Unit Rate Comparisons — Use unit rates to compare quantities. - SHSAT Math

SECTION 1

Why Do We Compare Rates?

People have been comparing quantities for thousands of years. Imagine you are a trader in ancient Egypt trying to figure out which merchant offers a better deal on grain. You need a way to compare prices fairly, even when the amounts are different. That is exactly the problem that unit rates solve.

A unit rate tells you how much of something you get for one unit of something else. It is one of the oldest and most useful math ideas in history.

~3000 BCE • Ancient Trade

Egyptian and Mesopotamian traders compared prices of goods like grain and cloth using simple ratios.

~300 BCE • Greek Mathematics

Greek mathematicians like Euclid studied ratios and proportions in a formal way, writing rules that still guide us today.

1600s • Science & Speed

Galileo used rates like distance per unit of time to study motion. This led to ideas like miles per hour.

Today • Everyday Comparisons

We use unit rates every day — price per ounce at the store, miles per gallon in a car, and points per game in sports.

So here is the big question: when two quantities are written as different ratios, how do you figure out which one is the better deal, the faster speed, or the higher score? That is what this lesson will teach you.

SECTION 2

Core Principles & Definitions

Before we start comparing, let's make sure we understand the key vocabulary. A ratio is a comparison of two quantities, like 3 apples to 5 oranges. A rate is a special ratio that compares two quantities with different units, like 120 miles in 2 hours. A unit rate is a rate where the second quantity is exactly 1, like 60 miles per 1 hour.

Ratio

A comparison of two numbers. Example: 4 to 5, or 4 : 5, or 4/5.

Rate

A ratio that compares quantities with different units. Example: 150 miles in 3 hours.

Unit Rate

A rate simplified so the denominator (bottom number) is 1. Example: 50 miles per 1 hour.

"Per"

The word "per" means "for each one." It signals that you are looking at a unit rate.

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KEY TAKEAWAY

Think of a unit rate like slicing a pizza so everyone gets exactly one slice. If you know how much topping is on one slice, you can easily compare any two pizzas. Finding a unit rate means dividing so one quantity equals exactly 1.

SECTION 3

Seeing Unit Rates in Action

Let's look at a picture that shows how unit rates help us compare. Suppose Store A sells 6 notebooks for $9 and Store B sells 4 notebooks for $5. Which store gives you the better price per notebook?

We divided the total cost by the number of notebooks at each store. Store B's unit rate ($1.25 per notebook) is lower than Store A's ($1.50 per notebook), so Store B is the better deal.

Notice how the bars make it easy to see the difference at a glance. The shorter bar means a lower cost per notebook. On the SHSAT, you won't draw bars, but you will calculate unit rates and compare the numbers. The idea is the same!

SECTION 4

The Math Behind Unit Rates

Finding a unit rate is all about division. You take the total amount and divide it by the number of units. Here is the formula you need.

UNIT RATE FORMULA

Unit Rate = Total Quantity ÷ Number of Units

Total Quantity = the amount you are measuring (dollars, miles, points, etc.). Number of Units = how many of the other item (hours, items, games, etc.).

For example, if a car travels 240 miles in 4 hours, you divide 240 by 4 to get 60 miles per hour. The words "per hour" tell you the second quantity has been reduced to 1.

COMPARISON RULE

If Unit Rate A > Unit Rate B, then A has the greater rate.

After finding the unit rates, simply compare them. A higher unit rate means more per unit (faster, more expensive, etc.). A lower unit rate means less per unit (slower, cheaper, etc.).

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SHSAT Tip

Always check what the question is asking. If it asks "which is cheaper," you want the lower unit rate. If it asks "which is faster," you want the higher unit rate.

SECTION 5

Common Types of Unit Rates

Unit rates pop up everywhere. Let's look at the most common types you will see on the SHSAT and in daily life. The diagram below organizes them by category.

This tree diagram shows three major categories of unit rates: price rates, speed rates, and other rates like scoring or reading speed. Each one uses the same division strategy.

Situation	What You Divide	Unit Rate
Shopping	Total price ÷ number of items	Price per item
Driving	Total miles ÷ total hours	Miles per hour
Typing test	Total words ÷ total minutes	Words per minute
Basketball	Total points ÷ total games	Points per game

Common unit rate situations

SECTION 6

Worked Example: Who Runs Faster?

Aisha runs 3 miles in 24 minutes. Ben runs 5 miles in 35 minutes. Who runs faster? Let's solve this step by step using unit rates.

Comparing Running Speeds

1
Step 1 — Write each rate as a fraction

Aisha: 3 miles ÷ 24 minutes. Ben: 5 miles ÷ 35 minutes. We want to find the number of miles per 1 minute for each runner.
2
Step 2 — Divide to find Aisha's unit rate

Aisha: 3 ÷ 24 = 0.125 miles per minute.

Aisha's unit rate = 0.125 miles per minute
3
Step 3 — Divide to find Ben's unit rate

Ben: 5 ÷ 35 = 0.1428… ≈ 0.143 miles per minute.

Ben's unit rate ≈ 0.143 miles per minute
4
Step 4 — Compare the unit rates

0.143 > 0.125. Since Ben covers more miles in each minute, Ben runs faster.

Ben is faster.

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Check Your Work

You could also compare by finding minutes per mile instead. Aisha: 24 ÷ 3 = 8 minutes per mile. Ben: 35 ÷ 5 = 7 minutes per mile. A lower time per mile means faster. Ben's 7 < Aisha's 8, so Ben is faster. Same answer!

SECTION 7

Common Mistakes & Helpful Tips

Unit rate problems seem simple, but there are some traps that catch students on test day. Here is a quick guide to what works and what doesn't.

❌ Common Mistake	✅ What to Do Instead
Dividing the wrong way (e.g., hours ÷ miles instead of miles ÷ hours)	Read the question carefully. Put the quantity you want "per 1" in the denominator.
Comparing rates that have different units (e.g., $/ounce vs. $/pound)	Make sure both rates use the same units before comparing.
Forgetting to simplify or rounding too early	Keep at least 2–3 decimal places while working. Only round at the end.
Saying a higher rate is always "better"	It depends! Higher speed = better if you want fast. Higher price = worse if you want cheap.

Mistakes vs. best practices

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REMEMBER THIS

Imagine you and a friend each pour water into a bucket. You pour 10 cups in 2 minutes. Your friend pours 15 cups in 4 minutes. To figure out who is pouring faster, you need to know how many cups each person pours per one minute. Always bring it back to "per 1" — that is the whole point of a unit rate.

SECTION 8

From Unit Rates to Proportions

Unit rates are your first step into a bigger world of math called proportional reasoning. Once you know how to find a unit rate, you can set up and solve proportions, work with percentages, and even graph straight lines. Here is how these ideas connect.

Concept	What It Means	How Unit Rates Help
Unit Rates	A rate where the denominator is 1	This is the foundation for all comparison problems.
Proportions	Two equivalent ratios set equal to each other	If you know the unit rate, you can scale up to any amount.
Slope of a Line	Rise over run on a graph	The slope IS a unit rate — it tells you how much y changes per 1 unit of x.
Percent Problems	A rate per 100	A percent is a special unit rate: amount per 100.

How unit rates connect to future topics

On the SHSAT, unit rate questions can show up as straightforward comparisons or inside word problems about speed, price, or work. Mastering this skill now will make proportions and algebra much easier later.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

In your own words, what is the difference between a rate and a unit rate? Give one example of each.

Show answer

A rate compares two quantities with different units, such as 150 miles in 3 hours. A unit rate is a special rate where the second quantity equals 1, such as 50 miles per 1 hour. You find the unit rate by dividing the first quantity by the second. The word "per" signals a unit rate.

Question 2: BASIC CALCULATION

A pack of 8 markers costs $6.00. What is the unit price (cost per marker)?

Show answer

Divide the total cost by the number of markers: $6.00 ÷ 8 = $0.75. The unit price is $0.75 per marker. This means each marker costs seventy-five cents.

Question 3: INTERMEDIATE

Carla types 432 words in 9 minutes. David types 510 words in 12 minutes. Who types faster, and by how many words per minute?

Show answer

Carla's unit rate: 432 ÷ 9 = 48 words per minute. David's unit rate: 510 ÷ 12 = 42.5 words per minute. Since 48 > 42.5, Carla types faster. The difference is 48 − 42.5 = 5.5 words per minute. We found each unit rate by dividing total words by total minutes, then compared them.

Question 4: APPLIED

At the grocery store, Brand X orange juice costs $3.60 for a 48-ounce bottle. Brand Y costs $2.70 for a 36-ounce bottle. Which brand is the better deal, and how much would you save per ounce by choosing it?

Show answer

Brand X: $3.60 ÷ 48 oz = $0.075 per ounce. Brand Y: $2.70 ÷ 36 oz = $0.075 per ounce. Surprise — both brands cost exactly the same per ounce! You would save $0.00 per ounce because the unit prices are equal. This problem shows why you should always calculate instead of guessing. The cheaper total price does not always mean a better deal.

Question 5: CRITICAL THINKING

Train A travels 210 miles in 3.5 hours. Train B travels 168 miles in 2.8 hours. Train C travels 275 miles in 4.5 hours. Rank the trains from fastest to slowest. Then explain: if all three trains left at the same time, how far ahead of the slowest train would the fastest train be after 2 hours?

Show answer

First find each unit rate. Train A: 210 ÷ 3.5 = 60 mph. Train B: 168 ÷ 2.8 = 60 mph. Train C: 275 ÷ 4.5 ≈ 61.11 mph. Ranking from fastest to slowest: Train C (≈61.11 mph), then Train A and Train B are tied (60 mph). After 2 hours, Train C would travel about 61.11 × 2 ≈ 122.22 miles. The slowest trains (A and B, tied at 60 mph) would travel 60 × 2 = 120 miles. So Train C would be about 122.22 − 120 = 2.22 miles ahead. This problem shows that even small differences in unit rates add up over time.

SUMMARY

Lesson Summary

Summary

A unit rate is a rate where the second quantity equals 1. You find it by dividing the total quantity by the number of units. Common examples include price per item, miles per hour, and points per game. To compare two rates, convert both to unit rates using the same units, then see which number is greater or smaller depending on what the question asks.

Watch out for common mistakes: dividing in the wrong order, comparing rates with different units, or assuming that a higher rate is always "better." On the SHSAT, always read the question carefully. This skill connects directly to proportions, slope, and percent problems, so mastering unit rates now will pay off throughout your math journey.