🎯 Digital SAT Math • Decimals & Fractions

Master Decimals for the Digital SAT

Decimals are simply another way to write fractions. This Digital SAT lesson focuses on decimal comparison, converting fractions and decimals, and decimal operations that frequently appear in College Board SAT Math practice tests. On the SAT, many problems become much easier when you can switch naturally between fractions, decimals, and percentages. This lesson will help you recognize equivalent forms, avoid common decimal traps, and solve confidently without guessing.

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Quick Check

Already Know This?

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?

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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The Big Idea

Decimals are simply another way to write fractions. The number has not changed — only the notation has.

$$\frac12 = 0.5 = 50\\%$$

Whenever you see a decimal, ask yourself:

“Would this problem become easier as a fraction?”

Strong SAT students switch naturally between fractions, decimals, and percentages.

🔗

Decimal, Fraction, and Percent Relationship

Fraction

$$\frac12$$

Decimal

$$0.5$$

Percent

$$50\\%$$

SAT Shortcut: If you know one of these forms, you should be able to move quickly to the other two forms without using a calculator.

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Why This Matters on the Digital SAT

Many SAT questions hide simple arithmetic behind different representations.

For example, these three expressions are exactly the same quantity:

$$\frac34 \qquad 0.75 \qquad 75\\%$$

Students who recognize these equivalences immediately can solve problems much faster and avoid unnecessary calculations.

## Decimals Chapter — Part 2 (Append Below Part 1)

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Decimal Place Value

Each place to the right of the decimal point is 10 times smaller than the place before it.

$$0.347 = 3\text{ tenths } + 4\text{ hundredths } + 7\text{ thousandths}$$

Tenths

$$0.3$$

Hundredths

$$0.04$$

Thousandths

$$0.007$$

Important: The number of digits after the decimal point matters because it tells you the place value of each digit.

➕

Trailing Zeros Do Not Change the Value

You may add zeros to the end of a decimal without changing its value.

$$0.7 = 0.70 = 0.700$$

This is extremely useful when comparing decimals because it allows you to give both numbers the same number of decimal places.

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Guided Example — Comparing Decimals

Which Number Is Greater?

$$0.7 \quad \text{or} \quad 0.68$$

Step 1: Give both numbers the same number of decimal places.

$$0.70 \qquad 0.68$$

Step 2: Compare the digits from left to right.

Tenths: 7 = 6? No. The first number has 7 tenths, the second has 6 tenths.

$$70\text{ hundredths } > 68\text{ hundredths}$$

$$0.70 > 0.68$$

Answer: 0.7 is greater.

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Another Comparison Example

Which Number Is Greater?

$$0.405 \quad \text{or} \quad 0.45$$

Add a trailing zero to the second number:

$$0.405 \qquad 0.450$$

Compare place values:

Tenths: 4 = 4
Hundredths: 0 < 5

$$0.405 < 0.450$$

Answer: 0.45 is greater.

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SAT Trap — 0.09 vs 0.1

Many students think 0.09 is greater than 0.1 because 9 is larger than 1. This is a place value mistake.

Rewrite both numbers with the same number of decimal places:

$$0.09 \qquad 0.10$$

Now compare:

0.09 = 9 hundredths
0.10 = 10 hundredths

$$9\text{ hundredths } < 10\text{ hundredths}$$

$$0.09 < 0.10$$

Remember: Compare the place value, not just the digits.

✏️

Try These Yourself

Question 1

Which is greater?

$$0.83 \quad \text{or} \quad 0.803$$

Question 2

Which is greater?

$$0.299 \quad \text{or} \quad 0.30$$

Question 3

Write with the same number of decimal places:

$$0.6 \quad \text{and} \quad 0.58$$

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Solutions

Solution 1

$$0.830 \qquad 0.803$$

$$0.830 > 0.803$$

Answer: 0.83

Solution 2

$$0.299 \qquad 0.300$$

$$0.299 < 0.300$$

Answer: 0.30

Solution 3

$$0.60 \qquad 0.58$$

Now they are easy to compare.

SAT Shortcut: When comparing decimals, add trailing zeros first. It reduces mistakes and makes the comparison almost automatic.

## Decimals Chapter — Part 3 (Append Below Part 2)

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Converting Fractions to Decimals

To convert a fraction to a decimal, divide the numerator by the denominator.

$$\frac34 = 3 \div 4 = 0.75$$

Think of the fraction bar as a division symbol.

Example 1

$$\frac12 = 1 \div 2 = 0.5$$

Example 2

$$\frac45 = 4 \div 5 = 0.8$$

Example 3

$$\frac18 = 1 \div 8 = 0.125$$

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Guided Example — A More Challenging Fraction

Convert 7/8 to a decimal.

Step 1: Divide 7 by 8.

$$7 \div 8 = 0.875$$

Step 2: Check the result.

$$0.875 \times 8 = 7$$

Answer: 0.875

➡️

Converting Decimals to Fractions

Use the place value of the last digit.

Tenths

$$0.3 = \frac3{10}$$

Hundredths

$$0.47 = \frac{47}{100}$$

Thousandths

$$0.125 = \frac{125}{1000}$$

Always simplify the fraction if possible.

✂️

Simplifying Fractions

Simplify 125/1000

Both numbers are divisible by 125.

$$\frac{125}{1000}=\frac{125\div125}{1000\div125}=\frac18$$

Final Answer: 1/8

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Common SAT Decimal Equivalents

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

SAT Pattern: Many College Board questions use these exact values because they convert cleanly between fractions, decimals, and percentages.

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SAT-Style Conversion Practice

Question 1

Convert 3/5 to a decimal.

$$\frac35 = 3 \div 5 = 0.6$$

Answer: 0.6

Question 2

Write 0.45 as a fraction in simplest form.

$$0.45 = \frac{45}{100}$$

$$\frac{45}{100}=\frac9{20}$$

Answer: 9/20

Question 3

Which fraction is equal to 0.375?

A) 1/4
B) 3/8
C) 2/5
D) 5/8

$$0.375 = \frac{375}{1000}$$

$$\frac{375}{1000}=\frac38$$

Answer: B) 3/8

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Mixed Representation Challenge

Arrange these numbers from least to greatest:

$$0.6,\\; \frac58,\\; 62\\%,\\; \frac34$$

Step 1: Convert everything to decimals.

$$0.6 = 0.60$$

$$\frac58 = 0.625$$

$$62\\% = 0.62$$

$$\frac34 = 0.75$$

Step 2: Compare the decimals.

$$0.60 < 0.62 < 0.625 < 0.75$$

Final Order: 0.6 < 62% < 5/8 < 3/4

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Key Idea to Remember

For SAT problems involving fractions, decimals, and percentages:

Use fractions when denominators are simple (2, 4, 5, 8, 10, 20).
Use decimals when comparing numbers.
Use percentages when the problem talks about parts out of 100.

The strongest SAT students are flexible. They do not stay in one form — they choose the form that makes the problem easiest.

## Decimals Chapter — Part 4 (Append Below Part 3)

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Adding Decimals

The most important rule is:

Always line up the decimal points before adding.

$$\begin{array}{r} 2.35\\ +\;1.70\\ \hline 4.05 \end{array}$$

Notice that the decimal points stay in the same vertical line.

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Subtracting Decimals

Example

$$\begin{array}{r} 5.00\\ -\;2.75\\ \hline 2.25 \end{array}$$

Writing 5.00 instead of 5 makes the subtraction much easier.

Answer: 2.25

✖️

Multiplying Decimals

Ignore the decimal points first, multiply normally, and then place the decimal point in the final answer.

Example: 0.25 × 8

Step 1: Ignore the decimal point.

$$25 \times 8 = 200$$

Step 2: The original number 0.25 has 2 decimal places, so move the decimal point two places left.

$$2.00 = 2$$

Answer: 2

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Dividing Decimals

Example: 3.6 ÷ 0.6

Dividing by a decimal is inconvenient, so make the divisor a whole number.

Multiply both numbers by 10.

$$\frac{3.6}{0.6}=\frac{36}{6}$$

$$36 \div 6 = 6$$

Answer: 6

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When Decimals Meet Fractions

Sometimes converting the decimal to a fraction is faster.

Example: 0.5 × 18

Instead of multiplying decimals, use:

$$0.5 = \frac12$$

$$\frac12 \times 18 = 9$$

Answer: 9

This is often the fastest calculator-free SAT method.

🛒

Word Problem — Money

Example

A notebook costs $3.75 and a pen costs $1.45. What is the total cost?

Step 1: Add the decimals.

$$\begin{array}{r} 3.75\\ +\;1.45\\ \hline 5.20 \end{array}$$

Step 2: Interpret the result.

Total cost: $5.20

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Word Problem — Measurement

Example

A rope is 12.8 meters long. A piece of 4.35 meters is cut off. How much rope remains?

Step 1: Write the numbers with the same number of decimal places.

$$12.80 – 4.35$$

Step 2: Subtract.

$$\begin{array}{r} 12.80\\ -\;4.35\\ \hline 8.45 \end{array}$$

Answer: 8.45 meters

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SAT Speed Shortcuts

0.5 of a Number

$$0.5n = \frac{n}{2}$$

Half of the number.

0.25 of a Number

$$0.25n = \frac{n}{4}$$

One fourth of the number.

0.75 of a Number

$$0.75n = \frac{3n}{4}$$

Three fourths of the number.

0.2 of a Number

$$0.2n = \frac{n}{5}$$

One fifth of the number.

## Decimals Chapter — Part 5 (Corrected Version)

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SAT-Style Practice Set A

These questions are designed to feel similar to the short calculator and non-calculator style questions that appear in recent Digital SAT Math practice tests.

Question 1 — Decimal Addition

A student buys a notebook for $4.75 and a pen for $2.48. What is the total cost?

Solution

Line up the decimal points.

$$\begin{array}{r} 4.75\\ +\;2.48\\ \hline 7.23 \end{array}$$

Answer: $7.23

Question 2 — Comparing Decimals

Which number is greatest?

A) 0.58
B) 0.605
C) 0.59
D) 0.580

Solution

Rewrite them with the same number of decimal places.

$$0.580 \qquad 0.605 \qquad 0.590 \qquad 0.580$$

Compare from left to right:

$$0.605 > 0.590 > 0.580$$

Answer: B) 0.605

Question 3 — Fraction to Decimal

What decimal is equivalent to 7/20?

Solution

$$\frac7{20} = 7 \div 20$$

$$\frac7{20}=0.35$$

Answer: 0.35

Question 4 — Decimal to Fraction

Write 0.625 as a fraction in simplest form.

Solution

$$0.625 = \frac{625}{1000}$$

Divide numerator and denominator by 125.

$$625 \div 125 = 5$$

$$1000 \div 125 = 8$$

$$\frac{625}{1000}=\frac58$$

Answer: 5/8

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Mixed Operations Practice

Question 5 — Multiplication Shortcut

Compute without a calculator:

$$0.75 \times 32$$

Solution

Use the fraction form:

$$0.75=\frac34$$

$$\frac34 \times 32$$

First divide:

$$32 \div 4 = 8$$

Then multiply:

$$8 \times 3 = 24$$

Answer: 24

Question 6 — Division with Decimals

Compute:

$$4.2 \div 0.7$$

Solution

Multiply both numbers by 10.

$$\frac{4.2}{0.7}=\frac{42}{7}$$

$$42 \div 7 = 6$$

Answer: 6

🧩

Bluebook-Style Challenge Question

Challenge

Which expression has the greatest value?

A) 0.4 × 15
B) 0.25 × 24
C) 3.6 ÷ 0.6
D) 0.75 × 8

Solution

Evaluate each choice.

$$0.4\times15=6$$

$$0.25\times24=6$$

$$3.6\div0.6=6$$

$$0.75\times8=6$$

All four expressions are equal.

Answer: All four choices have the same value.

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What the SAT Is Really Testing

Notice that most of these questions were not about difficult arithmetic. The SAT is usually testing whether you can:

recognize equivalent forms,
choose an efficient representation,
avoid place-value mistakes,
and organize decimal operations correctly.

Smart SAT Habit: Before calculating, ask: “Can I rewrite this decimal as a simple fraction?”

## Decimals Chapter — Part 6 (Append Below Part 5) ## Decimals Chapter — Part 6 (Corrected Version) Replace your current **Part 6** with this updated version. This version fixes: * all `\frac` rendering problems, * the mixed representation challenge, * the final order display, * all multiplication/division symbols, * and any remaining escaped LaTeX issues. — “`html

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SAT-Style Practice Set B

These questions combine decimals with fractions, percentages, and multi-step reasoning — the type of thinking that appears frequently on the Digital SAT.

Question 7 — Percent and Decimal Connection

A store reduces the price of a $80 item by 25%. What is the sale price?

Solution

Convert the percent to a decimal.

$$25\% = 0.25$$

Find the discount.

$$0.25 \times 80 = 20$$

Subtract from the original price.

$$80 – 20 = 60$$

Answer: $60

Question 8 — Multi-Step Decimal Problem

A student studies for 2.5 hours on Monday and 1.75 hours on Tuesday. If the student studies for 0.5 hours less on Wednesday than on Tuesday, how many total hours did the student study?

Solution

Find Wednesday’s study time.

$$1.75 – 0.5 = 1.25$$

Add all three days.

$$2.5 + 1.75 + 1.25$$

$$2.5 + 1.75 = 4.25$$

$$4.25 + 1.25 = 5.5$$

Answer: 5.5 hours

Question 9 — Choosing the Efficient Representation

Compute without a calculator:

$$0.125 \times 64$$

Solution

Recognize that:

$$0.125 = \frac18$$ if you could not remember start with $$\frac{125}{1000}$$ and simplify

Then:

$$\frac18 \times 64 = 64 \div 8 = 8$$

Answer: 8

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Mixed Representation Challenge

Arrange these numbers from least to greatest:

$$0.72,\; \frac34,\; 68\%,\; \frac58$$

Step 1: Convert everything to decimals.

$$0.72 = 0.72$$

$$\frac34 = 0.75$$

$$68\% = 0.68$$

$$\frac58 = 0.625$$

Step 2: Compare the decimals.

$$0.625 < 0.68 < 0.72 < 0.75$$

Final Order:

$$\frac58 < 68\% < 0.72 < \frac34$$

🧩

Digital SAT Challenge Problem

Challenge

Evaluate:

$$(0.5 \times 18) + (0.25 \times 36) – (4.2 \div 0.7)$$

Step 1 — First Product

$$0.5 \times 18 = \frac12 \times 18 = 9$$

Step 2 — Second Product

$$0.25 \times 36 = \frac14 \times 36 = 9$$

Step 3 — Division

$$4.2 \div 0.7 = 42 \div 7 = 6$$

Step 4 — Combine

$$9 + 9 – 6 = 12$$

Final Answer: 12

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Speed Test — Can You Do These Mentally?

1

$$0.5 \times 14$$

Answer: 7

2

$$0.25 \times 20$$

Answer: 5

3

$$0.75 \times 16$$

Answer: 12

4

$$0.2 \times 45$$

Answer: 9

Mental Math Pattern: Convert familiar decimals to fractions first: 0.5 = 1/2, 0.25 = 1/4, 0.75 = 3/4, 0.2 = 1/5.

📌

Common SAT Decimal Mistakes

Mistake 1

Thinking 0.09 > 0.1

$$0.09 < 0.10$$

Mistake 2

Adding without aligning decimals

$$2.5 + 0.75 = 3.25$$

Mistake 3

Forgetting to simplify a decimal fraction

$$\frac{250}{1000}=\frac14$$

🏆

Mini Practice Quiz

Quiz 1

Which is greater?

$$0.705 \quad \text{or} \quad 0.75$$

$$0.705 \qquad 0.750$$

Answer: 0.75

Quiz 2

Write 0.48 as a simplified fraction.

$$0.48 = \frac{48}{100}$$

$$\frac{48}{100}=\frac{12}{25}$$

Answer: 12/25

🌟

The Real SAT Skill

Notice how the hardest questions became easier when we changed the representation:

0.5 → 1/2
0.25 → 1/4
0.75 → 3/4
0.125 → 1/8

Digital SAT Strategy: Before doing any long decimal calculation, ask: “Would this be easier as a fraction?”

## Decimals Chapter — Part 7 (Corrected Final Version)

📚

Quick Reference — Fractions, Decimals, and Percents

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%

4/5
0.8
80%

1/8
0.125
12.5%

3/8
0.375
37.5%

🎯

Final SAT Challenge

Solve without a calculator:

$$(0.75 \times 40) + (0.2 \times 35) – (5.4 \div 0.9)$$

Step 1 — First Product

$$0.75 = \frac34$$

$$\frac34 \times 40 = 30$$

Step 2 — Second Product

$$0.2 = \frac15$$

$$\frac15 \times 35 = 7$$

Step 3 — Division

$$5.4 \div 0.9 = 54 \div 9 = 6$$

Step 4 — Combine

$$30 + 7 – 6 = 31$$

Final Answer: 31

🧠

What You Should Remember for the Digital SAT

Before you finish this chapter, make sure you can do all of the following confidently:

Compare decimals by aligning decimal places.
Add and subtract decimals by lining up decimal points.
Multiply decimals by counting decimal places.
Divide decimals by making the divisor a whole number.
Convert between fractions, decimals, and percentages quickly.
Recognize common SAT decimal equivalents such as 0.25, 0.5, 0.75, 0.2, and 0.125.

The SAT tests flexibility, not just computation. The fastest students choose the representation that makes the problem easiest: fraction, decimal, or percent.

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30-Second Review

Compare

$$0.70 > 0.68$$

Convert

$$0.625 = \frac58$$

Multiply

$$0.25 \times 36 = 9$$

Divide

$$4.2 \div 0.7 = 6$$

If you can solve these four problems quickly and explain why each step works, you are already performing at a strong SAT foundation level for decimal questions.

🏆

Chapter Mastery Checklist

Check each skill that you can now do confidently:

☐ Convert a fraction to a decimal
☐ Convert a decimal to a simplified fraction
☐ Compare decimals correctly
☐ Add and subtract decimals without errors
☐ Multiply and divide decimals efficiently
☐ Use fraction shortcuts for common decimals
☐ Solve multi-step decimal word problems
☐ Move easily between fractions, decimals, and percentages

Goal: You should be able to complete most basic and intermediate Digital SAT decimal questions in under one minute each.

Latest Updates

Categories

Master Decimals for the Digital SAT

Quick Check

Already Know This?

Warm-Up

The Big Idea

Decimal, Fraction, and Percent Relationship

Fraction

Decimal

Percent

Why This Matters on the Digital SAT

Decimal Place Value

Tenths

Hundredths

Thousandths

Trailing Zeros Do Not Change the Value

Guided Example — Comparing Decimals

Which Number Is Greater?

Another Comparison Example

Which Number Is Greater?

SAT Trap — 0.09 vs 0.1

Try These Yourself

Question 1

Question 2

Question 3

Solutions

Solution 1

Solution 2

Solution 3

Converting Fractions to Decimals

Example 1

Example 2

Example 3

Guided Example — A More Challenging Fraction

Convert 7/8 to a decimal.

Converting Decimals to Fractions

Tenths

Hundredths

Thousandths

Simplifying Fractions

Simplify 125/1000

Common SAT Decimal Equivalents

SAT-Style Conversion Practice

Question 1

Question 2

Question 3

Mixed Representation Challenge

Key Idea to Remember

Adding Decimals

Subtracting Decimals

Example

Multiplying Decimals

Example: 0.25 × 8

Dividing Decimals

Example: 3.6 ÷ 0.6

When Decimals Meet Fractions

Example: 0.5 × 18

Word Problem — Money

Example

Word Problem — Measurement

Example

SAT Speed Shortcuts

0.5 of a Number

0.25 of a Number

0.75 of a Number

0.2 of a Number

SAT-Style Practice Set A

Question 1 — Decimal Addition

Question 2 — Comparing Decimals

Question 3 — Fraction to Decimal

Question 4 — Decimal to Fraction

Mixed Operations Practice

Question 5 — Multiplication Shortcut

Question 6 — Division with Decimals

Bluebook-Style Challenge Question

Challenge

What the SAT Is Really Testing

SAT-Style Practice Set B

Question 7 — Percent and Decimal Connection