantities measured in different units. Whenever you see "per," you're looking at a rate: miles per hour, dollars per pound, words per minute.

Rates let us describe how fast, how cheap, or how efficient something is by connecting two different measurements.

What Is a Unit Rate?

A unit rate is a rate where the denominator equals 1. It answers: "How much of one quantity corresponds to exactly one of the other?"

Rate
\(\frac{150 \text{ miles}}{3 \text{ hours}}\)
 denominator ≠ 1 Unit Rate
\(\frac{50 \text{ miles}}{1 \text{ hour}}\)
 denominator = 1

Why Unit Rates Matter

You can't compare rates directly when their denominators differ. Unit rates give every option the same denominator of 1, making comparison instant.

Shopping
Which brand of cereal costs less per ounce?
Speed
Which runner is faster in miles per hour?
Efficiency
Which car gets more miles per gallon?

The Unit Rate Formula

Unit Rate
\(\text{Unit Rate} = \dfrac{\text{Total Quantity}}{\text{Number of Units}}\)

To find a unit rate, divide the total quantity by the number of units. The result tells you how much of the first quantity corresponds to exactly one of the second.

Comparing With Unit Rates

To compare two or more rates, convert each to a unit rate (denominator = 1), then compare the numerators directly.

Option A
\($12\) for \(4\) lbs → \($3.00\)/lb
 Option B
\($10\) for \(5\) lbs → \($2.00\)/lb
 ✓ Option B is cheaper per pound

Rate vs. Unit Rate vs. Ratio

Rate Unit Rate Ratio
Units Different units Different units Same or different
Denominator Any value Always 1 Any value
Example \(\frac{120 \text{ mi}}{2 \text{ hr}}\) \(60\) mi/hr \(3 : 5\)
Best for Describing Comparing Part-to-part

How to Solve Unit Rate Problems

1
Identify the two quantities and their units (e.g., dollars and pounds, miles and hours).
2
Decide which unit goes in the denominator. Look for the word "per": the unit after "per" is the denominator.
3
Divide the numerator quantity by the denominator quantity to get the unit rate.
4
Repeat for each option you need to compare, then select the highest or lowest unit rate as the problem requires.
5
Label your answer with the correct units (e.g., "miles per hour," "dollars per ounce").

Example 1: Best Buy

Problem: Brand A sells 16 oz of juice for \($3.20\). Brand B sells 20 oz for \($3.80\). Which is the better deal?

Brand A
\($3.20 \div 16\) oz = \($0.20\)/oz
 Brand B
\($3.80 \div 20\) oz = \($0.19\)/oz
 Cost per ounce: $0.20 $0.19

Example 2: Faster Runner

Problem: Alex runs \(6\) miles in \(48\) minutes. Jordan runs \(8\) miles in \(56\) minutes. Who runs faster?

1
Alex's unit rate: \(\dfrac{6 \text{ miles}}{48 \text{ min}} = 0.125\) miles/min
2
Jordan's unit rate: \(\dfrac{8 \text{ miles}}{56 \text{ min}} \approx 0.143\) miles/min
3
Compare: \(0.143 > 0.125\), so Jordan covers more miles per minute.

Example 3: Three-Way Comparison

Problem: Three printers print the following: Printer X prints 240 pages in 8 minutes, Printer Y prints 350 pages in 10 minutes, and Printer Z prints 180 pages in 5 minutes. Rank them from fastest to slowest.

Printer Calculation Unit Rate
X \(240 \div 8\) 30 pages/min
Y \(350 \div 10\) 35 pages/min
Z \(180 \div 5\) 36 pages/min

Example 4: Fractional Units

Problem: A recipe uses \(\frac{3}{4}\) cup of sugar for every \(\frac{1}{2}\) batch. How much sugar is needed per full batch?

1
Set up the rate: \(\dfrac{\frac{3}{4} \text{ cups}}{\frac{1}{2} \text{ batch}}\)
2
Divide fractions: \(\dfrac{3}{4} \div \dfrac{1}{2} = \dfrac{3}{4} \times \dfrac{2}{1} = \dfrac{6}{4} = \dfrac{3}{2}\)
3
Result: \(\dfrac{3}{2}\) cups per batch, or \(1.5\) cups per batch.

Unit Rates on Graphs

On a coordinate graph, the unit rate is the slope of the line. A steeper line means a higher rate.

Hours Miles 0 50 100 150 200 1 2 3 4
Car A (\(50\) mph)
Car B (\(40\) mph)

Common Mistakes

✕ Flipping the Ratio
Writing \(\frac{\text{hours}}{\text{miles}}\) instead of \(\frac{\text{miles}}{\text{hours}}\). Always check which unit goes on top.
✕ Comparing Raw Totals
Saying "\($3.20\) is cheaper than \($3.80\)" without converting to unit rates. Different package sizes require per-unit comparison.
✕ Forgetting to Simplify
Leaving the rate as \(\frac{150}{3}\) instead of computing \(50\). Always divide to get the denominator down to 1.

Summary Checklist

Step Action Example
1 Identify quantities and units \($4.50\), \(3\) lbs
2 Place "per" unit in denominator \(\frac{\$4.50}{3 \text{ lbs}}\)
3 Divide to get unit rate \($1.50\) per lb
4 Repeat for all options \($1.50\) vs. \($1.25\)
5 Compare and conclude \($1.25\)/lb wins

Question 1 of 5

A store sells 5 notebooks for \($8.75\). What is the unit price per notebook?

Question 2 of 5

Maria types 240 words in 4 minutes. Ben types 325 words in 5 minutes. Who types faster?

Question 3 of 5

Pack A contains 12 batteries for \($9.00\). Pack B contains 8 batteries for \($5.60\). Which pack has a lower cost per battery?

Question 4 of 5

A car uses \(\frac{3}{4}\) gallon of gas to travel \(22\frac{1}{2}\) miles. What is the car's fuel efficiency in miles per gallon?

Question 5 of 5

Three workers paint fences: Worker P paints 7 fences in 3.5 hours, Worker Q paints 10 fences in 4 hours, and Worker R paints 9 fences in 3 hours. Who is the fastest?