Converting Fractions & Decimals — I can convert between fractions and decimals and explain the equivalence. - Pre Algebra

SECTION 1

Where Did Fractions and Decimals Come From?

People have needed to split things into parts for thousands of years. Imagine sharing three loaves of bread among five friends. You can't give each person a whole loaf, so you need a way to describe "part of a whole." That's exactly the problem that led to fractions — and later, decimals.

Ancient civilizations each found their own clever way to write parts of numbers. Over time, mathematicians realized that fractions and decimals are just two notations for the same idea.

~1800 BCE • Egyptian Fractions

Ancient Egyptians wrote fractions with a numerator of 1, like ½ and ¼. They carved these symbols into papyrus scrolls for trade and building the pyramids.

~500 BCE • Babylonian Base-60 System

Babylonians used a base-60 number system (which is why we have 60 minutes in an hour). They wrote parts of numbers using place value — an early step toward decimals.

~800 CE • Modern Fraction Bar Appears

Arab mathematicians began writing fractions with a numerator on top and a denominator on the bottom, separated by a horizontal bar — the style we still use today.

1585 CE • Simon Stevin Introduces Decimals

Flemish mathematician Simon Stevin published a booklet showing how to write fractions using a base-10 place-value system. This made calculations much easier for merchants and scientists.

Today • Both Notations Everywhere

We use decimals on calculators, price tags, and sports stats. We use fractions in recipes, music, and measuring tools. Knowing how to switch between them is a key skill.

So here's the big question: if ¾ and 0.75 mean exactly the same thing, how do we go back and forth between the two? That's what this lesson is all about.

SECTION 2

Core Principles — What You Need to Know First

Before we start converting, let's lock down the key ideas. A fraction and a decimal are two different outfits for the same number. Understanding these principles will make every conversion feel natural.

A Fraction Is a Division Problem

The fraction bar means "divided by." So ³⁄₄ literally means 3 ÷ 4. Every fraction is a division problem waiting to happen.

Decimals Use Place Value

Each place after the decimal point represents a power of 10: tenths, hundredths, thousandths, and so on. The digit's position tells you its value.

Equivalence Means Equal Value

When we say ½ = 0.5, we mean they land on the exact same spot on a number line. Neither form is "better" — they're just different ways to write the same amount.

Terminating vs. Repeating Decimals

Some fractions, like ¼ = 0.25, produce decimals that stop (terminate). Others, like ¹⁄₃ = 0.333…, repeat forever. Both are still valid decimal forms.

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KEY TAKEAWAY

Think of fractions and decimals like English and Spanish. The sentence "I have three dogs" and "Tengo tres perros" carry the same meaning — just in different languages. Fractions and decimals carry the same value, just in different notation.

SECTION 3

Seeing the Connection

A picture is worth a thousand numbers. The diagram below shows how a single shaded region can be described as both a fraction and a decimal. Look at the bar model, the number line, and the place-value chart to see the same value represented three ways.

This diagram shows three representations of ¾: a shaded bar model, a number-line position at 0.75, and a place-value chart. The lower flowcharts show the steps for converting in each direction.

Notice how the shaded bar, the dot on the number line, and the digits 0.75 all describe exactly the same amount. The bar model splits a whole into 4 equal parts and shades 3 of them. The number line marks that same point between 0 and 1. The place-value chart tells us we have 7 tenths and 5 hundredths, which adds up to 0.75.

SECTION 4

The Math Behind the Conversion

Let's look at the formulas and rules that make conversions work every time. These aren't complicated — once you see the pattern, you'll be able to convert any fraction or decimal.

Fraction → Decimal

FRACTION TO DECIMAL

decimal = numerator ÷ denominator

The numerator (top number) is divided by the denominator (bottom number). For example, ⁵⁄₈ means 5 ÷ 8 = 0.625.

Decimal → Fraction

DECIMAL TO FRACTION

fraction = decimal digits ⁄ place-value denominator

Count the digits after the decimal point. If there is 1 digit, use 10 as the denominator. If 2 digits, use 100. If 3 digits, use 1,000. Then simplify by dividing both the numerator and denominator by their greatest common factor (GCF).

Simplifying a Fraction

SIMPLIFY USING GCF

a⁄b = (a ÷ GCF) ⁄ (b ÷ GCF)

The greatest common factor (GCF) is the largest number that divides evenly into both the numerator and the denominator. For example, for 75/100, the GCF is 25, so 75 ÷ 25 = 3 and 100 ÷ 25 = 4, giving us ³⁄₄.

ℹ

What About Repeating Decimals?

Some fractions produce decimals that never end and keep repeating the same digit or group of digits. For example, ¹⁄₃ = 0.333… (the 3 repeats forever). We write this as 0.3̄ with a bar over the repeating part. Not every fraction gives a clean, terminating decimal — and that's okay!

SECTION 5

Common Fraction-Decimal Equivalents

Some fraction-decimal pairs show up so often that it's worth memorizing them, like knowing your multiplication facts. The table and diagram below collect the ones you'll see most frequently in math class and everyday life.

Fraction	Division	Decimal	Type
½	1 ÷ 2	0.5	Terminating
¼	1 ÷ 4	0.25	Terminating
¾	3 ÷ 4	0.75	Terminating
⅕	1 ÷ 5	0.2	Terminating
⅛	1 ÷ 8	0.125	Terminating
¹⁄₃	1 ÷ 3	0.333…	Repeating
²⁄₃	2 ÷ 3	0.666…	Repeating
¹⁄₆	1 ÷ 6	0.1666…	Repeating

Common fraction-decimal equivalents

The number line shows common fractions and their decimal equivalents. Below, the two panels explain when a decimal terminates (stops) and when it repeats forever.

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Quick Rule: Will It Terminate?

Simplify the fraction first. Then look at the denominator. If the only prime factors of the denominator are 2 and/or 5, the decimal will terminate. If there is any other prime factor (like 3, 7, or 11), the decimal will repeat.

SECTION 6

Worked Examples — Step by Step

Example 1: Convert ⁷⁄₈ to a Decimal

Fraction → Decimal: ⁷⁄₈

1
Step 1 — Set Up the Division

The fraction ⁷⁄₈ means 7 ÷ 8. Write it as a long division problem: 8 goes into 7.000.
2
Step 2 — Divide 70 by 8

8 goes into 70 eight times (8 × 8 = 64). Subtract: 70 − 64 = 6. Bring down the next 0 to get 60.

First decimal digit: 8
3
Step 3 — Divide 60 by 8

8 goes into 60 seven times (8 × 7 = 56). Subtract: 60 − 56 = 4. Bring down the next 0 to get 40.

Second decimal digit: 7
4
Step 4 — Divide 40 by 8

8 goes into 40 exactly five times (8 × 5 = 40). Subtract: 40 − 40 = 0. The division is complete!

Third decimal digit: 5
5
Step 5 — State the Answer

Putting the digits together: ⁷⁄₈ = 0.875. This is a terminating decimal because the remainder reached 0.

⁷⁄₈ = 0.875

Example 2: Convert 0.35 to a Fraction

Decimal → Fraction: 0.35

1
Step 1 — Read the Place Value

The last digit (5) is in the hundredths place. So 0.35 means "35 hundredths."
2
Step 2 — Write as a Fraction

Write the digits over the place value: 35/100.

35/100
3
Step 3 — Find the GCF

Find the greatest common factor of 35 and 100. The factors of 35 are 1, 5, 7, 35. The factors of 100 include 1, 2, 4, 5, 10, 20, 25, 50, 100. The largest number they share is 5.

GCF = 5
4
Step 4 — Simplify

Divide both the numerator and denominator by 5: 35 ÷ 5 = 7 and 100 ÷ 5 = 20.

0.35 = 7/20

SECTION 7

When to Use Fractions vs. Decimals

Both fractions and decimals have their strengths. Sometimes one form is easier to work with than the other. The table below can help you decide which notation to use in different situations.

Situation	Fractions Are Better	Decimals Are Better
Exact values with repeating parts	Yes — ¹⁄₃ is exact	0.333… must be rounded
Comparing sizes quickly	Harder (different denominators)	Easy — just compare digits
Using a calculator	You must convert first	Calculators use decimals naturally
Cooking / measuring	Cups and spoons use fractions	Scales may show decimals
Money	Rarely used	$2.50 is standard

Choosing the best notation

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KEY TAKEAWAY

Think of fractions and decimals like sneakers and dress shoes. They both cover your feet, but each is better for certain occasions. Fractions shine when you need exact values (like ¹⁄₃). Decimals shine when you need to compare numbers or use a calculator. The best math students know when to switch!

SECTION 8

Connection to Percents and Ratios

Once you master fractions and decimals, you're ready for a third form: percents. A percent is just a special fraction with a denominator of 100. So the same value can appear in three outfits!

Concept	What You Know Now	What Comes Next
Notation	Fractions (¾) and decimals (0.75)	Percents (75%)
Operations	Add, subtract, multiply fractions & decimals	Find percent of a number, percent change
Applications	Recipes, measurements, basic math	Sales tax, tips, statistics, probability
Algebra connection	Fraction = numerator ÷ denominator	Ratios, proportions, and solving equations

From fractions & decimals to percents and beyond

Every time you convert a fraction to a decimal, you're practicing the same division and place-value thinking that powers algebra, statistics, and science. These skills only get more useful from here!

SECTION 9

Practice Problems

Try these five problems on your own. They start simple and get trickier. Work through each one, then check the answer to see if you're on the right track.

Question 1: CONCEPTUAL

In your own words, explain why the fraction ³⁄₅ and the decimal 0.6 represent the same number.

Show answer

The fraction ³⁄₅ means 3 divided by 5. When you perform that division (3.0 ÷ 5), you get 0.6. Both expressions describe the same point on a number line — the spot that is 3 out of 5 equal parts of a whole. They are two different ways to write the same value, just like "one half" and "0.5" mean the same thing.

Question 2: BASIC CALCULATION

Convert the fraction ⁵⁄₈ to a decimal by dividing.

Show answer

Divide 5 by 8. Set it up as 5.000 ÷ 8. First, 8 goes into 50 six times (48), remainder 2. Then 8 goes into 20 twice (16), remainder 4. Then 8 goes into 40 five times (40), remainder 0. The decimal is 0.625. Since the remainder reached 0, this is a terminating decimal.

Question 3: INTERMEDIATE

Convert the decimal 0.45 to a fraction in simplest form. Show your work.

Show answer

The last digit is in the hundredths place, so 0.45 = 45/100. To simplify, find the GCF of 45 and 100. The factors of 45 include 1, 3, 5, 9, 15, 45. The factors of 100 include 1, 2, 4, 5, 10, 20, 25, 50, 100. The GCF is 5. Divide both by 5: 45 ÷ 5 = 9 and 100 ÷ 5 = 20. So 0.45 = 9/20 in simplest form.

Question 4: APPLIED

A recipe calls for ⁷⁄₁₆ of a cup of sugar. Your digital kitchen scale shows decimals. How many cups is that as a decimal? Round to the nearest hundredth if needed.

Show answer

Divide 7 by 16. 16 goes into 70 four times (64), remainder 6. 16 goes into 60 three times (48), remainder 12. 16 goes into 120 seven times (112), remainder 8. 16 goes into 80 five times (80), remainder 0. So 7 ÷ 16 = 0.4375. Rounded to the nearest hundredth, that is 0.44 cups. The decimal terminates because 16 = 2⁴, and its only prime factor is 2.

Question 5: CRITICAL THINKING

Without doing long division, predict whether ⁹⁄₄₀ will produce a terminating or repeating decimal. Explain your reasoning, then verify by converting.

Show answer

First, check that 9/40 is already in simplest form (GCF of 9 and 40 is 1, so yes). Now look at the denominator: 40 = 2³ × 5. Since the only prime factors are 2 and 5, the decimal must terminate. To verify: 9 ÷ 40 = 0.225. Indeed it stops after three decimal places. This rule works because our number system is base 10, and 10 = 2 × 5, so any denominator built from only 2s and 5s divides evenly into a power of 10.

SUMMARY

Lesson Summary

Summary

Fractions and decimals are two notations for the same value. To convert a fraction to a decimal, divide the numerator by the denominator. To convert a decimal to a fraction, write the decimal digits over the appropriate power of 10 (10, 100, 1,000, etc.) and simplify by dividing both parts by their greatest common factor (GCF).

Some fractions produce terminating decimals (they stop), while others produce repeating decimals (a digit or group of digits repeats forever). A simplified fraction terminates only when the denominator's prime factors are limited to 2 and 5. Understanding these conversions prepares you for percents, ratios, and algebra — all of which build on the same ideas of division and place value.