Convert between Fractions, Decimals, and Percents - ISEE Middle Level Quantitative Reasoning

SECTION 1

Where Did These Three Forms Come From?

People have been splitting things into parts for thousands of years. Ancient Egyptians needed to divide bread fairly among workers. They invented some of the earliest fractions (numbers that show a part of a whole) to do it. Over time, mathematicians found other ways to write parts of a whole, giving us decimals and percents.

1800 BC • Egyptian Fractions

Ancient Egyptians wrote fractions using unit fractions like ½, ⅓, and ¼ to divide food and land among workers.

500 AD • Indian Decimal System

Mathematicians in India developed the base-10 number system we use today, laying the groundwork for decimal notation.

1585 • Decimal Point Introduced

Simon Stevin published a book showing how to use decimals for everyday math, making calculations much easier.

1600s • Percent Becomes Standard

Italian merchants popularized the word "per cento" (per hundred) for calculating taxes, tips, and interest rates.

Today, fractions, decimals, and percents are three different ways to write the same value. On the ISEE, you will need to switch between them quickly. Let's learn how!

SECTION 2

Core Principles: Three Names, One Number

Think of fractions, decimals, and percents as three languages for the same idea. The number ½ can also be written as 0.5 or 50%. They all mean half of something. Learning to translate between them is like being fluent in all three languages.

Fraction

A fraction shows a part over a whole, like ¾. The top number (numerator) tells how many parts you have. The bottom number (denominator) tells how many equal parts the whole is divided into.

Decimal

A decimal uses place value and a decimal point to show parts of a whole, like 0.75. Each place to the right of the point stands for tenths, hundredths, thousandths, and so on.

Percent

A percent means "per hundred." Writing 75% means 75 out of 100. Percents make it easy to compare parts because everything is based on the same denominator: 100.

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KEY TAKEAWAY

Imagine you have a pizza. Saying you ate "three-eighths," "0.375 of the pizza," or "37.5% of the pizza" all describe the exact same amount. Fractions, decimals, and percents are just three outfits for the same number!

SECTION 3

Seeing the Connection

The diagram below shows how the same value looks in all three forms. Study the arrows to see how you convert from one form to another.

The Conversion Triangle shows the six possible conversions. 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. These shortcuts will save you time on the ISEE.

Notice that every conversion follows a simple rule. The triangle is your roadmap. If you ever get stuck, just think: "Where am I, and where do I need to go?" Then follow the arrow.

SECTION 4

The Conversion Formulas

Here are the key formulas you need. Each one is a simple rule that works every time.

FRACTION → DECIMAL

Decimal = Numerator ÷ Denominator

Divide the top number by the bottom number. For example, 3 ÷ 4 = 0.75.

DECIMAL → PERCENT

Percent = Decimal × 100

Move the decimal point two places to the right and add the % sign. For example, 0.75 → 75%.

PERCENT → DECIMAL

Decimal = Percent ÷ 100

Move the decimal point two places to the left and drop the % sign. For example, 40% → 0.40.

DECIMAL → FRACTION

Write the decimal over the matching power of 10, then simplify

For 0.75, write 75/100. Then simplify: 75 ÷ 25 = 3, and 100 ÷ 25 = 4, so the fraction is ¾.

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ISEE Test Tip

On the ISEE, there is no penalty for wrong answers, so never leave a question blank! If you're unsure about a conversion, try plugging the answer choices back in to see which one matches.

SECTION 5

Must-Know Equivalents for the ISEE

Some conversions show up so often on the ISEE that you should memorize them. The table below lists the most common ones. Knowing these by heart will save you valuable time.

Fraction	Decimal	Percent
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%
1/3	0.333...	33⅓%
1/8	0.125	12.5%
1/10	0.1	10%

Common fraction-decimal-percent equivalents

This number line shows common fractions placed at their exact position between 0 and 1, with their decimal and percent equivalents labeled. Notice how all three forms mark the same spot on the line.

SECTION 6

Worked Example: Converting 3/8

Let's walk through a full conversion, step by step. We'll convert the fraction 3/8 into a decimal and then into a percent.

Convert 3/8 to a decimal and a percent

1
Step 1 — Identify the fraction

We start with the fraction 3/8. The numerator is 3 and the denominator is 8.
2
Step 2 — Divide numerator by denominator

Divide 3 by 8. You can do long division: 8 goes into 3.000 as follows. 8 goes into 30 three times (24), remainder 6. Bring down a zero to get 60. 8 goes into 60 seven times (56), remainder 4. Bring down a zero to get 40. 8 goes into 40 exactly 5 times.

3 ÷ 8 = 0.375
3
Step 3 — Convert the decimal to a percent

Multiply 0.375 by 100. This is the same as moving the decimal point two places to the right.

0.375 × 100 = 37.5%
4
Step 4 — Verify (optional but smart!)

Check: 37.5% means 37.5 out of 100. Write that as 37.5/100, which simplifies to 375/1000 = 3/8. It matches!

3/8 = 0.375 = 37.5% ✓

SECTION 7

When to Use Each Form

On the ISEE, different question types favor different forms. Knowing when to use each one helps you solve problems faster.

Form	Best For	Watch Out For
Fraction	Exact answers, multiplying parts of a number, comparing with common denominators	Hard to compare fractions with different denominators without converting
Decimal	Ordering numbers, adding and subtracting, comparing sizes quickly	Repeating decimals (like 0.333...) can be tricky to work with
Percent	Discounts, tips, tax problems, data interpretation, comparing ratios	Must convert to a decimal or fraction before doing most calculations

🎯

ISEE STRATEGY

If an ISEE question gives you a fraction but the answer choices are percents, convert the fraction right away. Match the form of your answer to the form of the choices — it's like making sure you're speaking the same language as the test!

SECTION 8

Connecting to Ratios and Proportions

Converting between fractions, decimals, and percents is the foundation for harder ISEE topics like ratios, proportions, and probability. Once you're comfortable switching forms, those topics become much easier.

This Lesson	What Comes Next
Convert 1/4 to 25%	Find 25% of a number (e.g., 25% of 60)
Know that 0.5 = 1/2	Probability: a 0.5 chance means "equally likely"
Write 3/5 as 60%	Use proportions: if 3 out of 5 students play sports, how many out of 30?

Think of this skill as a building block. Every time you see a ratio or probability question on the ISEE, there's a good chance you'll need to convert at some point. Mastering conversions now makes everything else smoother later.

SECTION 9

Practice Problems

Try these five problems. They mix standard multiple-choice questions with quantitative comparisons, just like the real ISEE. Remember: there's no penalty for guessing, so always pick an answer!

Question 1: CONCEPTUAL

Which of the following is equal to 0.6?

A. 1/6
B. 3/5
C. 2/3
D. 6/8

Show answer (B)

The correct answer is B. To check, divide 3 by 5: 3 ÷ 5 = 0.6. Choice A is wrong because 1 ÷ 6 = 0.1666..., which is not 0.6. Choice C is wrong because 2 ÷ 3 = 0.666..., a repeating decimal that is not equal to 0.6. Choice D is wrong because 6 ÷ 8 = 0.75.

Question 2: BASIC CALCULATION

What is 7/20 expressed as a percent?

A. 3.5%
B. 14%
C. 35%
D. 70%

Show answer (C)

The correct answer is C. First, convert 7/20 to a decimal: 7 ÷ 20 = 0.35. Then multiply by 100 to get the percent: 0.35 × 100 = 35%. Alternatively, since 20 × 5 = 100, multiply both numerator and denominator by 5: 7 × 5 = 35, so 7/20 = 35/100 = 35%. Choice A (3.5%) is too small — you may have divided by 200 instead of 20. Choice B (14%) could result from a calculation error. Choice D (70%) would be 14/20, not 7/20.

Question 3: INTERMEDIATE

Quantitative Comparison: Column A: 4/5 expressed as a decimal Column B: 85% expressed as a decimal

A. Column A is greater
B. Column B is greater
C. The two quantities are equal
D. Cannot be determined

Show answer (B)

The correct answer is B. Column A: 4 ÷ 5 = 0.80. Column B: 85% = 85 ÷ 100 = 0.85. Since 0.85 > 0.80, Column B is greater. Choice A is wrong because 0.80 is less than 0.85. Choice C is wrong because the two values are not equal. Choice D is wrong because both columns contain only numbers with no variables, so the relationship can always be determined.

Question 4: APPLIED

A store marks a $40 shirt as "30% off." To find the discount amount, Maya first converts 30% to a fraction. Which fraction should she use?

A. 1/3
B. 3/10
C. 3/8
D. 7/10

Show answer (B)

The correct answer is B. Convert 30% to a fraction: 30/100. Simplify by dividing both numbers by 10: 30 ÷ 10 = 3, 100 ÷ 10 = 10, giving 3/10. Choice A (1/3) equals 33.3%, not 30%. Choice C (3/8) equals 37.5%, not 30%. Choice D (7/10) equals 70%, which is what you'd pay after the discount, not the discount itself. As a check, 3/10 × $40 = $12 discount, so the shirt costs $28.

Question 5: CRITICAL THINKING

Quantitative Comparison: A jar contains some red and blue marbles. 60% of the marbles are red. Column A: The fraction of marbles that are blue Column B: 3/8

A. Column A is greater
B. Column B is greater
C. The two quantities are equal
D. Cannot be determined

Show answer (A)

The correct answer is A. If 60% of the marbles are red, then 100% − 60% = 40% are blue. Convert 40% to a fraction: 40/100 = 2/5. Now compare 2/5 and 3/8. Find a common denominator of 40: 2/5 = 16/40 and 3/8 = 15/40. Since 16/40 > 15/40, Column A (2/5) is greater than Column B (3/8). Choice B is wrong because 15/40 < 16/40. Choice C is wrong because the values are not equal. Choice D is wrong because the centered information tells us exactly what fraction of marbles are blue — there are no unknowns.

SUMMARY

Summary: Fractions, Decimals, and Percents

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. To go from a percent to a fraction, place the number over 100 and simplify.

Memorize the common equivalents (like 1/4 = 0.25 = 25%) to save time on the ISEE. Always match the form of your answer to the answer choices. On quantitative comparisons, convert both columns to the same form before comparing. And remember — never leave a question blank!