Convert between forms (fraction, decimal, percent) as...

SECTION 1

Where Did Fractions, Decimals, and Percents Come From?

People have always needed ways to talk about parts of a whole. Imagine splitting a loaf of bread or dividing a harvest fairly. Ancient civilizations created fractions to solve exactly that problem. Over time, mathematicians invented decimals and percents as other ways to express the same idea.

~1800 BCE

Egyptian Fractions

Ancient Egyptians used unit fractions (fractions with 1 on top, like ½ and ¼) to divide food and land.

~500 CE

Decimal System in India

Indian mathematicians developed the base-ten number system we use today, which later made decimals possible.

1585

Decimal Point Introduced

Simon Stevin, a Dutch mathematician, published a book showing how to use decimal notation for everyday calculations.

1600s–1700s

Percents in Business

European merchants began using "per centum" (Latin for "per hundred") to calculate interest, taxes, and profit.

Today, fractions, decimals, and percents show up everywhere—in test scores, recipes, store discounts, and sports stats. The big question is: how do you move between these three forms quickly and correctly? That's exactly what this lesson teaches you.

SECTION 2

Core Principles: Three Forms, One Value

Here is the most important idea in this lesson: a fraction, a decimal, and a percent can all represent the exact same amount. Think of them as three different outfits for the same number. The number ½ is the same as 0.5, which is the same as 50%. They look different, but they mean the same thing.

Fraction

A fraction shows a part out of a whole using a numerator (top number) and a denominator (bottom number). Example: ¾ means 3 parts out of 4.

Decimal

A decimal uses a decimal point to show parts of a whole in base ten. Example: 0.75 means 75 hundredths.

Percent

A percent means "out of 100." The symbol % tells you how many hundredths you have. Example: 75% means 75 out of 100.

They Are Equivalent

¾ = 0.75 = 75%. Converting just changes the form, never the value.

✦ KEY TAKEAWAY

Imagine you have a pizza cut into 4 equal slices, and you eat 3 of them. You could say you ate ¾ of the pizza, or 0.75 of the pizza, or 75% of the pizza. All three describe the same amount of pizza in your stomach!

SECTION 3

The Conversion Triangle

The diagram below shows all six possible conversions. Each arrow tells you the operation (what you do) to change one form into another. Study the arrows carefully—they are your roadmap for every conversion.

Each arrow shows the operation needed. For example, to go from a fraction to a decimal, divide the numerator by the denominator. To go from a decimal to a percent, multiply by 100 and add the % sign.

Notice that every conversion uses simple math you already know: division, multiplication by 100, or writing over a power of ten. Keep this triangle in mind as a quick-reference guide.

SECTION 4

The Conversion Formulas

Let's write down the exact steps for each direction. These formulas are simple, but memorizing them will save you a lot of time on tests.

FRACTION → DECIMAL

Decimal = Numerator ÷ Denominator

Divide the top number by the bottom number. For example, ³⁄₈ → 3 ÷ 8 = 0.375.

DECIMAL → PERCENT

Percent = Decimal × 100%

Move the decimal point two places to the right and add the % sign. For example, 0.375 → 37.5%.

PERCENT → DECIMAL

Decimal = Percent ÷ 100

Move the decimal point two places to the left and drop the % sign. For example, 37.5% → 0.375.

DECIMAL → FRACTION

Read the decimal, write it over the place value, then simplify.

For example, 0.375 is "375 thousandths," so write 375/1000. Simplify by dividing top and bottom by 125 to get ³⁄₈.

⚡ Shortcut: Fraction ↔ Percent

You can go straight from a fraction to a percent by dividing the numerator by the denominator, then multiplying by 100. Going the other way, drop the % and write the number over 100, then simplify. For example, 60% = 60/100 = ³⁄₅.

SECTION 5

Must-Know Equivalents

Some fractions, decimals, and percents come up so often that you should try to memorize them. The table below shows the most common ones. Knowing these will help you work faster on the TACHS.

Common fraction-decimal-percent equivalents

Fraction	Decimal	Percent
½	0.5	50%
¼	0.25	25%
¾	0.75	75%
⅓	0.333...	33.3̄%
⅔	0.666...	66.6̄%
⅕	0.2	20%
⅖	0.4	40%
⅗	0.6	60%
⅘	0.8	80%
⅛	0.125	12.5%
1	1.0	100%

Three number lines stacked to show that equivalent fractions, decimals, and percents always line up at the same point. The dashed vertical lines connect matching values.

Notice how the colored dots and dashed lines always line up. That's visual proof that these three forms are just different labels for the same spot on the number line.

SECTION 6

Worked Example: Converting ⅜ to All Three Forms

Let's walk through a full conversion. We'll start with the fraction ⅜ and find its decimal and percent equivalents.

Convert ⅜ to a decimal and a percent

Step 1 — Identify the fractionWe start with the fraction ⅜. The numerator is 3 and the denominator is 8.

Step 2 — Fraction → Decimal (divide top by bottom)Divide 3 by 8. You can use long division: 3 ÷ 8 = 0.375. Think of it as 3.000 ÷ 8. Eight goes into 30 three times (24), remainder 6. Eight goes into 60 seven times (56), remainder 4. Eight goes into 40 five times (40), remainder 0.

⅜ = 0.375

Step 3 — Decimal → Percent (multiply by 100)Take the decimal 0.375 and multiply by 100. That moves the decimal point two places to the right: 0.375 × 100 = 37.5. Then add the % sign.

0.375 = 37.5%

Step 4 — Check your answerTo verify, convert 37.5% back to a fraction: 37.5 ÷ 100 = 0.375, and 0.375 = 375/1000. Simplify by dividing numerator and denominator by 125: 375 ÷ 125 = 3, and 1000 ÷ 125 = 8. We get ⅜. ✓

⅜ = 0.375 = 37.5% ✓

SECTION 7

When to Use Each Form

Each form has situations where it's the easiest to work with. Part of being great at math is choosing the best form for the job.

Choosing the right form for the job

Form	Best Used When...	Example Situation
Fraction	You need exact values, especially with repeating decimals, or when multiplying/dividing parts.	A recipe calls for ⅔ cup of flour.
Decimal	You are adding, subtracting, or comparing numbers. Decimals line up nicely in columns.	Comparing prices: $2.49 vs. $2.53.
Percent	You want to describe a rate, a discount, a score, or a change relative to 100.	"You scored 85% on the quiz."

✦ KEY TAKEAWAY

Think of fractions, decimals, and percents like languages. Some sentences sound better in Spanish, and others in English. You say the same thing, but one way might be clearer. On a test, convert to whichever form makes the math simplest, then convert back to the form the question asks for.

SECTION 8

Tricky Cases and Common Mistakes

Some conversions are straightforward. Others can trip you up if you're not careful. Let's look at common mistakes and how to avoid them.

Watch out for these common errors

Tricky Case	Common Mistake	Correct Approach
Repeating decimals	Writing ⅓ = 0.3 instead of 0.333...	Use a bar over the repeating digit (0.3̄) or write "0.333..." to show it repeats forever.
Percents less than 1%	Thinking 0.5% = 0.5 (forgetting to divide by 100)	0.5% = 0.5 ÷ 100 = 0.005. Always divide by 100 when removing the % sign.
Percents greater than 100%	Thinking percents can't go above 100	150% = 1.5 = 3/2. A percent greater than 100 just means more than one whole.
Not simplifying fractions	Writing 0.4 as 4/10 instead of 2/5	Always simplify by dividing the numerator and denominator by their greatest common factor (GCF).

🔭 Looking Ahead

In high school, you'll use these same conversion skills when solving equations with proportions, working with probability, and analyzing data in statistics. Mastering conversions now gives you a strong foundation for algebra and beyond.

SECTION 9

Practice Problems

Try these five problems on your own. Each one gets a little harder. After you work through each problem, check the answer to see if you're on track.

PROBLEM 1 — CONCEPTUAL

True or false: 0.6 and 60% represent the same value. Explain your reasoning.

PROBLEM 2 — BASIC CALCULATION

Convert the fraction ⁷⁄₂₀ to a decimal and a percent.

PROBLEM 3 — INTERMEDIATE

Write 0.125 as a fraction in simplest form, then express it as a percent.

PROBLEM 4 — APPLIED

A store advertises "⅖ off all shoes." Your friend says that's a 25% discount. Is your friend correct? Show your work.

PROBLEM 5 — CRITICAL THINKING

Arrange these numbers from least to greatest: 33%, ⁷⁄₂₀, 0.34. (Hint: convert them all to the same form first.)

SUMMARY

Lesson Summary

Fractions, decimals, and percents are three ways to write the same value. To go from a fraction to a decimal, divide the numerator by the denominator. To go from a decimal to a percent, multiply by 100 and add the % sign. To go from a percent to a decimal, divide by 100 and drop the % sign. To go from a decimal to a fraction, write the decimal over its place value and simplify.

On the TACHS, choose the form that makes the problem easiest. To compare numbers, convert them all to the same form (usually decimals). Memorize common equivalents like ½ = 0.5 = 50% and ¼ = 0.25 = 25% to save time. Always simplify fractions to their lowest terms, and watch out for repeating decimals and percents less than 1% or greater than 100%.