⏪ Quick Check

If these feel comfortable, skim the explanations and spend more time on the SAT Practice and Challenge questions.

• Which is larger? 3/4 or 2/3
• What is 1/2 + 1/2?
• Write 2/5 as a decimal.
• Simplify 12/18
• Which is larger? 5/8 or 6/8

📋 Essential Ideas

On the SAT, fractions appear in all three forms.

Recognizing which meaning the fraction represents is often the first step toward solving the problem.

🧐 First Thought

Whenever you see a fraction, don’t panic.

Simply ask yourself:

“Is this showing part of a whole, a division, or a ratio?”

That one question often tells you how to begin.

📊 Common Fraction Equivalents

🚀 High-Frequency Fraction Equivalents

These aren’t facts to memorize overnight.

The more you work with fractions, the more naturally these values will come to mind—and that can save valuable time on the SAT.

📱 Different Representations, Same Number

Figure 5.1 — Different Representations of the Same Number

Different representation. Same quantity.

As you solve problems, you’ll become comfortable moving from one form to another.

🦋 Guided Example

🧠 First Thought: There are several correct ways to compare fractions. Here we’ll convert them to decimals because it’s quick and convenient.

3/5 = 0.6
5/8 = 0.625

Since 0.625 > 0.6, the greater fraction is 5/8.

👀 Read Carefully

Students sometimes compare only the numerators.

For example, 3/8 and 2/7. They see 3 > 2 and stop thinking.

Not so fast!

When denominators are different, you need a valid comparison method.

Convert to decimals, use equivalent fractions, compare with common denominators, or use another method that makes sense to you.

🚀 Speed Boost

When you see common values such as 25%, 50%, 75%, 20%, 0.5, 0.25, 0.2, try recognizing them immediately instead of converting every time.

For example:

• 25% = 1/4
• 50% = 1/2
• 75% = 3/4
• 20% = 1/5

As you practice, you’ll naturally become faster at moving between fractions, decimals, and percentages.

Choose whichever representation helps you think most clearly.

⏪ Quick Check

• Convert 1/2 to a decimal.
• Convert 0.75 to a fraction.
• Simplify 15/20
• Which is larger? 4/5 or 5/6?
• Write 25% as a fraction.

🟢 SAT Practice

Question 1

A water tank is 3/4 full. After some water is used, it is 1/2 full. What fraction of the tank was used?

Question 2

A student answered 18 out of 24 questions correctly. Which fraction is equivalent to the student’s score?

Question 3

A recipe uses 2/3 cup of milk. Emma wants to make half of the recipe. How much milk should she use?

🎯 SAT Connection

Fractions appear throughout SAT Math.

You’ll see them in:

• Algebra
• Functions
• Geometry
• Ratios
• Probability
• Statistics
• Word Problems

The more comfortable you become with fractions, the easier many later topics will feel.

✅ Answers & Solutions

Quick Check

SAT Practice

⭐ Key Idea & Looking Ahead

Fractions, decimals, and percentages are simply different ways of representing the same quantity.

As you solve more problems, you’ll naturally begin choosing the representation that feels most convenient.

There isn’t one “best” method.

Choose the one that helps you understand the mathematics most clearly.

🌟 Looking Ahead

You’ve met fractions and learned that they’re much less mysterious than they first appear.

Next, we’ll focus on one of the SAT’s favorite fraction skills:

Simplifying fractions quickly and correctly.

A simple fraction can often make a difficult problem feel surprisingly easy.

📝 Quick Check

1/2 = □/10

Which fraction is equivalent to 3/4?

• A) 6/8
• B) 5/7
• C) 8/9

True or False? 3/6 = 1/2

If these feel comfortable, skim the explanations and spend more time on the SAT Practice and Challenge questions.

📌 Essential Idea

• ✔ Two fractions are equivalent if they represent exactly the same value.
• ✔ You can create an equivalent fraction by multiplying both the numerator and denominator by the same nonzero number, or dividing both by the same nonzero number.
• ✔ That’s all there is to it.

📖 Picture It

Imagine the same pizza.

If it’s cut into 2 slices, eating 1 slice means you’ve eaten 1/2.

Now imagine cutting the same pizza into 4 slices. Eating 2 slices means you’ve eaten 2/4.

Different fractions. Exactly the same amount of pizza. That’s what equivalent means.

📊 Examples

⚠️ Watch Out!

Never change only one part of a fraction.

For example, 1/2 does not become 2/2.

The numerator and denominator are teammates. They always move together.

🎯 Guided Example 1

Which fraction is equivalent to 3/5?

• A) 6/10
• B) 6/11
• C) 9/20
• D) 12/25

🧠 First Thought

Don’t start converting everything to decimals. Ask yourself: Did the numerator and denominator change by the same factor?

From 3/5 to 6/10, both numbers were multiplied by 2.

✅ Solution

Answer: A

🎯 Guided Example 2

Complete the missing value: 4/7 = □/21

🧠 First Thought

The denominator changed 7 → 21. That’s multiplication by 3.

Multiply the numerator by the same factor: 4 × 3 = 12.

✅ Solution

12

🚀 Speed Boost

When the new denominator is a multiple of the original denominator, you often don’t need cross multiplication.

Example: 5/8 = □/40

Since 8 → 40 is multiplication by 5, multiply the numerator by 5 as well: 5 × 5 = 25.

So 25/40. Much faster.

👀 Read Carefully

The SAT rarely asks, “Which fraction is equivalent?”

Instead, it may ask:

• Which expression has the same value?
• Which ratio represents the same relationship?
• Which probability is equal to…?

The wording changes. The mathematics doesn’t.

🚀 High-Frequency Equivalent Fractions

Notice something interesting? Many familiar percentages come directly from equivalent fractions. We’ll build on this idea throughout the rest of the chapter.

⚠️ Watch Out!

A very common mistake is adding the same number instead of multiplying.

For example, some students write 3/5 = 6/8. Why? Because they added 3 to both numbers.

✘ That doesn’t preserve the value.

Equivalent fractions are created by multiplying or dividing, not by adding or subtracting.

📝 Quick Check

1) Find the missing number: 1/2 = □/8

2) Find the missing number: 5/7 = 15/□

3) Find the missing number: 4/9 = □/27

4) Which fraction is equivalent to 3/4?

• A) 8/10
• B) 9/12
• C) 10/15
• D) 12/20

🟢 SAT Practice

Question 1

A recipe uses 2/3 cup of yogurt. Which measurement is equivalent?

• A) 4/9
• B) 6/8
• C) 8/9
• D) 10/15

Question 2

A survey found that 18 out of 30 students preferred online homework. Which fraction represents the same proportion in simplest form?

Question 3

A map uses the scale 3 cm for every 15 km. Which ratio represents the same scale?

• A) 2 : 8
• B) 1 : 5
• C) 6 : 20
• D) 9 : 30

Question 4

Which expression has the same value as 15/20?

• A) 3/4
• B) 4/5
• C) 2/3
• D) 3/5

🔴 Challenge

Without converting to decimals, decide which fraction is larger: 7/9 or 21/28.

Can you justify your answer?

Hint: Try rewriting one or both fractions as equivalent fractions—or simplify them first.

🎯 SAT Connection

Equivalent fractions quietly appear throughout SAT Math, including:

• Ratios
• Percentages
• Similar figures
• Probability
• Unit conversions
• Algebraic fractions
• Rational equations

The better you become at recognizing equivalent fractions, the easier many later topics will feel.

✅ Answers & Solutions

Quick Check Solutions

SAT Practice Solutions

🔴 Challenge Solution

Simplify 21/28 = 3/4. Now compare 7/9 and 3/4. Since 7/9 ≈ 0.778 and 3/4 = 0.75, we conclude 7/9 > 3/4.

💡 Key Idea

Equivalent fractions aren’t just about rewriting numbers—they’re about making comparisons, simplifying calculations, and recognizing familiar values more quickly. That’s why this idea appears again and again throughout SAT Math.

📝 Quick Check

6/8, 15/20, 18/24, 42/56, 45/60

If these feel comfortable, skim the explanations and spend more time on the SAT Practice and Challenge questions.

📌 Essential Idea

• ✔ A fraction is in simplest form when the numerator and denominator have no common factor greater than 1.
• ✔ To simplify a fraction, divide both the numerator and denominator by the same factor.
• ✔ Repeat until no further simplification is possible.

🧐 First Thought

Before doing any calculations, ask yourself:

“What is the greatest number that divides both?”

Finding that number often lets you simplify in a single step.

📊 Examples

🚀 Pattern Recognition

After enough practice, you’ll begin recognizing common factor pairs almost instantly.

You don’t need to memorize this table. The goal is simply to become familiar with these patterns through practice.

🎯 Guided Example 1

Simplify 36/48

🧠 First Thought

Can both numbers be divided by 12? Yes.

36 ÷ 12 = 3, 48 ÷ 12 = 4

✅ Solution

3/4

🎯 Guided Example 2

Simplify 63/81

🧠 First Thought

Both numbers are divisible by 9.

63 ÷ 9 = 7, 81 ÷ 9 = 9

✅ Solution

7/9

⚠️ Watch Out!

Some students divide only the numerator.

For example, 12/18 becomes 2/9.

✘ Incorrect.

Always divide both numbers by the same factor.

Think of a fraction as a balanced scale. If one side changes, the other must change in exactly the same way.

👀 Read Carefully

The SAT rarely asks, “Simplify this fraction.”

Instead, simplifying is often a hidden step inside questions about:

• Ratios
• Probabilities
• Slopes
• Algebra
• Functions
• Geometry

Whenever you see a fraction, ask yourself: “Can I simplify this first?”

⭐ SAT Habit #1

This is one of the best habits you can develop for SAT Math.

Before multiplying, before dividing, before solving an equation, take one quick look.

Can anything be simplified?

A few seconds spent simplifying can save you much more time later—and help prevent arithmetic mistakes.

🚀 Speed Boost

Sometimes you don’t even need to find the greatest common factor. You can simplify in steps.

Example: 24/36

Divide both by 2 → 12/18. Now divide both by 6 → 2/3.

You reached the same answer. Choose whichever method feels most natural to you.

📝 Quick Check

1) 8/12

2) 28/42

3) 30/45

4) 54/72

5) 81/108

🟢 SAT Practice

Question 1

A classroom has 24 girls and 36 boys. What fraction of the class is made up of girls? Write your answer in simplest form.

Question 2

A recipe calls for 18 ounces of juice and 30 ounces of water. What is the ratio of juice to water in simplest form?

Question 3

A bag contains 14 red marbles and 21 blue marbles. What fraction of the marbles are red? Express your answer in simplest form.

Question 4

Which expression is equivalent to 45/60?

• A. 3/4
• B. 4/5
• C. 2/3
• D. 5/6

🔴 Challenge

Without simplifying every fraction completely, identify the fraction that is already in simplest form:

• A. 42/63
• B. 32/48
• C. 25/36
• D. 54/72

Can you explain how you knew?

🎯 SAT Connection

Simplifying fractions quietly appears throughout SAT Math, including:

• Slope calculations
• Ratios
• Probability
• Similar triangles
• Rational expressions
• Algebraic equations
• Functions

Students who automatically simplify fractions tend to make fewer arithmetic mistakes and solve problems more efficiently.

✅ Answers & Solutions

Quick Check Solutions

SAT Practice Solutions

🔴 Challenge Solution

Answer: C

Let’s check for common factors:

• 42/63 → divisible by 21 ❌
• 32/48 → divisible by 16 ❌
• 25/36 → no common factors greater than 1 ✅
• 54/72 → divisible by 18 ❌

Therefore, 25/36 is already in simplest form.

💡 Key Idea

A fraction is in simplest form when the numerator and denominator share no common factor greater than 1. Before simplifying, quickly ask yourself whether the two numbers have any common divisors. With practice, you’ll recognize many of these relationships almost instantly.

📚 Looking Ahead

You’ve now learned how to:

• ✔ Recognize fractions
• ✔ Convert between fractions, decimals, and percentages
• ✔ Find equivalent fractions
• ✔ Simplify fractions

These aren’t separate skills—they’re closely connected and often work together in SAT problems.

Next, you’ll learn how to compare fractions efficiently, even when they have different denominators.

⏪ Already Know This?

• Which is larger? 3/4 or 4/5
• Which is larger? 1/2 or 3/5
• Which is larger? 7/9 or 8/9
• Which is larger? 3/8 or 5/8
• Are these equal? 2/4 and 3/6

If these feel comfortable, skim the explanations and spend more time on the practice questions.

📋 Essential Ideas

• ✔ There is more than one correct way to compare fractions.
• ✔ The goal isn’t to memorize one method.
• ✔ The goal is to recognize which method is quickest for the problem in front of you.

📊 Comparison Strategies (Cases 1 – 3)

Case 1 — Same Denominator

Compare the numerators.

Example: 3/8 < 5/8 because 3 < 5.

Case 2 — Same Numerator

Compare the denominators. The smaller denominator gives the larger fraction because the whole is divided into fewer equal pieces.

Example: 3/7 > 3/9.

Case 3 — Different Numerators and Denominators

Choose the method that makes the comparison easiest. You might:

• Rewrite the fractions with a common denominator
• Think of familiar decimal values
• Recognize equivalent fractions
• Use cross multiplication

There is no single “correct” method. Strong SAT students choose the one that saves the most time.

🧐 First Thought

Before calculating anything, pause for one second and ask yourself:

“Do these fractions already have something in common?”

Many students immediately use cross multiplication—even when the answer is obvious. Looking first often saves both time and mistakes.

📊 Examples

💡 Which Method Should I Use?

Notice what we’re doing. We’re not memorizing one procedure. We’re learning to choose the smartest one. That’s exactly how experienced SAT students think.

❌ Use Cross Multiplication

Sometimes neither fraction has a familiar denominator. Neither converts easily to a decimal. Nothing stands out. This is a good time to use cross multiplication.

Suppose we want to compare 4/7 and 5/9. Instead of finding a common denominator directly, think about what that denominator would be. Both fractions can be rewritten with a denominator of 7 × 9 = 63.

So, 4/7 = (4 × 9)/63 and 5/9 = (5 × 7)/63. Now both fractions have the same denominator, meaning we only need to compare the numerators. Since 4 × 9 > 5 × 7, we know 4/7 > 5/9.

Notice what happened. We never actually needed to write the denominator 63 in our calculations. We simply compared the numerators that each fraction would have after being rewritten with the same denominator. That’s exactly why cross multiplication works.

🎯 Guided Example 1

Which fraction is larger? 5/8 or 3/4

🧠 First Thought

Both are familiar fractions. Think mentally.

5/8 = 0.625, 3/4 = 0.75

Therefore, 3/4 is larger.

🎯 Guided Example 2

Which fraction is larger? 4/7 or 5/9

🧠 First Thought

Nothing stands out immediately. This is a good time to use cross multiplication.

Compare 4 × 9 and 5 × 7.

Since 36 > 35, 4/7 > 5/9.

⚠️ Trap Alert!

Many students compare only the numerators. For example, 2/9 and 3/8. Since 3 > 2, they choose 3/8.

This time they happen to be correct… but the reasoning is not. On another question, it could lead to the wrong answer. Always use a valid comparison strategy.

⭐ SAT Habit #2

Before you write anything, pause for one second. Ask yourself, “Is there a quicker way?”

Look for the same denominator, the same numerator, familiar fractions, or equivalent fractions before deciding to use cross multiplication. Choosing the right strategy is part of solving the problem.

📝 Warm-Up

• 2/5 □ 3/5
• 4/9 □ 4/7
• 6/8 □ 3/4
• 5/7 □ 5/8
• 10/15 □ 2/3

🟢 SAT-Style Practice

Question 1

A class completed 5/8 of a project. Another class completed 3/4. Which class completed a larger fraction of its project?

Question 2

Which fraction is greatest?

• A) 4/5
• B) 5/6
• C) 7/8
• D) 11/12

Hint: All four fractions are close to 1. Which one is missing the smallest piece?

Question 3

A recipe uses 2/3 cup of flour. Another recipe uses 3/4 cup. How much more flour does the second recipe use?

🔴 Challenge

Without converting to decimals, order the fractions from least to greatest:

4/5, 5/6, 7/8, 11/12

Which comparison did you make first? Explain why your strategy was efficient.

🎯 SAT Connection

Comparing fractions appears throughout the SAT in ratios, probability, statistics, data analysis, graph interpretation, and word problems. Sometimes the SAT never asks, “Which fraction is larger?” Instead, it hides the comparison inside a real-world situation. Recognizing the underlying mathematics is an important SAT skill.

✅ Answers & Solutions

Quick Check / Warm-Up

SAT-Style Practice Solutions

🔴 Challenge Solution

Notice that all four fractions are close to 1. Compare what each fraction is missing from 1:

• 4/5 is missing 1/5
• 5/6 is missing 1/6
• 7/8 is missing 1/8
• 11/12 is missing 1/12

The smaller the missing piece, the larger the fraction. Therefore, 4/5 < 5/6 < 7/8 < 11/12.

🎉 Congratulations!

You’ve completed the first part of your mathematical toolbox. Now it’s time to combine those ideas.

Unlike the previous spreads, these questions won’t tell you which skill to use. Some involve integers. Some involve fractions. Some require simplification. Some require comparing fractions.

Your first task is to decide what kind of problem you’re looking at. That’s exactly what happens on the SAT.

🌟 Before You Begin

Ask yourself three questions for every problem:

• What is the question really asking?
• Which idea fits this problem best?
• Is there a quicker way?

Remember:

“Observe → Identify → Reason → Solve → Verify”

🟢 Warm-Up

1. Which of the following is not an integer?

• A) 0
• B) 81
• C) -15
• D) 92

2. Arrange from least to greatest:

4, −3, 0, −8, 6

3. Simplify:

24/36

4. Which fraction is larger?

5/8 or 2/3. Choose the quickest method.

🟡 Mixed SAT Practice

Question 5

A diver starts 18 meters below sea level. She rises 11 meters. She then descends another 4 meters. What is her final position?

• A) 3
• B) –3
• C) –11
• D) –25

Question 6

Which fraction is equivalent to 18/30?

• A) 3/5
• B) 2/3
• C) 5/6
• D) 4/9

Question 7

Without using a calculator, which fraction is greatest?

• A) 3/4
• B) 5/6
• C) 7/8
• D) 4/5

Question 8

A recipe calls for 3/4 cup of sugar. Emma prepares only half of the recipe. How much sugar should she use?

• A) 3/8
• B) 1/4
• C) 1/2
• D) 5/8

Question 9

Which statement is true?

• A) 6/8 > 3/4
• B) –5 > –2
• C) 15/20 = 3/4
• D) 0 is positive

🔴 Challenge Problems

10

Without converting every fraction into decimals, order these from least to greatest:

2/3, 3/4, 5/6, 7/8

Explain your strategy.

11

Without using a calculator, find the value of −8 + 24/6 − 3.

Can you simplify before calculating?

12 ★

A class completed 3/4 of a science project on Monday. On Tuesday, they completed half of the remaining work. What fraction of the entire project has now been completed?

• A) 7/8
• B) 3/4
• C) 5/8
• D) 5/6

🌟 Reflection

Before checking the answers, ask yourself:

• Which questions felt easy?
• Which questions took the longest?
• Did you choose efficient methods?
• Did you simplify before calculating?
• Did you use cross multiplication only when it was actually helpful?

Improving these habits is just as important as getting the correct answer.

✅ Answers & Solutions

💡 Dr. Aytekin Says

If you solved every question correctly, that’s excellent—but don’t just move on.

Ask yourself why each method worked.

If you missed a question, don’t think of it as a setback. Think of it as a clue pointing to the next skill to strengthen.

Remember:

“The SAT doesn’t reward students who memorize the most. It rewards students who recognize patterns, choose efficient strategies, and apply the fundamentals with confidence.”

🏆 Checkpoint Complete!

You have successfully completed the first checkpoint challenge. Keep refining your approach and building confidence across all fundamental topics!

⏪ Quick Check

• 1/4 + 2/4
• 3/7 − 1/7
• 1/2 + 1/4
• 5/6 − 1/3
• 2/3 + 1/6

If these feel comfortable, skim the explanations and spend more time on the practice questions.

📋 Essential Rules

Rule 1 — Same Denominator

If the denominators are the same, add or subtract only the numerators. The denominator stays the same.

Example: 3/8 + 2/8 = 5/8

Rule 2 — Different Denominators

If the denominators are different, make them the same first. This is called finding a common denominator. Only then can you add or subtract.

🧐 First Thought

Before writing anything, ask yourself:

“Can I use one of the denominators as the common denominator?”

If not, look for the least common denominator. Smaller numbers usually mean less work.

📊 Examples

💡 Choosing a Common Denominator

Many books simply tell students: “Multiply the denominators.” That always works. But it isn’t always the smartest choice.

Example: 1/2 + 1/6

Some students use 12 as the common denominator. It works. But since 6 is already a multiple of 2, we can simply use 6. Much less work. Whenever possible, choose the smallest common denominator.

⭐ SAT Habit #3

Strong SAT students don’t just know procedures. They look for the simplest path.

Before multiplying denominators, ask:

• Is one denominator already a multiple of the other?
• Is there a smaller common denominator?

A few seconds of thinking often saves many seconds of calculation.

🎯 Guided Examples

Guided Example 1

Find 1/2 + 1/3

🧠 First Thought: The denominators are different. The least common denominator of 2 and 3 is 6. Rewrite both fractions: 1/2 = 3/6, 1/3 = 2/6. Now add: 3/6 + 2/6 = 5/6.

Guided Example 2

Find 3/4 − 1/6

🧠 First Thought: The least common denominator of 4 and 6 is 12. Rewrite: 3/4 = 9/12, 1/6 = 2/12. Subtract: 9/12 − 2/12 = 7/12.

⚠️ Trap Alert! & Read Carefully

A very common mistake is 1/2 + 1/3 = 2/5.

❌ Never add numerators and denominators. Always make the denominators the same first.

👀 Read Carefully: The SAT rarely says, “Add these fractions.” Instead, it may ask things like:

• What fraction of the project has been completed?
• How much of the tank is still full?
• What portion of the class chose science?

The story changes. The mathematics does not.

📝 Warm-Up

• 5/8 + 1/8
• 7/9 − 2/9
• 1/2 + 1/6
• 3/4 + 1/8
• 5/6 − 1/3

🟢 SAT-Style Practice

Question 1

Emma completed 2/5 of a book on Monday and 1/5 on Tuesday. What fraction of the book has she completed?

Question 2

A water tank is 5/6 full. Later, 1/3 of the tank is emptied. What fraction of the tank is still full?

Question 3

A runner completed 3/4 of a race before stopping. She later completed another 1/8 of the race. How much of the race has she completed altogether?

Question 4

A recipe calls for 3/4 cup of milk and 1/6 cup of cream. How much liquid is used altogether?

🔴 Challenge

Without using a calculator, find 3/4 + 1/6 + 1/12.

Show your reasoning. (Hint: Think about the least common denominator first.)

🚀 Speed Boost & SAT Connection

🚀 Speed Boost: After every fraction calculation, ask yourself one final question: Can I simplify my answer? That last step often turns a correct answer into the answer choice that appears on the SAT.

🎯 SAT Connection: Adding and subtracting fractions appear throughout SAT Math, especially in:

• Ratios
• Probability
• Percent problems
• Algebraic fractions
• Geometry
• Statistics
• Multi-step word problems

The better you become at fraction arithmetic, the more mental energy you’ll have for the ideas that make SAT questions truly challenging.

💡 Dr. Aytekin Says

Mastering fraction addition and subtraction is not about memorizing mechanical steps; it is about recognizing relationships between numbers instantly.

When you look at denominators like 2, 4, 6, and 12, your mind should immediately map their common multiples without tedious multiplication. Build this fluency now, and your speed on test day will naturally follow.

✅ Answers & Solutions

Quick Check Answers

Warm-Up Answers

SAT-Style Practice Solutions

🔴 Challenge Solution

The least common denominator is 12.

3/4 = 9/12,   1/6 = 2/12,   1/12 = 1/12

Now add: 9/12 + 2/12 + 1/12 = 12/12 = 1

Answer: 1

⏪ Quick Check

• 1/2 × 3/4
• 2/3 × 3/5
• 3/4 ÷ 1/2
• 4/5 ÷ 2/5
• 6/7 × 14/15

If these feel comfortable, spend most of your time on the SAT Practice section.

📋 Essential Rules

Rule 1 — Multiplication

• Multiply the numerators.
• Multiply the denominators.
• Then simplify if necessary.

Formula:

Rule 2 — Division

To divide by a fraction, multiply by its reciprocal.

The reciprocal simply flips the fraction.

Formula:

🧐 First Thought

Before multiplying, always ask yourself:

“Can I simplify first?”

Small simplifications often save a lot of arithmetic.

📊 Examples

🚀 Speed Boost

Example: 4/9 × 3/8

• Cancel common factors.
• 4 and 8 divide by 4.
• 3 and 9 divide by 3.

Now multiply: 1/3 × 1/2 = 1/6.

Much easier than multiplying first.

🦋 Guided Example 1

Find 2/3 × 9/10

🧠 First Thought: Can anything simplify first? Yes.

• 2 and 10 divide by 2.
• 9 and 3 divide by 3.

Now multiply: 1 × 3 = 3, 1 × 5 = 5.

Answer: 3/5

🦋 Guided Example 2

Find 5/6 ÷ 4/9

🧠 First Thought: Flip the second fraction: 5/6 × 9/4.

• Now simplify: 6 and 9 divide by 3.
• 5 and 4 share no factor.

Multiply: 15/8

⚠️ Trap Alert!

Many students write 3/4 ÷ 1/2 = 3/8.

❌ Incorrect.

Division does not mean multiplying the denominators.

Always flip the second fraction first.

👀 Read Carefully

SAT questions rarely ask, “Multiply these fractions.”

Instead, they often hide multiplication inside:

• probability
• area
• scale drawings
• proportions
• geometry
• percent increase and decrease

Recognizing the operation is often the hardest part.

⭐ SAT Habit #4

Don’t rush into multiplying.

Always pause for one second and ask: Can anything simplify first?

Professional SAT students do this automatically.

🟢 Warm-Up

• 2/3 × 3/4
• 5/7 × 14/15
• 3/4 ÷ 2/3
• 5/6 ÷ 5/12
• 8/9 × 3/16

🟡 SAT-Style Practice

Question 1

A recipe uses 3/4 cup of sugar. Emma makes half the recipe. How much sugar does she need?

Question 2

A class completed 4/5 of a project. One-half of that work was completed today. What fraction of the entire project was completed today?

Question 3

A map uses the scale 1 inch = 12 miles. Two towns are 2 1/2 inches apart on the map. How many miles apart are they?

(Students must convert the mixed number to 5/2 before multiplying.)

Question 4

A tank is 5/6 full. One-third of the water is drained. What fraction of the entire tank is drained?

🔴 Checkpoint Challenge

Without using a calculator, find: (3/4) × (2/3) + (1/6)

Show each step clearly.

🎯 SAT Connection

Multiplication and division of fractions appear frequently in:

• Probability
• Geometry
• Ratios
• Scale drawings
• Percent problems
• Word problems
• Functions

These skills are usually part of a larger problem, not the main challenge.

✅ Answers & Solutions

Quick Check Answers

Warm-Up Answers

SAT-Style Practice Solutions

🔴 Challenge Solution

(3/4) × (2/3) = 1/2.

Then, 1/2 + 1/6 = 3/6 + 1/6 = 4/6 = 2/3.

Answer: 2/3