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Unit Rate
Comparisons

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

SSAT Middle Level: Quantitative

What Is a Rate?

A rate is a ratio that compares two quantities measured in different units. Whenever you see "per," you are looking at a rate.

Speed
\( 60 \) miles per \( 1 \) hour
Price
\( \$4.50 \) per \( 1 \) pound
Typing
\( 45 \) words per \( 1 \) minute

What Is a Unit Rate?

A unit rate is a special rate where the second quantity is exactly \( 1 \). It tells you the amount of the first quantity for each single unit of the second.

General Rate
\( \$12 \text{ for } 3 \text{ lbs} \)
\( \$12 : 3 \text{ lbs} \)
\( \div 3 \)
 Unit Rate
\( \$4 \text{ per } 1 \text{ lb} \)
\( \$4 / \text{lb} \)

Why Unit Rates Matter

Without unit rates, comparing quantities with different totals is confusing. Unit rates give you one number to compare, making decisions quick and clear.

Store A 6 apples for $3.00
\( \$3.00 \div 6 = \)
\( \$0.50 \text{ / apple} \)
 Store B 8 apples for $4.80
\( \$4.80 \div 8 = \)
\( \$0.60 \text{ / apple} \)

The Unit Rate Formula

Unit Rate Formula Divide the total of the first quantity by the total of the second quantity

$$ \text{Unit Rate} = \frac{\text{Total Quantity}}{\text{Number of Units}} $$

Using Unit Rates to Compare

To compare two rates, convert both to unit rates with the same denominator unit, then compare the numerators directly.

Runner A 12 miles in 2 hours
\( 12 \div 2 = 6 \text{ mph} \)
 Runner B 20 miles in 4 hours
\( 20 \div 4 = 5 \text{ mph} \)
Compare: \( 6 \text{ mph} > 5 \text{ mph} \)
 Runner A is faster!

Rate vs. Unit Rate

Rate Unit Rate
Definition Compares two quantities Rate with denominator = 1
Example \( \$6 \) for \( 3 \) lbs \( \$2 \) per \( 1 \) lb
Format \( a \) for \( b \) units \( \frac{a}{b} \) per \( 1 \) unit
Use Describes a situation Compares two situations
Easy to compare? Not directly Yes, just compare numbers

How to Find and Compare Unit Rates

1
Identify the two quantities and their units (dollars and pounds, miles and hours, etc.).
2
Decide which unit goes in the denominator. This is the "per" unit: per mile, per hour, per item.
3
Divide the first quantity by the second to get the unit rate.
4
Repeat for each option you need to compare.
5
Compare the unit rates. Lower cost per unit = better deal. Higher distance per time = faster speed.

Example 1: Best Price

Problem: Brand X sells \( 5 \) notebooks for \( \$8.75 \). Brand Y sells \( 3 \) notebooks for \( \$4.80 \). Which brand is cheaper per notebook?

Brand X 5 notebooks for $8.75
\( \$8.75 \div 5 \)
\( = \$1.75 \text{ each} \)
 Brand Y 3 notebooks for $4.80
\( \$4.80 \div 3 \)
\( = \$1.60 \text{ each} \)

Example 2: Faster Cyclist

Problem: Alex cycles \( 24 \) miles in \( 3 \) hours. Jordan cycles \( 35 \) miles in \( 5 \) hours. Who is faster?

Alex
\( 24 \text{ mi} \div 3 \text{ hr} \)
\( = 8 \text{ mph} \)
 Jordan
\( 35 \text{ mi} \div 5 \text{ hr} \)
\( = 7 \text{ mph} \)
 Unit Rate Comparison (mph) 8 Alex 7 Jordan

Example 3: Three-Way Comparison

Problem: Three printers print pages at different rates. Which is fastest?

Printer Total Pages Minutes Unit Rate
Printer A \( 120 \) \( 4 \) \( 30 \) pages/min
Printer B \( 200 \) \( 8 \) \( 25 \) pages/min
Printer C \( 84 \) \( 3 \) \( 28 \) pages/min

Example 4: Unit Price with Decimals

Problem: A \( 12 \)-ounce bottle of juice costs \( \$2.64 \). A \( 16 \)-ounce bottle costs \( \$3.20 \). Which is the better deal?

1
12-oz bottle: \( \frac{\$2.64}{12 \text{ oz}} = \$0.22 \) per ounce
2
16-oz bottle: \( \frac{\$3.20}{16 \text{ oz}} = \$0.20 \) per ounce
3
Compare: \( \$0.20 < \$0.22 \), so the 16-oz bottle is the better deal.

Lower or Higher: Which Wins?

Whether a lower or higher unit rate is "better" depends on what you are measuring.

Lower is better
Cost per item: you want to pay less per unit. Time per task: you want fewer minutes per mile.
Higher is better
Miles per hour: you want more distance per hour. Points per game: you want more points per game.

Common Mistakes to Avoid

✕ Dividing Backwards
Dividing units by dollars instead of dollars by units. Always put the quantity you want "per one of" in the denominator.
✕ Comparing Totals
Saying "$3 is cheaper than $5" without checking quantities. The total cost does not tell you the unit price.
✕ Rounding Too Early
Rounding \( \$0.216 \) to \( \$0.20 \) before comparing. Keep at least 2 decimal places until you finish comparing.

Summary: Unit Rate Comparison Checklist

Step Action Watch Out
1. Identify Name both quantities and units Don't mix up numerator and denominator
2. Divide Compute \( \frac{\text{quantity}}{\text{units}} \) for each Keep decimals: don't round yet
3. Match units Both unit rates must use the same "per" unit Comparing $/lb to $/oz is invalid
4. Compare Lower cost = better deal; higher speed = faster Context decides: lower or higher?
5. Answer State the winner and the unit rate Include units in your answer

Question 1 of 5

A store sells 4 oranges for \( \$3.00 \). What is the unit price per orange?

Question 2 of 5

Sam reads \( 45 \) pages in \( 3 \) hours. Mia reads \( 56 \) pages in \( 4 \) hours. Who reads faster?

Question 3 of 5

A \( 6 \)-pack of water costs \( \$4.20 \). A \( 10 \)-pack costs \( \$6.50 \). Which pack has the lower unit price?

Question 4 of 5

A car travels \( 210 \) miles on \( 6 \) gallons of gas. What is its fuel efficiency as a unit rate?

Question 5 of 5

Three runners finish a race:
Anna: \( 5 \) km in \( 25 \) min. Ben: \( 3 \) km in \( 18 \) min. Carla: \( 4 \) km in \( 20 \) min.
Who ran the fastest?