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Mixed Numbers & Improper Fractions
Two Ways to Write the Same Number
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⏩ Quick Check (1 minute)
Can you answer these without using a calculator?
Convert 2 1/3to an improper fraction.
Convert 11/4to a mixed number.
Which is greater?
2 1/2 “or” 5/2

Write 9/4as a mixed number.
True or False?
3 1/4=13/4

If these feel comfortable, skim the explanations and spend more time on the SAT Practice section.
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📋 Essential Ideas
There are two common ways to write numbers greater than 1.
Mixed Number
A mixed number combines a whole number with a fraction.
Example:
2 1/3

means
2+1/3.

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Improper Fraction
An improper fraction has a numerator that is greater than (or equal to) the denominator.
Example:
7/3.

Despite the name, there is nothing “improper” about it. It simply represents a value greater than or equal to one.
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😊 Picture It
Imagine you ordered pizza.
🍕
You ate two whole pizzas and one-third of another.
Naturally, you’d say,
“I ate 2⅓ pizzas.”
Mathematically, that’s exactly the same as saying
7/3

Different notation.
Exactly the same amount.
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🧠 First Thought
Whenever you see a mixed number, think
Whole number + fraction
instead of trying to remember a formula.
Understanding always beats memorizing.
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🎯 Guided Example 1
Convert
3 2/5

to an improper fraction.
Think First
Three wholes equal
15/5. because (3*5)/5.

Now add the remaining
2/5.
15/5+2/5=17/5.

The Shortcut
Multiply.
3×5=15

Add the numerator.
15+2=17

Keep the denominator.
▭(17/5)

Now students see why the shortcut works instead of simply memorizing it.
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🎯 Guided Example 2
Convert
19/6

to a mixed number.
Think First
How many groups of six fit into nineteen?
Three.
3×6=18

One remains.
Therefore
19/6=3 1/6.

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💡 Which Form Should You Use?
Strong SAT students don’t always leave numbers in the form they were given.
Sometimes a mixed number is easier to understand.
Sometimes an improper fraction is easier to calculate.
For example,
1 3/4+1/2

is usually easier after converting
1 3/4=7/4.

Choose the representation that makes your work simpler.
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⚠️ Trap Alert!
A common mistake is
2 3/5=5/5.

❌ Incorrect.
The whole number must first become fifths.
Think
2=10/5.

Then
10/5+3/5=13/5.

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🌟 SAT Habit #4
Whenever you need to add, subtract, multiply, or divide mixed numbers,
convert them to improper fractions first.
The arithmetic becomes much cleaner.
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🚀 Speed Boost
Notice something interesting.
When converting a mixed number,
the denominator never changes.
Only the numerator changes.
Remembering this small fact prevents many careless mistakes.
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🟢 Warm-Up
Convert each number.

1 3/4

2 4/5

15/4

17/3

4 1/6

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🟡 SAT-Style Practice
Question 1
Emma walked
2 1/2

miles in the morning and
1 1/4

miles in the afternoon.
How many miles did she walk altogether?
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Question 2
A ribbon is
23/5

meters long.
Write the length as a mixed number.
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Question 3
Recipe A uses
1 2/3

cups of flour.
Recipe B uses
5/3

cups.
Which recipe uses more flour?
Explain your reasoning without converting both numbers to decimals.
(I like this version because it encourages mathematical reasoning rather than routine computation.)
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⭐ Challenge
Without writing an improper fraction first, determine whether
3 1/4+3/4

is greater than, less than, or equal to
4.

Explain your reasoning.
Hint: What is
1/4+3/4?

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🎯 SAT Connection
Mixed numbers appear less frequently than basic fractions, but they still arise in
Geometry
Measurement
Unit conversions
Data analysis
Word problems
Students who convert confidently save both time and unnecessary arithmetic.
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✅ Answers & Solutions
Quick Check
7/3
2¾
Equal
2¼
True
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Warm-Up
7/4
14/5
3¾
5⅔
25/6
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SAT-Style Practice
1) Answer: 3¾ miles
Key Idea: Convert the mixed numbers to improper fractions (or fourths), add, and express the result as a mixed number.
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2) Answer: 4⅗
Key Idea: Divide 23 by 5. The quotient is the whole number, and the remainder becomes the numerator.
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3) Answer: Neither. They are equal.
Key Idea: Convert 1 2/3to an improper fraction:
1 2/3=5/3.

Both recipes use the same amount of flour.
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⭐ Challenge
Answer: Equal to 4
Solution
Notice that
1/4+3/4=1.

So,
3 1/4+3/4=3+1=4.

There is no need to convert to improper fractions. Looking for simple relationships first can save valuable time.

Spread 13
🏆 Chapter Challenge
Putting It All Together
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🎯 Your Mission
Congratulations!
You’ve reached the end of the Numbers chapter.
This isn’t a new lesson—it’s your opportunity to combine everything you’ve learned.
Some questions are straightforward.
Some require several steps.
Some reward choosing the smartest strategy instead of the longest one.
Throughout this challenge, you’ll use ideas from the entire chapter, including:
✅ Integers
✅ Fractions
✅ Equivalent fractions
✅ Simplifying
✅ Comparing
✅ Mixed numbers
✅ Order of operations
✅ Fraction reasoning
Remember: the SAT rarely tests just one skill at a time. Strong students recognize which ideas work together.
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🟢 Level 1 — Quick Wins
These questions check your understanding of the foundations.
1)
Simplify
24/36

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2)
Which fraction is equivalent to
4/5?

A)
6/8

B)
8/10

C)
12/16

D) Both B and C
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3)
Compare
7/9 “and” 8/9.

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4)
Convert
2 1/4

to an improper fraction.
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5)
Convert
18/5

to a mixed number.
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🟡 Level 2 — SAT-Style Practice
These require more than one idea.
6)
Emma completed
1/3

of her homework before dinner and
1/2

after dinner.
What fraction of the homework did she complete altogether?
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7)
A recipe requires
3/4

cup of sugar.
Sam accidentally used
5/8

cup.
How much less sugar did Sam use?
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8)
A class has
12 boys
and
18 girls.
What fraction of the class is girls?
Write your answer in simplest form.
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9)
Which fraction is greatest?
A)
5/6

B)
8/9

C)
11/12

D)
7/8

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10)
A runner completed
1 1/2

miles in the morning and
2 1/4

miles in the afternoon.
How many miles did the runner complete altogether?
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🔴 Level 3 — Challenge
These reward mathematical reasoning.
11)
Without converting to decimals,
order the fractions from least to greatest.
4/5,”  ” 7/8,”  ” 5/6,”  ” 11/12

Explain how you decided.
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12)
True or False?
If
a/b=c/d,

must
(a+c)/(b+d)

always have the same value?
Explain your reasoning.
(Hint: Test the statement with simple numbers before deciding.)
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13)
A tank is
3/4

full.
One-third of the water currently in the tank is used.
What fraction of the entire tank remains full?
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14)
Without converting to decimals,
decide which fraction is larger.
49/50 “or” 97/100

Can you explain your reasoning without doing any multiplication?
(Hint: Think about how far each fraction is from 1.)
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15)
Evaluate
(1/2+1/3)×6/5.

Can you predict the answer before finishing the calculation?
(This is a much stronger final question than repeating 1/2+1/3+1/6. It combines common denominators, parentheses, multiplication, and recognizing that 1/2+1/3=5/6, so the final multiplication becomes 5/6×6/5=1. It rewards insight rather than routine arithmetic.)
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💭 Think Like an SAT Student
Before checking the solutions, ask yourself:
Which question took the longest?
Could I have chosen a faster method?
Did I simplify whenever possible?
Did I recognize familiar fractions?
Did I make any careless arithmetic mistakes?
Which idea do I need to practice one more time?
Improving isn’t just about getting the right answer.
It’s also about understanding how you solved the problem.
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📊 Self-Assessment
Score What It Means
14–15 🌟 Excellent! Fractions have become one of your strengths.
11–13 👍 Strong understanding. Review the few questions you missed.
8–10 📖 Good progress. Revisit the examples and try again tomorrow.
Below 8 💪 Build your foundation by reviewing the fraction spreads before moving on.
Remember, this isn’t about passing or failing. Every mistake is useful because it tells you exactly what to practice next.
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🌱 Looking Ahead
Excellent work!
You’ve completed the first chapter and built a strong foundation in working with numbers.
In the next chapter, we’ll study decimals.
The good news is that you’ve already learned many of the underlying ideas. Decimals are simply another way of representing the same quantities you’ve been working with throughout this chapter.
You’ll discover that fractions, decimals, and percentages are not separate topics—they’re different languages describing the same mathematical ideas.
Spread 14
Decimals
Another Way to Write the Same Number
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😊 Already Know This?
Can you answer these mentally?
Write 1/2 as a decimal.
Write 0.75 as a fraction.
Which is greater?
0.8 or 0.78
Round 6.47 to the nearest tenth.
Which is greater?
0.49 or 1/2
If these feel easy, skim this spread and spend more time on the SAT Practice section.
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📋 Essential Rules
Decimals are simply another way to write fractions.
Nothing new.
Nothing scary.
For example,
1/2=0.5
3/4=0.75
1/5=0.2

The number hasn’t changed.
Only the notation has.
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🧠 First Thought
Whenever you see a decimal, ask yourself:
“Would this be easier as a fraction?”
Sometimes the answer is yes.
Sometimes it’s easier to stay with decimals.
Strong SAT students switch naturally between both forms.
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📊 Common Decimal Equivalents
Fraction Decimal Percent
1/2 0.5 50%
1/4 0.25 25%
3/4 0.75 75%
1/5 0.2 20%
2/5 0.4 40%
3/5 0.6 60%
4/5 0.8 80%
1/10 0.1 10%
These values appear so often that recognizing them instantly can save valuable time.
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😊 Think About It
Imagine two students saying:
“I ran half a mile.”
“I ran 0.5 miles.”
Did they run different distances?
Of course not.
Fractions and decimals are simply different ways of writing the same number.
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⚡ Math Memory
Try to recognize these immediately.
Decimal Fraction
0.5 1/2
0.25 1/4
0.75 3/4
0.2 1/5
0.8 4/5
0.125 1/8
0.375 3/8
0.625 5/8
You’ll see these values throughout the SAT.
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🎯 Guided Example
Which number is greater?
0.7
or
0.68
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🧠 First Thought
Write them with the same number of decimal places.
0.70
0.68
Now compare digit by digit.
The tenths are equal.
The hundredths decide.
Since
70 hundredths > 68 hundredths,
the answer is
0.7
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⚠️ Trap Alert!
Many students think
0.5
is smaller than
0.45
because
45 > 5.
Instead, rewrite them.
0.50
0.45
Now compare.
50 hundredths > 45 hundredths.
So
0.5 > 0.45
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🚀 Speed Boost
When comparing decimals,
add trailing zeros whenever they help.
For example,
0.8
=
0.80
Comparing
0.80
and
0.78
is much easier than comparing
0.8
and
0.78.
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🟢 Warm-Up
Convert 4/5 to a decimal.
Convert 0.25 to a fraction.
Which is larger?
0.62 or 0.7
Write 0.125 as a fraction.
Which is larger?
0.09 or 0.1
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🟡 SAT-Style Practice
Question 1
A student answered 18 out of 20 questions correctly.
Write the student’s score as
a fraction,
a decimal,
and a percent.
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Question 2
Which number is greatest?
A) 0.62
B) 3/5
C) 61%
D) 0.605
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Question 3
A store advertises a 25% discount.
Which decimal represents the discount?
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Question 4
Without using a calculator,
decide whether
0.375
is greater than or less than
3/8.
Explain your reasoning.
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⭐ Checkpoint Challenge
Without a calculator, order these numbers from least to greatest.
0.8,”  ” 3/4,”  ” 75%,”  ” 0.78,”  ” 4/5

Hint: Convert as little as possible.
Look for equivalent numbers first.
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🎯 SAT Connection
The SAT constantly switches between
fractions,
decimals,
and percentages.
Students who can move comfortably between these forms often solve problems much faster.
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✅ Answers & Solutions
Already Know This
0.5
3/4
0.8
6.5
1/2
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Warm-Up
0.8
1/4
0.7
1/8
0.1
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SAT-Style Practice
9/10, 0.9, 90%
A
0.25
They are equal because 3/8 = 0.375.
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⭐ Checkpoint Challenge
Answer
3/4=75%<0.78<0.8=4/5 Solution Notice that several numbers are already equivalent: 3/4 = 75% = 0.75 4/5 = 0.8 After recognizing these familiar values, only 0.78 needs to be compared. Since 0.75 < 0.78 < 0.80, the complete order is ▭(3/4=75%<0.78<0.8=4/5) ________________________________________ Spread 15 Converting Fractions and Decimals Learn the Method. Speed Comes Later. ________________________________________ ⏩ Already Know This? Can you convert these without a calculator? Convert 3/5 to a decimal. Convert 0.25 to a fraction. Convert 0.125 to a fraction. Convert 0.4 to a fraction. Which is greater? 3/8 or 0.4 If these feel easy, skim this spread and spend more time on the SAT Practice section. ________________________________________ 📋 Essential Rules There are only two methods to learn. Method 1 — Fraction → Decimal Divide. That's all. Example: 3/4=3÷4 ________________________________________ Method 2 — Decimal → Fraction Read the decimal using its place value. Example 0.25 means 25 hundredths So write 25/100 Then simplify. ________________________________________ 🧠 Why Does This Work? Think about money. One dollar has 100 cents. Twenty-five cents is 25/100 of a dollar. We also write it as 0.25 dollars. Decimals are simply another way of writing parts of a whole. ________________________________________ 🎯 Guided Example Convert 0.25 to a fraction. 🧠 First Thought Don't guess. Read the decimal. 0.25 means 25 hundredths. Write 25/100 Now simplify. 25/100 = 1/4 Answer: 1/4 ________________________________________ 🎯 Guided Example Convert 0.125 to a fraction. 🧠 First Thought Read the place value first. 0.125 means 125 thousandths. Write 125/1000 Now simplify. 125/1000 = 1/8 Answer: 1/8 Notice something? We didn't memorize 0.125. We created the fraction. ________________________________________ 🎯 Guided Example Convert 7/8 to a decimal. 🧠 First Thought Fractions become decimals by division. 7 ÷ 8 = 0.875 Answer: 0.875 Later you may remember this instantly. But even if you don't, the method always works. ________________________________________ ⚡ Speed Boost Some fractions appear so often that you'll eventually recognize them immediately. Fraction Decimal 1/2 0.5 1/4 0.25 3/4 0.75 1/5 0.2 2/5 0.4 4/5 0.8 1/8 0.125 3/8 0.375 5/8 0.625 7/8 0.875 Don't force yourself to memorize them. Understanding comes first. Recognition comes naturally through practice. ________________________________________ ⚠️ Trap Alert! Don't forget the final simplification. For example, 0.5 becomes 5/10 But that's not the final answer. Always simplify. 5/10 = 1/2 ________________________________________ 🌟 SAT Habit #6 Whenever you see a decimal, ask: "Would this problem become easier as a fraction?" Sometimes changing representations saves several steps. ________________________________________ 🟢 Warm-Up Convert each number. 0.6 0.04 2/5 3/8 0.875 ________________________________________ 🟡 SAT-Style Practice Question 1 A bottle is filled to 0.75 of its capacity. Write this amount as a fraction in simplest form. ________________________________________ Question 2 A student answered 0.8 of the questions correctly. Express this as a fraction. ________________________________________ Question 3 A ribbon is 5/8 meter long. Write the length as a decimal. ________________________________________ Question 4 Which number is greater? 0.375 or 1/3 Explain your reasoning. ________________________________________ ⭐ Checkpoint Challenge Without using a calculator, order these numbers from least to greatest. 0.4,"  " 3/8,"  " 2/5,"  " 0.375,"  " 1/2 Look for equivalent numbers before converting anything. ________________________________________ 🎯 SAT Connection The SAT often lets you choose between fractions and decimals. One representation is usually easier than the other. Strong SAT students don't ask, "Should I use fractions or decimals?" They ask, "Which representation makes this problem easier?" ________________________________________ ✅ Answers & Solutions Already Know This 0.6 1/4 1/8 2/5 0.4 ________________________________________ Warm-Up 3/5 1/25 0.4 0.375 7/8 ________________________________________ SAT-Style Practice 3/4 4/5 0.625 0.375 is greater. Solution: Since 1/3=0.333…, we have 0.375>0.333….
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⭐ Checkpoint Challenge
Answer
0.375=3/8<0.4=2/5<1/2 Solution Start by recognizing equivalent numbers: 3/8=0.375 2/5=0.4 Now only compare these familiar values with 1/2=0.5. Therefore, ▭(0.375=3/8<0.4=2/5<1/2) Spread 16 Decimal Operations The Decimal Point Isn't the Difficult Part ________________________________________ 😊 Already Know This? Try these mentally or with very little writing. 0.4 + 0.3 0.9 − 0.25 2.5 + 1.75 0.2 × 5 3.6 ÷ 0.6 If these feel comfortable, skim this spread and spend more time on the SAT Practice section. ________________________________________ 📋 Essential Rules Working with decimals is almost the same as working with whole numbers. The only extra step is keeping track of the decimal point. ________________________________________ Addition Line up the decimal points. 2.35 +1.70 ----- 4.05 ________________________________________ Subtraction Again, line up the decimal points. 5.00 -2.75 ----- 2.25 Adding trailing zeros often makes your work clearer. ________________________________________ Multiplication Multiply as if the numbers were whole numbers. Then place the decimal point in the answer. ________________________________________ Division On the SAT, many decimal division questions appear in the calculator section. When they appear without a calculator, they are usually designed to be manageable with mental math or a simple rewrite. ________________________________________ 🧠 First Thought Before calculating, ask yourself: "Which representation makes this problem easier?" Sometimes decimals are easiest. Sometimes fractions are easier. There is no single best method. Choose the one that makes the work simplest. ________________________________________ 🎯 Guided Example 1 Find 0.75 + 0.2 🧠 First Thought Line up the decimal points. Write 0.20 instead of 0.2. 0.75 0.20 ---- 0.95 Answer: 0.95 ________________________________________ 🎯 Guided Example 2 Find 0.25 × 8 Method 1 — Standard Method Multiply as usual. Since 25 × 8 = 200, place the decimal point correctly. Answer: 2 ________________________________________ Method 2 — A Faster Observation Notice that 0.25 = 1/4. So 1/4 × 8 = 2. Both methods are correct. Choose the one that feels more natural. ________________________________________ 🎯 Guided Example 3 Find 3.6 ÷ 0.6 🧠 First Thought The divisor has one decimal place. Multiply both numbers by 10. 3.6 ÷ 0.6 = 36 ÷ 6 = 6 We didn't change the value. We only made the numbers easier to work with. ________________________________________ 🚀 Speed Boost Some decimal values appear so often that recognizing them immediately can save time. 0.5 = 1/2 0.25 = 1/4 0.75 = 3/4 0.2 = 1/5 0.125 = 1/8 You don't have to use these forms. They're simply options whenever they make a problem easier. ________________________________________ ⚠️ Trap Alert! Many students think 0.09 is greater than 0.1. Instead, write 0.09 0.10 Now compare. Since 9 hundredths < 10 hundredths, 0.09<0.10. ________________________________________ 🌟 SAT Habit #7 Don't start calculating immediately. First ask yourself, "Can I make this problem simpler?" Sometimes that means using decimals. Sometimes that means using fractions. Strong SAT students make these decisions naturally. ________________________________________ 🟢 Warm-Up Find each value. 0.6 + 0.8 2.4 − 1.9 0.5 × 16 0.75 × 20 4.2 ÷ 0.7 ________________________________________ 🟡 SAT-Style Practice Question 1 A bottle contains 0.75 liter of water. Emma drinks 0.25 liter. How much water remains? ________________________________________ Question 2 A notebook costs $4.50. You buy two notebooks. How much do you pay before tax? ________________________________________ Question 3 A runner completes 0.40 of a race in the first half-hour and 0.35 in the second half-hour. What fraction of the race remains? Express your answer as either a decimal or a fraction. ________________________________________ Question 4 Without performing any calculations, decide whether 0.375 and 3/8 represent the same value. Explain your reasoning. ________________________________________ ⭐ Checkpoint Challenge Without using a calculator, evaluate 0.5×12+0.25×8-0.2×5. Look for simple relationships before multiplying. ________________________________________ 😊 Math Smile Someone says, "Decimals are harder than fractions." You smile and reply, "They're just fractions wearing a different outfit." 😄 ________________________________________ 🎯 SAT Connection On the digital SAT, a calculator is available for many questions. But the fastest students don't reach for it immediately. They first ask, "Can I solve this mentally?" Often, a few seconds of thinking saves much more time than using a calculator. ________________________________________ ✅ Answers & Solutions Already Know This 0.7 0.65 4.25 1 6 ________________________________________ Warm-Up 1.4 0.5 8 15 6 ________________________________________ SAT-Style Practice 1) Answer: 0.50 liter Solution: 0.75-0.25=0.50. ________________________________________ 2) Answer: $9.00 Solution: Multiply 4.50×2. ________________________________________ 3) Answer: 0.25 (or 1/4) Solution: 0.40+0.35=0.75. The remaining part is 1-0.75=0.25. ________________________________________ 4) Answer: Yes, they are equal. Solution: From the previous spread, 3/8=0.375. ________________________________________ ⭐ Checkpoint Challenge Answer: 7 Solution Recognize the familiar values. 0.5×12=6 0.25×8=2 0.2×5=1 Therefore, 6+2-1=7. Notice that converting to familiar fractions makes the arithmetic almost effortless. Spread 17 Percentages The Language of "Out of 100" ________________________________________ ⏩ Quick Check (45 seconds) Can you answer these without a calculator? 1) Write 50% as a fraction. 2) Write 25% as a decimal. 3) Write 0.8 as a percentage. 4) Which is greater? 40% or 2/5 5) What is 10% of 80? If these feel easy, skim the explanations and spend more time on the SAT Practice section. ________________________________________ 📋 Essential Rules A percent simply means "out of 100." The word percent literally means per hundred. For example, 50% = 50/100 = 1/2 The number hasn't changed. Only the way we write it has. ________________________________________ 🧠 First Thought Whenever you see a percentage, ask yourself: Would this problem become easier as a fraction or a decimal? Sometimes the answer is yes. Sometimes it isn't. Strong SAT students choose the representation that makes the work easiest. ________________________________________ 📊 Three Ways to Write the Same Number Fraction Decimal Percent 1/2 0.5 50% 1/4 0.25 25% 3/4 0.75 75% 1/5 0.2 20% 2/5 0.4 40% 3/5 0.6 60% 4/5 0.8 80% Notice something? You've already learned almost everything on this page. The only new language is the percentage. ________________________________________ 💡 Three Conversion Methods Percent → Decimal Divide by 100. Example 75% ↓ 0.75 Moving the decimal point two places left is simply a shortcut for dividing by 100. ________________________________________ Decimal → Percent Multiply by 100. Example 0.37 ↓ 37% Again, moving the decimal point is simply a shortcut. ________________________________________ Fraction → Percent Choose whichever method is easier. Method 1 Convert to a decimal. 3/4 ↓ 0.75 ↓ 75% Method 2 Rewrite with denominator 100 when possible. 1/4 ↓ 25/100 ↓ 25% Choose the method that requires the least work. ________________________________________ 🌟 SAT Habit #8 There is no single "correct" representation. Sometimes percentages are easiest. Sometimes fractions are. Sometimes decimals save time. The strongest SAT students switch naturally between all three. ________________________________________ 🎯 Guided Example 1 Convert 35% to a decimal. First Thought Percent means out of 100. 35% = 35/100 = 0.35 Answer 0.35 ________________________________________ 🎯 Guided Example 2 Convert 0.125 to a percentage. First Thought Multiply by 100. 0.125 ×100 =12.5 Answer 12.5% ________________________________________ 🎯 Guided Example 3 What is 25% of 80? First Thought There are several good approaches. Method 1 25% = 1/4 So 1/4 of 80 =20 Method 2 0.25 ×80 =20 Both methods are equally correct. Choose whichever feels more natural. ________________________________________ ⚠️ Trap Alert! Many students think 0.5% = 0.5 ❌ Remember what percent means. 0.5% = 0.5/100 = 0.005 Never ignore the percent sign. ________________________________________ 💡 Efficiency Note You don't have to memorize every percentage. But after enough practice, these should become familiar. Percent Fraction Decimal 50% 1/2 0.5 25% 1/4 0.25 75% 3/4 0.75 20% 1/5 0.2 10% 1/10 0.1 5% 1/20 0.05 1% 1/100 0.01 12.5% 1/8 0.125 You aren't memorizing isolated facts. You're recognizing relationships you've already learned. ________________________________________ 🟢 Warm-Up Convert each number. 1) 40% to a decimal. 2) 0.08 to a percentage. 3) 1/5 to a percentage. 4) 125% to a decimal. 5) 0.6 to a percentage. ________________________________________ 🟡 SAT-Style Practice Question 1 A student answered 80% of 40 questions correctly. How many questions did the student answer correctly? ________________________________________ Question 2 A jacket originally costs $80. It is on sale for 25% off. What is the sale price? ________________________________________ Question 3 A school's attendance rate is 95%. Out of 400 students, how many are present? ________________________________________ Question 4 Which is greater? 35% or 1/3 Explain your reasoning. ________________________________________ Question 5 A container is 60% full. One-half of the water currently inside is removed. What percentage of the container is still full? (This is a nice SAT-style reasoning question because "half of the water" is not the same as "half of the container.") ________________________________________ 🔴 Challenge Without using a calculator, decide which is larger. 12.5% or 1/8 Can you justify your answer in more than one way? ________________________________________ ✅ Answers & Solutions Warm-Up 1) 40% = 0.4 2) 0.08 = 8% 3) 1/5 = 20% 4) 125% = 1.25 5) 0.6 = 60% ________________________________________ SAT Practice Question 1 80% of 40 =0.8×40 =32 Answer: 32 ________________________________________ Question 2 25% of $80 =20 Sale price =80−20 =60 Answer: $60 ________________________________________ Question 3 95% of 400 =0.95×400 =380 Answer: 380 students ________________________________________ Question 4 35% =0.35 1/3≈0.333... Therefore, 35% is greater. ________________________________________ Question 5 60% full. Half of the water remains after removing half. 60% ÷2 =30% Answer: 30% ________________________________________ Challenge 12.5% =0.125 1/8 =0.125 They are equal. ________________________________________ 🎯 Looking Ahead Percentages appear everywhere—in discounts, taxes, probability, data analysis, and growth problems. In the next spread, you'll learn one of the SAT's favorite applications: finding a percentage of a number, calculating percentage increases and decreases, and solving multi-step percentage problems efficiently. Spread 18 Finding a Percentage of a Quantity The SAT Loves Percent Problems ________________________________________ ⏩ Quick Check (45 seconds) Can you answer these without much writing? 1) What is 10% of 60? 2) What is 50% of 48? 3) What is 25% of 80? 4) A shirt costs $40. It is discounted by 50%. What is the sale price? 5) Which is greater? 20% of 60 or 30% of 40? If these feel easy, skim the explanations and spend more time on the SAT Practice section. ________________________________________ 📋 Essential Idea Nearly every percentage problem begins with the same question: What is the whole? Once you've identified the whole, the rest becomes much easier. A useful relationship is Part = Percent × Whole Think of this as a relationship, not a formula to memorize. Whole = the total amount Percent = the portion (written as a decimal or fraction) Part = the amount represented by that percentage ________________________________________ 🧠 First Thought Before touching your calculator—or even your pencil—ask yourself: What is the whole? Many SAT mistakes happen because students identify the wrong whole. ________________________________________ 🎯 Guided Example 1 What is 25% of 80? First Thought 25% = 1/4 A quarter of 80 is Answer: 20 Notice that we never needed 0.25 × 80. ________________________________________ 🎯 Guided Example 2 Find 15% of First Thought Build it from percentages you already know. 10% of 200 = 20 5% is half of 10% =10 Therefore, 15% =20+10 =30 Answer: 30 ________________________________________ 🎯 Guided Example 3 Find 35% of First Thought There isn't just one good method. Method A 35% =0.35 0.35 ×80 =28 Method B 35% =30%+5% 30% of 80 =24 5% of 80 =4 24+4=28 Both methods work. Choose whichever feels easier. ________________________________________ 💡 Efficiency Note Many percentages can be built from simpler ones. Percentage Think of it as... 5% Half of 10% 15% 10% + 5% 20% Double 10% 25% One-fourth 50% One-half 75% Three-fourths The goal isn't to memorize tricks. It's to recognize useful relationships. ________________________________________ ⚠️ Trap Alert! A shirt costs $80 and is 25% off. Many students answer $20. That's only the discount. The question asks for the sale price. 80−20=60 Answer: $60 Always read the final sentence carefully. ________________________________________ 🌟 SAT Habit #9 After solving a percentage problem, ask yourself: Did I find the amount the question asked for? Many students calculate the correct intermediate value but stop too soon. ________________________________________ 🟢 Warm-Up 1) Find 10% of 250. 2) Find 25% of 96. 3) Find 50% of 78. 4) A book costs $24 and is discounted by 25%. What is the sale price? 5) Find 5% of 300. ________________________________________ 🟡 SAT-Style Practice Question 1 A school has 800 students. If 35% participate in at least one club, how many students participate? ________________________________________ Question 2 A concert hall sold 240 tickets. If 75% were purchased online, how many tickets were purchased online? ________________________________________ Question 3 A store advertises a 20% discount on a backpack that originally costs $65. What is the sale price? ________________________________________ Question 4 A science class contains 24 students. If 3 students are absent, what percent of the class is absent? ________________________________________ Question 5 (Student-Produced Response) A bicycle originally costs $240. It is discounted by 15%. What is the amount of the discount? ________________________________________ 🔴 Challenge A jacket is discounted by 20%. After the discount, the price is $72. What was the original price? Try to solve it without using an equation if you can. Then solve it using an equation. Which method feels more natural? (This is very close to the kind of reasoning the SAT expects.) ________________________________________ ✅ Answers & Solutions Warm-Up 1) 10% of 250 = 25 2) 25% of 96 = 24 3) 50% of 78 = 39 4) Discount = 25% of $24 = $6 Sale price = $18 5) 5% of 300 = 15 ________________________________________ SAT Practice Question 1 35% of 800 =0.35×800 =280 Answer: 280 ________________________________________ Question 2 75% =3/4 3/4 of 240 =180 Answer: 180 ________________________________________ Question 3 20% of 65 =13 65−13 =52 Answer: $52 ________________________________________ Question 4 3 out of 24 =1/8 =12.5% Answer: 12.5% ________________________________________ Question 5 15% of 240 =10%+5% =24+12 =36 Answer: $36 ________________________________________ Challenge After a 20% discount, the customer pays 80% of the original price. 80% =0.8 Original price =72÷0.8 =90 Answer: $90 ________________________________________ 🎯 Looking Ahead You've learned how to find a percentage of a quantity. The next spread introduces one of the SAT's favorite topics: Percent increase Percent decrease Percent change These questions appear frequently in real-world contexts such as finance, science, business, and data analysis. Spread 19 Percent Increase & Percent Decrease One of the SAT's Favorite Question Types ________________________________________ ⏩ Quick Check (45 seconds) Don't calculate yet. Just think. 1) A shirt costs $80 and is discounted by 25%. Will the sale price be more or less than $80? ________________________________________ 2) A population grows by 10%. Will the new population be larger or smaller? ________________________________________ 3) A phone costs $500. Its price increases by 20%. Will the increase be more or less than $100? ________________________________________ 4) Which is larger? 50% of 40 or 40% of 50? ________________________________________ 5) A number increases by 100%. Does it A) Stay the same B) Double C) Triple ________________________________________ 📋 Essential Idea Almost every percent increase or decrease problem follows the same two steps. Step 1 Find the amount of the increase (or decrease). Step 2 Add or subtract that amount from the original value. Many students stop after Step 1. The SAT often counts on that mistake. ________________________________________ 🧠 First Thought Before calculating anything, ask yourself: Am I finding the change, or the final amount? Those are not always the same. ________________________________________ 🎯 Guided Example 1 A jacket costs $120. It is on sale for 25% off. Find the sale price. First Thought 25% = 1/4 One-fourth of 120 is That is the discount, not the final price. 120−30=90 Answer: $90 ________________________________________ 🎯 Guided Example 2 A school's enrollment was 400 students. This year it increased by 15%. How many students are enrolled now? First Thought 10% of 400 =40 5% of 400 =20 15%=40+20=60 400+60=460 Answer: 460 ________________________________________ 🎯 Guided Example 3 A price increases from $80 to $100. By what percent did it increase? First Thought The increase is 20 dollars. But percentages compare the change with the original amount. 20/80 =1/4 =25% Answer: 25% ________________________________________ 💡 Strategy Spotlight There is rarely just one correct method. For example, to find 25% of 96, you could convert 25% to 0.25, think of 25% as 1/4, or divide 96 by 4. All three methods are correct. Choose the one that requires the least work. ________________________________________ ⚠️ Trap Alert! A price increases from $80 to $100. Some students answer 20%. The increase is $20, but 20 is compared with the original price: 20÷80 =25% Always compare with the original amount unless the problem tells you otherwise. ________________________________________ 🌟 SAT Habit #10 Words such as increased decreased discounted reduced grew dropped rose are clues that percentages are involved. Recognizing those clues quickly saves valuable time. ________________________________________ 🟢 Warm-Up 1) A book costs $40. It is discounted by 10%. Find the sale price. ________________________________________ 2) A town has 800 residents. Its population grows by 5%. Find the new population. ________________________________________ 3) A phone costs $600. Its price decreases by 20%. Find the amount of the decrease. ________________________________________ 4) A bicycle costs $240. Its price increases by 25%. Find the new price. ________________________________________ 5) A test score increases from 60 to 72. By what percent did it increase? ________________________________________ 🟡 SAT-Style Practice Question 1 A science club has 120 members. Membership increases by 15%. How many members does the club have after the increase? ________________________________________ Question 2 A store advertises 30% off a backpack that originally costs $90. What is the sale price? ________________________________________ Question 3 A company's profits decreased from $500,000 to $425,000. What was the percent decrease? ________________________________________ Question 4 A survey found that 240 of 300 students own a graphing calculator. What percentage of the students own a graphing calculator? ________________________________________ Question 5 (Student-Produced Response) The price of a concert ticket increases from $48 to $60. By what percent did the price increase? ________________________________________ Question 6 (Thinking Question) A store advertises "15% off" and another store advertises "$12 off." A backpack costs $80. Which store gives the larger discount? Explain how you decided. (Strong SAT students compare the actual amounts before choosing.) ________________________________________ 🔴 Challenge A store offers 20% off, then an additional 20% off the sale price. Is this the same as 40% off the original price? Explain your reasoning. ________________________________________ ✅ Answers & Solutions Warm-Up 1) Discount = $4 Sale price = $36 2) 5% of 800 =40 New population = 840 3) 20% of 600 = 120 4) 25% of 240 =60 New price = $300 5) Increase =12 12/60=20% Answer: 20% ________________________________________ SAT Practice Question 1 15% of 120 =18 120+18=138 Answer: 138 ________________________________________ Question 2 30% of 90 =27 90−27=63 Answer: $63 ________________________________________ Question 3 Decrease =75,000 75,000÷500,000=0.15 Answer: 15% ________________________________________ Question 4 240/300 =4/5 =80% Answer: 80% ________________________________________ Question 5 Increase =12 12÷48 =1/4 =25% Answer: 25% ________________________________________ Question 6 15% of $80 = $12 Both discounts are the same. ________________________________________ Challenge Suppose the original price is $100. After the first 20% discount: $100 → $80 A second 20% discount is taken from $80, not from $100. 20% of $80 = $16 Final price = $64 The total discount is $36, which is 36%, not 40%. ________________________________________ 🎯 Looking Ahead You've learned how to: find a percentage of a quantity, calculate percent increases and decreases, distinguish between a change and a final value, and work backward from percentage information. These ideas appear frequently throughout the SAT, especially in algebra, data analysis, finance, and real-world modeling. Spread 20 🧠 Think Before You Calculate The Fastest Solution Isn't Always the Longest One. ________________________________________ ⏩ Already Know This? Don't calculate immediately. Instead ask yourself, "What is the smartest way?" 1) Which is larger? 49% of 200 or 50% of 196 ________________________________________ 2) Which is easier to compute? 0.25 × 48 or 1/4 × 48 ________________________________________ 3) Without calculating, which is larger? 7/8 or 0.86 ________________________________________ 4) A shirt costs $80. It is discounted by 25%. Would you rather think 25% or 1/4 ? ________________________________________ 5) Which would you do first? 3/4+1/6 A) Convert to decimals B) Find a common denominator C) Estimate ________________________________________ If these feel easy, you're beginning to think like an SAT student. ________________________________________ 📋 The Big Idea Most SAT questions can be solved in more than one way. The best students don't always calculate first. They first choose the easiest path. Sometimes that means • rewriting • estimating • simplifying • recognizing a familiar pattern before touching the calculator. ________________________________________ 🧠 Dr. Aytekin's Question Whenever you see a problem, pause for two seconds and ask "Can I make this problem easier before I solve it?" Those two seconds often save twenty. ________________________________________ 🌟 SAT Habit #11 Before writing anything, ask yourself these four questions. 1) Can I rewrite the numbers? Examples 50% → 1/2 25% → 1/4 0.2 → 1/5 0.75 → 3/4 ________________________________________ 2) Can I simplify first? Example 24/36 is much easier after simplifying. ________________________________________ 3) Can I estimate? Suppose the answer choices are 18 180 1800 18,000 A quick estimate may eliminate three answers instantly. ________________________________________ 4) Is there a pattern? Many SAT problems reward recognition more than computation. ________________________________________ 🎯 Guided Example 1 Which is larger? 49% of 200 or 50% of 196 ________________________________________ 🧠 First Thought Don't multiply immediately. 50% of 200 is 100. 49% of 200 must be slightly less. 50% of 196 is exactly 98. Now only one quick calculation remains. 49% of 200 = 98 Answer: They are equal. ________________________________________ 🎯 Guided Example 2 Find 0.125 × 64 ________________________________________ 🧠 First Thought Recognize 0.125 as 1/8. Now the problem becomes 1/8 × 64 = 64 ÷ 8 Answer: 8 Almost no arithmetic. ________________________________________ 🎯 Guided Example 3 Which is larger? 7/8 or 0.86 ________________________________________ 🧠 First Thought Don't convert 7/8 by long division. Recognize it. 7/8 = 0.875 Now compare 0.875 and 0.860 Answer: 7/8 Recognition beats computation. ________________________________________ 🚀 Speed Boost Many SAT questions become easier after one small rewrite. Instead of 0.5 × 18 think 1/2 × 18 Instead of 25% of 64 think 1/4 of 64 Instead of 75% think 3/4 Small changes often create big shortcuts. ________________________________________ ⚠️ Trap Alert! Many students believe "More work means fewer mistakes." Usually, the opposite is true. Every extra step creates another opportunity for an arithmetic error. The shortest correct solution is often the safest one. ________________________________________ 🟢 Warm-Up Choose the fastest strategy, then solve. 1) 50% of 90 ________________________________________ 2) 25% of 120 ________________________________________ 3) 0.5 × 18 ________________________________________ 4) 75% of 40 ________________________________________ 5) 0.2 × 55 ________________________________________ 6) 3/8 or 0.4 Which is larger? ________________________________________ 🟡 SAT-Style Practice Question 1 A bookstore offers a 25% discount on a novel that costs $36. Without using a calculator, what is the sale price? ________________________________________ Question 2 A survey reports that 80% of 250 participants preferred Product A. How many participants preferred Product A? ________________________________________ Question 3 A scientist records that 0.25 of the plants in an experiment flowered during the first week. If 64 plants were observed, how many flowered? ________________________________________ Question 4 A school has 480 students. Exactly 75% participate in at least one extracurricular activity. How many students participate? ________________________________________ Question 5 A classroom contains 32 students. 25% are left-handed. Of the remaining students, 50% wear glasses. How many students are right-handed and wear glasses? ________________________________________ Question 6 Without calculating every option, which is greatest? A) 49% of 200 B) 98 C) 1/2 of 196 D) 0.49 × 200 (A nice SAT-style equivalence question. Students should realize all four expressions are equal.) ________________________________________ 🔴 Challenge A store increases the price of a bicycle by 20%. Later, it offers a 20% discount on the new price. Without choosing numbers first, decide whether the final price is • greater than • less than • equal to the original price. Explain your reasoning. ________________________________________ 💬 Dr. Aytekin's Corner One habit separates many high-scoring SAT students from everyone else. Average students ask, "How do I solve this?" Top students ask, "Can I make this easier before I solve it?" That simple question saves time, reduces mistakes, and builds confidence. It is one of the most valuable habits you can develop—not only for the SAT, but for mathematics in general. Spread 21 🏁 Chapter Challenge The SAT Doesn't Label Topics. Neither Will We. ________________________________________ 🎯 Before You Begin Every question in this challenge comes from this chapter. But there's one difference. The topics are mixed. Just like the SAT, your first job is to recognize what kind of problem you're looking at. ________________________________________ 🟢 Level 1 — Quick Review 1) Which number is largest? A) 0.75 B) 75% C) 3/4 D) They are all equal. ________________________________________ 2) Simplify 36/48 ________________________________________ 3) Find 25% of 96 ________________________________________ 4) Which is larger? 5/8 or 0.6 ________________________________________ 5) Convert 0.125 to a fraction in simplest form. ________________________________________ 🟡 Level 2 — Mixed SAT Practice 6) A school has 480 students. If 25% participate in the robotics club, how many students are in the club? ________________________________________ 7) A backpack costs $80. The store offers a 15% discount. What is the sale price? ________________________________________ 8) Order these from least to greatest. 0.39 0.4 40% 2/5 ________________________________________ 9) A tank is 3/4 full. One-third of the water currently in the tank is used. What fraction of the entire tank remains full? ________________________________________ 10) A student answers 36 out of 45 questions correctly. What percentage of the questions did the student answer correctly? ________________________________________ 🔴 Level 3 — Think Like the SAT 11) A company's revenue increases by 20%. The next year it decreases by 20%. Compared with the original revenue, is the final revenue greater than, less than, or equal to the original? Explain. ________________________________________ 12) Without calculating every value, determine which is greatest. 49% of 400 50% of 392 1/2 × 396 ________________________________________ 13) A survey reports that 60% of students prefer online homework. If 240 students prefer online homework, how many students took the survey? ________________________________________ 14) Without using a calculator, explain why 0.375 equals 3/8. Use a method from this chapter. ________________________________________ 15) A class has 40 students. 30% play a musical instrument. Of the remaining students, 25% play a sport. How many students do neither? ________________________________________ 🌟 Chapter Reflection Before checking your answers, ask yourself: ✓ Which question did you solve the fastest? ✓ Which one made you stop and think? ✓ Did you rewrite any fractions, decimals, or percentages? ✓ Did you ever simplify before calculating? Those habits matter just as much as the answers. ________________________________________ 🧰 Your Mathematical Toolbox You can now confidently ✅ work with integers ✅ simplify and compare fractions ✅ convert between fractions, decimals, and percentages ✅ perform operations with fractions and decimals ✅ solve percentage problems ✅ recognize efficient shortcuts ✅ choose the best representation for a problem These skills will appear again and again throughout the SAT. ________________________________________ 💡 One Number, Many Representations One of the biggest ideas in this chapter is that a number can wear many different "outfits." 1/2=0.5=50% 1/4=0.25=25% 1/5=0.2=20% The number never changes. Only the representation does. The more naturally you move between these forms, the easier many SAT questions become. ________________________________________ 🎯 A Final Thought Strong SAT students don't memorize more. They recognize more. They notice that 25% is one-fourth. 0.5 is one-half. 0.125 is one-eighth. Instead of asking, "Which formula should I use?" they ask, "Can I make this problem simpler?" That habit is far more valuable than memorizing another rule. ________________________________________