Fractions, Decimals, and Percents

SECTION 1

Historical Context & Motivation

Have you ever wondered why we have three different ways to write the same number? A score of 3 out of 4 on a quiz can be written as the fraction ¾, the decimal 0.75, or as 75 percent. These three forms didn't all appear at the same time in history. Each one was invented to solve real-world problems.

~1800 BCE

Ancient Egyptian Fractions

Egyptians used unit fractions (fractions with 1 on top, like ½ and ⅓) to divide bread, land, and supplies fairly among workers.

~500 CE

Indian Decimal System

Mathematicians in India developed the base-10 number system we use today. This made it possible to write parts of a whole using a decimal point.

1585

Simon Stevin Promotes Decimals

The Flemish mathematician Simon Stevin published a book showing European merchants how decimals make calculations much easier than fractions.

1600s–1700s

Percent Becomes Standard

Italian traders used the phrase 'per cento' (meaning 'for every hundred') to compare prices and interest rates. The % symbol soon became standard.

Today, you'll see all three forms on the SHSAT. A question might give you a fraction and ask you to compare it to a decimal or a percent. The key question is: How do you move smoothly between fractions, decimals, and percents? That's exactly what this lesson teaches you.

SECTION 2

Core Principles & Definitions

Before we start converting, let's make sure we understand what each form really means. All three are just different ways to express a part of a whole.

Fraction

A fraction shows a part over a whole. The top number (numerator) tells how many parts you have. The bottom number (denominator) tells how many equal parts make the whole.

Decimal

A decimal uses the base-10 place value system to show parts of a whole. Each digit after the decimal point represents tenths, hundredths, thousandths, and so on.

Percent

A percent means 'out of 100.' So 75% means 75 out of every 100. The percent symbol (%) is a shortcut for dividing by 100.

They Are All the Same Value

The fraction ¾, the decimal 0.75, and 75% all sit at the exact same spot on a number line. Converting doesn't change the value — only the way it looks.

✦ KEY TAKEAWAY

Think of fractions, decimals, and percents like three languages describing the same thing. Saying 'half,' '0.5,' and '50%' is like saying 'water,' 'agua,' and 'eau.' Different words, same meaning. Converting is just translating between languages.

SECTION 3

The Conversion Triangle

The diagram below shows you the six possible conversions between fractions, decimals, and percents. Each arrow tells you the operation you need to perform. Study this triangle — it's your roadmap for every conversion.

The six conversion paths. Going from Fraction → Decimal, divide the numerator by the denominator. Going from Decimal → Percent, multiply by 100.

Notice the pattern. Going clockwise from Fraction to Decimal, you divide the numerator by the denominator. Going from Decimal to Percent, you multiply by 100. Going from Percent back to Fraction, you put the number over 100 and simplify. Every reverse path just does the opposite operation.

SECTION 4

The Conversion Formulas

Here are the main formulas you need. Don't just memorize them — understand what each one is really doing.

FRACTION → DECIMAL

Decimal = Numerator ÷ Denominator

Divide the top number by the bottom number. For example, 3 ÷ 4 = 0.75. You can use long division if you don't have a calculator.

DECIMAL → PERCENT

Percent = Decimal × 100

Move the decimal point two places to the right. For example, 0.75 × 100 = 75%. The word 'percent' literally means 'per hundred.'

PERCENT → FRACTION

Fraction = Percent ÷ 100, then simplify

Write the percent as a fraction over 100, then reduce to lowest terms. For example, 75% = 75/100 = 3/4.

DECIMAL → FRACTION

Read the decimal place value, write as a fraction, simplify

0.75 is 'seventy-five hundredths' = 75/100 = 3/4. The number of digits after the decimal tells you the denominator: 1 digit → 10, 2 digits → 100, 3 digits → 1000.

💡 SHSAT Tip

On the SHSAT, you won't have a calculator. Practice long division for tricky fractions like 5/6 or 7/8. Many test-takers memorize common conversions (like ⅓ = 0.333… ≈ 33.3%) to save time.

SECTION 5

Common Conversions You Should Know

Some conversions come up so often that it's worth memorizing them. The table and diagram below show the most common values tested on the SHSAT.

Common fraction-decimal-percent equivalents for the SHSAT

Fraction	Decimal	Percent
1/2	0.5	50%
1/3	0.333…	33.3…%
2/3	0.666…	66.6…%
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%
1/8	0.125	12.5%
3/8	0.375	37.5%
1/10	0.1	10%

Two number lines showing common fractions and their decimal/percent equivalents. Notice how thirds and sixths produce repeating decimals, while halves, quarters, and fifths produce terminating decimals.

🔁 Repeating Decimals

When you divide and the digits start repeating forever (like 0.333…), that's called a repeating decimal. You can write it with a bar over the repeating part: 0.3̄. On the SHSAT, you'll often see these written with '…' or rounded.

SECTION 6

Worked Example: All Six Conversions

Let's walk through a complete problem that uses multiple conversions. Follow each step carefully.

Convert 7/8 to a decimal and a percent.

Step 1 — Set up the divisionTo convert the fraction 7/8 to a decimal, divide the numerator (7) by the denominator (8). Set up 7 ÷ 8. Since 8 doesn't go into 7, we write 7.000 and use long division.

Step 2 — Perform the long division8 goes into 70 eight times (8 × 8 = 64). Write 8 after the decimal point. Subtract: 70 − 64 = 6. Bring down the next 0 to get 60. 8 goes into 60 seven times (8 × 7 = 56). Subtract: 60 − 56 = 4. Bring down the next 0 to get 40. 8 goes into 40 exactly 5 times (8 × 5 = 40). No remainder.

7 ÷ 8 = 0.875

Step 3 — Convert the decimal to a percentMultiply the decimal by 100. This is the same as moving the decimal point two places to the right: 0.875 × 100 = 87.5.

0.875 = 87.5%

Step 4 — Quick checkDoes 87.5% make sense? Since 7/8 is just a little less than 8/8 (which is 100%), we'd expect a percent close to but less than 100. Yes, 87.5% checks out!

7/8 = 0.875 = 87.5% ✓

Convert 0.36 to a fraction in simplest form.

Step 1 — Read the place valueThe number 0.36 has two digits after the decimal point. That means it reaches the hundredths place. So 0.36 = 36/100.

Step 2 — Simplify the fractionFind the greatest common factor (GCF) of 36 and 100. The factors of 36 include 1, 2, 3, 4, 6, 9, 12, 18, 36. The factors of 100 include 1, 2, 4, 5, 10, 20, 25, 50, 100. The GCF is 4.

Step 3 — Divide both parts by the GCFDivide the numerator and denominator by 4: 36 ÷ 4 = 9, and 100 ÷ 4 = 25.

0.36 = 9/25

SECTION 7

When to Use Each Form

On the SHSAT, you'll need to decide which form is easiest to work with. Here's a comparison to help you choose.

Choosing the right form on the SHSAT

Form	Best For	Watch Out For
Fraction	Exact values, multiplying/dividing parts of a whole, finding common denominators to compare	Hard to compare fractions with different denominators at a glance
Decimal	Adding and subtracting, comparing sizes, ordering numbers from least to greatest	Repeating decimals can be tricky — rounding may lose accuracy
Percent	Comparing relative amounts, understanding discounts, tips, taxes, and probability	Some students forget that 5% = 0.05, not 0.5 (a common SHSAT trap!)

✦ KEY TAKEAWAY

When a SHSAT question asks you to compare numbers given in different forms (like 'Which is greatest: 3/5, 0.58, or 62%?'), convert them all to the same form first. Decimals are usually the easiest form for comparing. Think of it like measuring weight — you wouldn't compare pounds, kilograms, and ounces without converting to the same unit first!

SECTION 8

Connection to Advanced Topics

Mastering conversions isn't just useful for one type of question. These skills appear throughout the SHSAT and beyond. Let's see how this topic connects to bigger ideas.

How conversion skills build toward advanced math

What You Learn Now	Where It Leads
Convert percents to decimals (e.g., 15% = 0.15)	Solving percent word problems: 'What is 15% of 200?' → 0.15 × 200 = 30
Convert fractions to decimals for comparing	Ordering rational numbers, placing values on a number line, and probability
Recognize repeating decimals (0.333…)	Understanding rational vs. irrational numbers in algebra and high school math
Simplify fractions using GCF	Simplifying algebraic fractions and working with ratios and proportions

On the SHSAT specifically, these conversions show up in word problems, comparison questions, and data interpretation. If you can convert fluently, you save time and avoid careless mistakes on test day.

SECTION 9

Practice Problems

Try these five problems on your own before checking the answers. They go from easy to challenging.

PROBLEM 1 — CONCEPTUAL

True or False: 0.4, 2/5, and 40% all represent the same value. Explain your reasoning.

PROBLEM 2 — BASIC CALCULATION

Convert 5/8 to a decimal and then to a percent.

PROBLEM 3 — INTERMEDIATE

Which is the greatest: 7/12, 0.58, or 59%? Show your work.

PROBLEM 4 — APPLIED

A store marks a jacket as '3/10 off.' Another store offers '25% off' the same jacket. If both jackets originally cost $80, which store gives you the bigger discount? By how much?

PROBLEM 5 — CRITICAL THINKING

A student claims that 1/7 = 0.14 = 14%. Is this correct? If not, explain the error and find a better approximation.

SUMMARY

Lesson Summary

Fractions, decimals, and percents are three forms of 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. To go from a percent to a fraction, place the number over 100 and simplify. Memorize common equivalents like 1/4 = 0.25 = 25% to save time on the SHSAT.

When comparing values in different forms, convert them all to the same form — decimals are usually easiest. Watch out for repeating decimals (like 1/3 = 0.333…) and avoid rounding too early. Practice these conversions until they feel automatic, and you'll be ready for any SHSAT rational number question.