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ISEE Lower Level: Quantitative Reasoning • Ratios and Proportional Reasoning

Use a rate to solve a unit conversion or comparison problem.

Learn how rates help you compare, convert, and solve real-world problems like a math detective!

Explore the lesson 9 min read
ISEE Lower Level emblem with a school, foundational shapes, and open book
Inside this lesson 10 sections 5 questions 7 figures

SECTION 1

Why Do We Need Rates?

Have you ever wondered how fast a cheetah runs? Or how many slices of pizza your class eats in an hour? People have been asking questions like these for thousands of years! A rate is a special math tool that compares two different things.

Long ago, traders needed to figure out prices. Farmers needed to know how much seed to plant. Travelers needed to know how long a trip would take. Rates helped them solve all of these problems!

3000 BC • Ancient Traders

Merchants in ancient Egypt and Mesopotamia used rates to trade goods. For example, 3 goats for 1 cow!

300 BC • Greek Mathematicians

Greek thinkers studied ratios and rates to understand distance, speed, and time.

1700s • Miles Per Hour

As roads and coaches improved, people started measuring speed in miles per hour to plan travel.

Today • Rates Everywhere!

We use rates every day: price per pound, miles per gallon, words per minute, and so much more.

On the ISEE, you will see word problems that ask you to use rates. The big question is: How can you use a rate to convert or compare? Let's find out!

SECTION 2

Core Ideas About Rates

Before we solve problems, let's learn the key ideas. A rate compares two amounts that use different units. Think of it as a math recipe that connects two things.

1

What Is a Rate?

A rate tells you how much of one thing goes with another. Example: 60 miles in 1 hour means 60 miles per hour.
2

What Is a Unit Rate?

A unit rate is a rate where the second number is 1. "$3 per pound" means each pound costs $3. The word "per" means "for each one."
3

Multiply to Scale Up

If you know the rate for 1, multiply to find the amount for many. If 1 box has 8 crayons, then 5 boxes have 5 × 8 = 40 crayons.
4

Divide to Find the Rate

If you know the total, divide to find the rate for 1. 24 apples shared among 4 bags = 24 ÷ 4 = 6 apples per bag.
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KEY TAKEAWAY

Think of a rate like a vending machine. You put in one thing (like quarters) and get another thing out (like juice boxes). If 1 juice box costs 4 quarters, then the rate is 4 quarters per juice box. You always multiply or divide to figure out how many!

SECTION 3

See How Rates Work

Let's look at a picture that shows how a rate connects two things. Imagine a bakery sells 3 cookies for every $2. The diagram below shows how we can scale that rate up.

Rate Table: 3 Cookies per $2Dollars ($)Cookies$23 🍪$46 🍪$69 🍪$812 🍪Each row multiplies the rate by 1, 2, 3, and 4.Pattern: multiply BOTH sides by the same number!
This rate table shows that if 3 cookies cost $2, we can multiply both numbers by the same amount to find bigger orders. Notice how both columns grow together.

See the pattern? Each time we doubled the dollars, we also doubled the cookies. Each time we tripled the dollars, we tripled the cookies. That's the magic of rates: multiply or divide both sides by the same number.

SECTION 4

The Math Behind Rates

There are two main moves when you work with rates on the ISEE. Let's look at each one with simple formulas.

FINDING A UNIT RATE

UnitRate=TotalAmount÷NumberofUnitsUnit Rate = Total Amount ÷ Number of Units

Example: You read 20 pages in 4 hours. Unit rate = 20 ÷ 4 = 5 pages per hour.

SCALING A RATE UP

Total=UnitRate×NumberofUnitsTotal = Unit Rate × Number of Units

Example: You read 5 pages per hour. In 7 hours you read 5 × 7 = 35 pages.

COMPARING TWO RATES

Findtheunitrateforeach,thencompare.Find the unit rate for each, then compare.

Example: Store A sells 6 pencils for $3 (that's $0.50 each). Store B sells 8 pencils for $4 (that's $0.50 each). They're the same rate!

💡

ISEE Test Tip

When you see a rate problem, look for the word "per" or "each" or "every". These are clue words that tell you a rate is hiding in the problem! Also, remember: there is no penalty for guessing on the ISEE, so always pick an answer.

SECTION 5

Types of Rate Problems You'll See

The ISEE uses rates in different ways. Let's look at the three main types so you'll recognize them right away.

Three Types of Rate ProblemsTYPE 1Find the Unit Rate"12 muffins cost $6.How much is 1 muffin?"DIVIDE: $6 ÷ 12= $0.50 eachTYPE 2Scale Up a Rate"A car goes 50 miles perhour. How far in 3 hours?"MULTIPLY: 50 × 3= 150 milesTYPE 3Compare Two Rates"Amy reads 12 pages in 3hours. Ben reads 20 in 5."12÷3=4 vs 20÷5=4= Same speed!🔑 Quick Decision GuideNeed the amount for 1? → DIVIDE | Know the rate for 1 and want more? → MULTIPLYISEE StrategyStep 1: Read carefully. Circle the rate and what they're asking for.Step 2: Decide — divide to find "per 1" or multiply to scale up.
This diagram shows the three main types of rate problems. Type 1 uses division, Type 2 uses multiplication, and Type 3 uses both to compare.

Here's a simple trick. When the ISEE asks "how much for one?" you divide. When it asks "how much for many?" you multiply. When it says "who is faster (or cheaper)?" you find the unit rate for each and compare.

SECTION 6

Worked Example: Step by Step

Let's solve a problem together, just like you would on the real ISEE. Take your time and follow each step!

ℹ

Problem

A factory makes 36 toy cars in 4 hours. At this rate, how many toy cars does the factory make in 7 hours?

Solution: Toy Cars

  1. 1

    Step 1 — Find the Rate

    The factory makes 36 toy cars in 4 hours. We need to find how many it makes in 1 hour. Divide: 36 ÷ 4 = 9.
    Unit Rate = 9 toy cars per hour
  2. 2

    Step 2 — Multiply to Scale Up

    Now we know the factory makes 9 toy cars every hour. The question asks about 7 hours. Multiply: 9 × 7 = 63.
    Total = 63 toy cars
  3. 3

    Step 3 — Check Your Answer

    Does 63 make sense? In 4 hours the factory makes 36. In 7 hours it should make almost double, since 7 is almost double 4. And 63 is almost double 36. ✓ It makes sense!
    Answer: 63 toy cars
💡

ISEE Test Tip

After you solve, quickly check if your answer makes sense. If the time gets bigger, the total should get bigger too. If something seems way off, go back and try again!

SECTION 7

Helpful Strategies & Common Mistakes

Rate problems can be tricky! Let's look at what helps and what can trip you up.

Smart Strategies ✅Common Mistakes ❌
Always find the unit rate first (the amount for 1).Forgetting to divide first and jumping straight to multiplying.
Look for clue words: "per," "each," "every," "for each."Mixing up which number to divide by which.
Write the rate as a short label, like "5 miles per minute."Writing just a number without its label, like "5" instead of "5 miles."
Check: does your answer make sense? Bigger time = bigger amount.Getting an answer that is smaller when it should be bigger.
Use estimation: round numbers to check if your answer is close.Not reading the question carefully — it might ask for a different unit!

Strategies vs. Common Mistakes

🧠

REMEMBER THIS

Think of rate problems like sharing candy. If you have 20 candies for 5 friends, you divide to find how many each friend gets (20 ÷ 5 = 4 each). If you know each friend gets 4 and there are 8 friends, you multiply (4 × 8 = 32 total). Divide to go smaller, multiply to go bigger!

SECTION 8

Rates Now and Later

The rate skills you're learning now will help you in many future math classes. Here's a peek at how rates grow with you!

What You Learn Now (ISEE)What Comes Later
Unit rates with whole numbers (5 apples per bag)Unit rates with fractions and decimals
Comparing rates by finding the unit rateProportions — two equal rates set side by side
"Miles per hour" and "cost per item"Speed, density, unit pricing, and exchange rates
Multiply and divide to scale ratesCross-multiplication and algebraic equations

How rate skills grow over time

Every time you solve a rate problem now, you are building a strong foundation. These same ideas will pop up in science, cooking, sports stats, and even video games. You've got this!

SECTION 9

Practice Problems

Time to practice! Try each problem on your own first. Remember: find the unit rate (divide), then scale up (multiply) if needed. On the real ISEE, always pick an answer — there's no penalty for guessing!

Question 1: CONCEPTUAL

A store sells 4 notebooks for $8. What is the cost of 1 notebook?
  • A. $1
  • B. $2
  • C. $4
  • D. $8
Show answer (B)
To find the cost of 1 notebook, divide the total cost by the number of notebooks: $8 ÷ 4 = $2. Choice A ($1) is too low — that would mean 4 notebooks cost only $4. Choice C ($4) would mean 4 notebooks cost $16, which is too much. Choice D ($8) is the total cost, not the cost of one notebook.

Question 2: BASIC CALCULATION

A printer prints 10 pages every 2 minutes. How many pages does it print in 8 minutes?
  • A. 16
  • B. 20
  • C. 40
  • D. 80
Show answer (C)
First, find the unit rate: 10 pages ÷ 2 minutes = 5 pages per minute. Then multiply: 5 × 8 = 40 pages. Choice A (16) comes from adding 10 + 6, which is not the right operation. Choice B (20) comes from doubling 10 but forgetting that 8 is four times 2, not two times 2. Choice D (80) comes from multiplying 10 × 8 without first finding the unit rate.

Question 3: INTERMEDIATE

Emma walks 15 blocks in 5 minutes. Jake walks 24 blocks in 6 minutes. Who walks faster?
  • A. Emma, because 15 is less than 24
  • B. Jake, because 24 is more than 15
  • C. They walk at the same speed
  • D. Jake, because he walks 4 blocks per minute
Show answer (D)
Find each person's unit rate. Emma: 15 ÷ 5 = 3 blocks per minute. Jake: 24 ÷ 6 = 4 blocks per minute. Since 4 > 3, Jake walks faster. Choice A is wrong because comparing totals without looking at time doesn't tell you who is faster. Choice B makes the same mistake — just looking at the bigger total. Choice C is wrong because 3 blocks per minute is not equal to 4 blocks per minute.

Question 4: APPLIED

A recipe uses 6 cups of flour for every 3 batches of cookies. Maria wants to make 7 batches. How many cups of flour does she need?
  • A. 7
  • B. 12
  • C. 14
  • D. 21
Show answer (C)
Find the unit rate: 6 cups ÷ 3 batches = 2 cups per batch. For 7 batches: 2 × 7 = 14 cups. Choice A (7) just repeats the number of batches. Choice B (12) comes from doubling 6, which gives the flour for 6 batches, not 7. Choice D (21) comes from multiplying 7 × 3, which mixes up the numbers.

Question 5: CRITICAL THINKING

Pack A has 14 juice boxes for $7. Pack B has 8 juice boxes for $4. Pack C has 18 juice boxes for $6. Which pack gives you the most juice boxes per dollar?
  • A. Pack A only
  • B. Pack B only
  • C. Pack C only
  • D. All three packs give the same number of juice boxes per dollar
Show answer (C)
Find the unit rate (juice boxes per dollar) for each pack. Pack A: 14 ÷ 7 = 2 juice boxes per dollar. Pack B: 8 ÷ 4 = 2 juice boxes per dollar. Pack C: 18 ÷ 6 = 3 juice boxes per dollar. Pack C gives 3 juice boxes per dollar, which is more than Pack A's 2 and Pack B's 2. Pack C gives the most juice boxes per dollar. The answer is C.

SUMMARY

Putting It All Together

Summary

A rate compares two amounts with different units, like miles and hours or dollars and items. To find a unit rate, you divide to find the amount for just 1. To scale up a rate, you multiply the unit rate by the number you need. To compare rates, find the unit rate for each and see which is bigger or smaller.

On the ISEE, look for clue words like "per," "each," and "every" to spot rate problems. Always check that your answer makes sense. And remember — never leave a question blank! Use process of elimination to cross out wrong answers and make your best guess. You've got this!

Varsity Tutors • ISEE Lower Level • Use a rate to solve a unit conversion or comparison problem.