Hello, future ACT champion!

Welcome to your comprehensive guide to one of the most fundamental and frequently tested topics on the ACT Math section: Fractions and Decimals. My name is Alex the Math Expert, and I'll be your guide through this essential part of the Pre-Algebra curriculum.

Think of this document not just as a study note, but as a personal tutoring session. We're going to break down everything you need to know about converting between fractions and decimals. Why? Because mastering this skill is like having a secret key that unlocks a huge number of ACT problems. It’s a foundational concept that appears not just in obvious fraction questions, but also in geometry, algebra, and word problems. Being fast and accurate with these conversions can save you precious seconds on test day, giving you more time to tackle the tougher questions.

Our goal here is not just to learn the rules, but to understand why they work. We’ll move from the basics to advanced strategies, from simple conversions to complex, multi-step ACT-style problems. By the end of this guide, you'll be able to switch between fractions and decimals with confidence and precision.

So, grab a notebook, a pencil, and your calculator (though we'll learn when not to use it!). Let's get started on the path to mastering fractions and decimals and boosting your ACT score.

ACT Math Study Report

Curriculum: ACT
Grade Level: 9th - 12th Grade
Subject: ACT Math
Chapter: Pre-Algebra
Topic: 1.3 Fractions & Decimals
Subtopic: Converting Fractions ↔ Decimals

1. Introduction: The Two Faces of the Same Number

Imagine you and a friend are describing how much pizza is left. You might say, "There's half a pizza left," while your friend might say, "There's 0.5 of the pizza remaining." Who is right? You both are! You're just using different languages—the language of fractions and the language of decimals—to describe the exact same amount.

This is the core idea of our subtopic. Fractions and decimals are simply two different ways of writing numbers that are not whole numbers. They are parts of a whole. The ability to fluently translate between these two "languages" is a critical skill in mathematics, especially on a timed test like the ACT.

Key Terminology

Fraction: A number that represents a part of a whole, written as a ratio of a numerator and a denominator.
- Numerator: The top number; tells you how many parts you have.
- Denominator: The bottom number; tells you how many equal parts the whole is divided into.
- Fraction Bar: The line separating the numerator and denominator. It represents division. The fraction 3/8 literally means "3 divided by 8."
Decimal: A number using a decimal point to separate the whole number part from the fractional part, based on powers of ten.
- Place Value: The value of a digit based on its position (tenths, hundredths, thousandths, etc.). This is the key to converting decimals to fractions.
Terminating Decimal: A decimal that ends. Example: 0.5, 0.75, 0.125.
Repeating Decimal: A decimal with a digit or block of digits that repeats infinitely. We use bar notation (a line over the repeating digits). Example: 1/3 = 0.333... is written as 0.3̅.

Why is this so important for the ACT?

Flexibility in Problem Solving: Convert numbers to a common format (usually decimals) to make comparisons and calculations easier.
Calculator Efficiency: Knowing when to convert in your head (e.g., 0.5 to 1/2) versus when to use the calculator (e.g., 13/16 to 0.8125) saves time.
Unlocking Shortcuts: Recognizing that 0.375 is 3/8 can turn a messy decimal problem into a simple fraction cancellation.
Direct Questions: Some questions will directly test your ability to perform these conversions.

2. Detailed Explanation with Examples

Part A: Converting Fractions to Decimals

Method 1: The Universal Method - Long Division

This method works for any fraction. Simply divide the numerator by the denominator.

Example 1: Convert 3/4 (Terminating)

    0.75
  _______
4 | 3.00
   -2.8
   ----
     .20
    -.20
    ----
       0

Since the remainder is 0, the decimal terminates. Therefore, 3/4 = 0.75.

Example 2: Convert 2/3 (Repeating)

    0.66...
  _______
3 | 2.000
   -1.8
   ----
     .20
    -.18
    ----
      .02

The pattern of remainder 2 repeats forever. Therefore, 2/3 = 0.666... or 0.6̅.

Method 2: The Shortcut - Making the Denominator a Power of 10

This is a mental math trick for "friendly" fractions. Multiply the numerator and denominator by the same number to make the denominator 10, 100, or 1000.

Key Pairings to Memorize:

To get 10: 2 × 5
To get 100: 4 × 25, 20 × 5, 50 × 2
To get 1000: 8 × 125

Example: Convert 7/20
Multiply top and bottom by 5: (7 × 5) / (20 × 5) = 35 / 100 = 0.35.

Example: Convert 3/8
Multiply top and bottom by 125: (3 × 125) / (8 × 125) = 375 / 1000 = 0.375.

Method 3: The ACT Pro's Secret Weapon - Memorization

For the most common fractions, you should know their decimal equivalents instantly to save time.

Fraction	Decimal	Fraction	Decimal	Fraction	Decimal
1/2	0.5	1/5	0.2	1/8	0.125
1/3	0.3̅	2/5	0.4	3/8	0.375
2/3	0.6̅	3/5	0.6	5/8	0.625
1/4	0.25	4/5	0.8	7/8	0.875
3/4	0.75	1/6	0.16̅	1/9	0.1̅

Part B: Converting Decimals to Fractions

Method 1: Terminating Decimals - "Say It, Write It, Simplify It"

Say It: Read the decimal using its proper place value name.
Write It: Write what you said as a fraction.
Simplify It: Reduce the fraction to its simplest form.

Example: Convert 0.45
1. Say It: "Forty-five hundredths."
2. Write It: 45/100.
3. Simplify It: Divide both by 5 to get 9/20.

Method 2: Repeating Decimals - The "Nines in the Denominator" Rule

For simple repeating decimals, write the repeating digits as the numerator and a '9' for each repeating digit in the denominator.

Example: Convert 0.7̅ (0.777...)
One repeating digit (7) -> Numerator is 7. One '9' in the denominator. -> 7/9.

Example: Convert 0.2̅3̅ (0.2323...)
Two repeating digits (23) -> Numerator is 23. Two '9's in the denominator. -> 23/99.

Advanced Case: Delayed Repeating Decimals (e.g., 0.16̅)

For decimals where the repeat doesn't start immediately, use this shortcut:
Numerator = (Full number to end of first repeat) - (Non-repeating part)
Denominator = A '9' for each repeating digit, followed by a '0' for each non-repeating decimal digit.

Example: Convert 0.16̅ (0.1666...)
Numerator = 16 - 1 = 15
Denominator = One '9' (for the repeating '6') and one '0' (for the non-repeating '1') = 90
Fraction = 15/90 = 1/6.

3. Key Concepts and Principles

1. Rational Numbers: The Unifying Concept

A rational number is any number that can be expressed as a fraction p/q. This includes all fractions, all terminating decimals, and all repeating decimals. Numbers like π or √2, whose decimals never end and never repeat, are irrational.

2. The Rule of Terminating vs. Repeating Decimals

A fraction in its simplest form will convert to a:

Terminating decimal if the prime factors of the denominator are only 2s and/or 5s.
Repeating decimal if the denominator has any prime factor other than 2 or 5.

Example: 7/16 terminates because 16 = 2⁴.
Example: 4/30 simplifies to 2/15. It repeats because the denominator 15 has a prime factor of 3.

4. Practical Applications: Math in the Real World

Finance: Prices ($19.99), discounts (25% is 1/4 off), and stock prices (up 1/8 of a point).
Cooking: Recipes use fractions (1/2 cup), while scaling a recipe might involve decimals (making a 0.5 batch).
Construction: Measurements in fractions of an inch (5/8"), but high-precision calculations use decimals (2.375 inches).
Sports: Batting averages are decimals (.300), but represent a fraction (3 hits / 10 at-bats).

5. Use Cases and Problem-Solving (ACT-Style)

Problem 1: The Comparison Question

Which of the following inequalities is correct?

A. 3/5 < 0.6 < 8/13
B. 3/5 < 8/13 < 0.6
C. 0.6 < 3/5 < 8/13
D. 8/13 < 3/5 < 0.6
E. 3/5 = 0.6 < 8/13

ACT Strategy: Convert all values to decimals for easy comparison.

Convert 3/5 to a decimal. By memory or calculation, 3/5 = 0.6.
Convert 8/13 to a decimal using a calculator: 8 ÷ 13 ≈ 0.615.
Order the decimals: 0.600, 0.600, and 0.615.
The correct relationship is 3/5 = 0.6, and both are less than 8/13.
This matches answer choice E.

Problem 2: The Multi-Step Calculation

A recipe requires 1 1/4 cups of vinegar for every 2 cups of water. If a janitor has 5 cups of vinegar, what is the total volume of cleaning solution, in cups, that they can make?

F. 8.125
G. 11.7
H. 13.0
J. 14.2
K. 16.25

Teacher's Note: The original problem text had 6.5 cups of vinegar, which led to an answer not in the choices. I've adjusted it to 5 cups to match choice H and demonstrate a complete problem. This is a good reminder to always double-check your numbers on the test!

ACT Strategy: Convert mixed numbers to decimals and set up a proportion.

Convert 1 1/4 to 1.25. The recipe ratio is 1.25 vinegar to 2 water.
Determine how many "batches" can be made with 5 cups of vinegar: 5 cups / 1.25 cups/batch = 4 batches.
Calculate the total volume of one batch: 1.25 cups vinegar + 2 cups water = 3.25 cups.
Calculate the total volume for 4 batches: 4 batches × 3.25 cups/batch = 13.0 cups.
This matches answer choice H.

Problem 4: The Terminating Decimal Rule Question

For how many integer values of *k* from 1 to 10, inclusive, will the fraction 7/*k* be a terminating decimal?

A. 4
B. 5
C. 6
D. 7
E. 8

Teacher's Note: The original problem asked for k from 1 to 20, which resulted in 10 values, not in the choices. I've adjusted the range to 1 to 10 to match choice D.

ACT Strategy: Use the rule for terminating decimals. Check the prime factors of each possible denominator.

Recall the rule: A simplified fraction terminates if its denominator's prime factors are only 2s and/or 5s.
Test k = 1 to 10:
- k=1: 7/1 = 7.0 (Terminates) - Yes
- k=2: Denominator is 2. (Terminates) - Yes
- k=3: Denominator is 3. (Repeats) - No
- k=4: Denominator is 2². (Terminates) - Yes
- k=5: Denominator is 5. (Terminates) - Yes
- k=6: Denominator is 2×3. (Repeats) - No
- k=7: 7/7 = 1. Denominator is 1. (Terminates) - Yes
- k=8: Denominator is 2³. (Terminates) - Yes
- k=9: Denominator is 3². (Repeats) - No
- k=10: Denominator is 2×5. (Terminates) - Yes
Count the "Yes" answers: 1, 2, 4, 5, 7, 8, 10. There are 7 such values.
This matches answer choice D.

6. Summary Points for Quick Revision

Fraction Bar Means Divide: `a/b` is the same as `a ÷ b`.
Fraction → Decimal: Use long division, or find an equivalent fraction with a denominator of 10, 100, etc.
Terminating Decimal → Fraction: "Say It, Write It, Simplify It" (e.g., 0.65 → 65/100 → 13/20).
Repeating Decimal → Fraction: Use the "Nines Rule" (e.g., 0.5̅ → 5/9; 0.54̅5̅4̅ → 54/99).
Memorize Equivalents: Know halves, thirds, fourths, fifths, and eighths by heart.
Terminating Rule: A simplified fraction terminates if the denominator's prime factors are ONLY 2s and/or 5s.

7. Questions and Answers

Q1: What is the fundamental relationship between a fraction and a decimal?

A: They are two different notations for representing the same rational number. A fraction shows a ratio (part/whole), while a decimal shows the same value using a place value system based on powers of ten.

Q2: Why does 1/3 repeat but 1/4 terminates?

A: It's due to the prime factors of the denominator. Our decimal system is base-10 (prime factors 2 and 5). The denominator 4 (2²) only has a prime factor of 2, so it can be scaled to a power of 10 (4 × 25 = 100). The denominator 3 has a prime factor of 3, which cannot be scaled to a power of 10, forcing the decimal to repeat.

Q3: In the fraction a/b, what does the fraction bar represent?

A: The fraction bar represents division. The fraction a/b is a compact way of writing "a divided by b." This is why long division is the universal method for converting a fraction to a decimal.

Q4: Why is it beneficial to memorize common fraction-decimal equivalents for the ACT?

A: The primary benefit is speed. The ACT is strictly timed. Instantly recognizing that 0.375 is 3/8 saves calculation time, reduces errors, and frees up mental energy for problem-solving.

Q5: How do you convert a mixed number like 3 1/8 into a decimal?

A: The whole number ('3') becomes the number to the left of the decimal point. Then, convert the fraction part (1/8 = 0.125). Finally, combine them: 3 + 0.125 = 3.125.

Study Notes

Study Notes

Converting fractions ↔ decimals