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

SECTION 1

Historical Context & Motivation

Humans have represented parts of a whole for thousands of years, but the notation we use today evolved slowly across many civilizations. Ancient Egyptians wrote fractions using unit fractions exclusively—every fraction was expressed as a sum of fractions with a numerator of 1. The idea of a single, flexible fraction notation took centuries to mature. Understanding this history shows you that fractions, decimals, and percents are simply different languages for the same idea: a part of a whole.

1800 BCE • Egyptian Unit Fractions

The Rhind Papyrus shows Egyptians expressing parts as sums of unit fractions like ½ + ¼, laying the groundwork for fraction arithmetic.

500 CE • Indian Decimal System

Indian mathematicians developed the place-value decimal system, which eventually allowed fractions to be written as decimal expansions.

1585 • Simon Stevin's Decimal Notation

Flemish mathematician Simon Stevin published De Thiende, advocating a standardized decimal notation that replaced awkward fraction tables for merchants and scientists.

1700s • Percent Becomes Standard

The percent symbol (%) evolved from the Italian 'per cento' and became the universal language of commerce, taxes, and interest rates.

Today, the ISEE tests your fluency in all three representations because real-world problems mix them constantly. A store advertises 25% off, your calculator shows 0.75, and a recipe calls for ¾ of a cup—these are all the same value. The central question this lesson addresses is: how do you move seamlessly between fractions, decimals, and percents so that no conversion slows you down on test day?

SECTION 2

Core Principles & Definitions

Before diving into conversion techniques, you need a solid grip on what each representation actually means. A fraction expresses a ratio of two integers—numerator over denominator. A decimal uses place value to represent a number in base ten, where digits to the right of the decimal point stand for tenths, hundredths, thousandths, and so on. A percent literally means "per hundred" and tells you how many parts out of 100 you have. All three are just different lenses on the same number.

Fraction → Decimal

Divide the numerator by the denominator. For example, 3 ÷ 8 = 0.375. This works for every fraction, whether the result terminates or repeats.

Decimal → Percent

Multiply the decimal by 100 (shift the decimal point two places right) and attach the % symbol. So 0.375 becomes 37.5%.

Percent → Fraction

Write the percent over 100, then simplify. For instance, 37.5% = 37.5/100 = 375/1000 = 3/8.

Reverse Conversions

Decimal → Fraction: read the place value and simplify. Percent → Decimal: divide by 100 (shift left two places). Fraction → Percent: convert to decimal first, then multiply by 100.

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KEY TAKEAWAY

Think of fractions, decimals, and percents like different currencies for the same amount of money. Just as you can express $0.75 as "three quarters" or "75 cents out of a dollar," every rational number has a fraction face, a decimal face, and a percent face. Converting is simply translating between currencies—the value never changes, only the notation.

SECTION 3

Visual Explanation — The Conversion Cycle

The triangle shows all six conversion paths. Solid arrows trace the most common direction; dashed arrows show the reverse. Notice that every path ultimately relies on division or multiplication by powers of 10.

Look at the diagram above and trace the journey of the value 3/8. Starting from the Fraction box, dividing 3 by 8 sends you to the Decimal box (0.375). Multiplying that decimal by 100 takes you to the Percent box (37.5%). Going back from percent to fraction, you write 37.5 over 100 and simplify. Every conversion is just one or two arithmetic steps, which is why memorizing the cycle gives you a huge speed advantage on the ISEE.

SECTION 4

Mathematical Framework

Let's formalize the six conversion rules so you have a concise reference. Each formula below uses a/b for a fraction (where b ≠ 0), d for a decimal, and p for a percent value.

FRACTION → DECIMAL

d = a ÷ b

Divide the numerator (a) by the denominator (b). Example: 7/20 = 7 ÷ 20 = 0.35.

DECIMAL → PERCENT

p = d × 100

Shift the decimal point two places to the right and add the % sign. Example: 0.35 × 100 = 35%.

PERCENT → DECIMAL

d = p ÷ 100

Shift the decimal point two places to the left and drop the % sign. Example: 62.5% ÷ 100 = 0.625.

PERCENT → FRACTION

a/b = p / 100, then simplify

Write the percent value over 100 and reduce to lowest terms. Example: 62.5% = 62.5/100 = 625/1000 = 5/8.

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ISEE Tip: Repeating Decimals

Some fractions produce repeating decimals: 1/3 = 0.333… and 1/6 = 0.1666…. On the ISEE, answer choices usually include a fraction or a rounded decimal, so you often do not need to write out the full repeating expansion. If an answer says 33⅓%, recognize that this equals 1/3 exactly.

SECTION 5

Benchmark Conversions You Should Memorize

Speed matters on the ISEE. You have about 57 seconds per question in the Quantitative Reasoning section, so memorizing benchmark conversions eliminates time-consuming long division. The table below lists the conversions that appear most frequently. Study them until they become automatic—like knowing that a quarter is $0.25.

Fraction	Decimal	Percent
1/2	0.5	50%
1/3	0.333…	33⅓%
2/3	0.666…	66⅔%
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%
5/8	0.625	62.5%
7/8	0.875	87.5%

Common benchmark conversions for the ISEE

This number line shows how common fractions, their decimal equivalents, and their percent equivalents all represent the same position. The amber dots mark quarter positions, while green and violet dots mark eighths and fifths.

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ISEE Strategy: Eliminate with Benchmarks

If a question asks you to compare 7/12 with 58%, quickly note that 7/12 is a bit more than 1/2 (which is 6/12). Since 7/12 ≈ 0.583, and 58% = 0.58, you can see that 7/12 is slightly larger. Benchmark values like 1/2 let you estimate without doing full division.

SECTION 6

Worked Example

Let's walk through a multi-step problem similar to what you might encounter on the ISEE. Pay attention to how we decide which conversion direction to use at each step.

Ordering Mixed Representations

1
Step 1 — Read the Problem

Arrange these values from least to greatest: 7/20, 0.38, 33⅓%, 3/8. We need a common format to compare all four values, and converting everything to decimals is the most efficient approach.
2
Step 2 — Convert Each Value to a Decimal

7/20: Divide 7 by 20. Since 20 × 0.35 = 7.00, we get 0.35. The value 0.38 is already a decimal. For 33⅓%, divide by 100 to get 0.333… (repeating). For 3/8, divide 3 by 8: 8 × 0.375 = 3.000, so 3/8 = 0.375.

7/20 = 0.35, 0.38 = 0.38, 33⅓% = 0.333…, 3/8 = 0.375
3
Step 3 — Order the Decimals

Line up the decimal places: 0.333…, 0.350, 0.375, 0.380. Comparing digit by digit from the tenths place, we see the order from least to greatest.

33⅓% < 7/20 < 3/8 < 0.38
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Step 4 — Verify with Benchmarks

Quick sanity check: 33⅓% is one-third, which is the smallest since it's less than 0.35. The value 0.38 is the largest because it exceeds 3/8 = 0.375. Our ordering is confirmed.

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STRATEGY NOTE

When comparing values in different formats, convert everything to the same format—usually decimals—before you compare. Decimals let you line up place values for quick, reliable ordering. This strategy works every time, regardless of how tricky the numbers look.

SECTION 7

Common Traps & How to Avoid Them

The ISEE is designed to test not just whether you know the conversions, but whether you can avoid careless mistakes under time pressure. Here are the most common traps students fall into and how to sidestep them.

Trap	What Goes Wrong	How to Fix It
Moving the decimal the wrong way	Converting 0.06 to percent and writing 0.06% instead of 6%.	Remember: decimal → percent means multiply by 100, so the decimal moves RIGHT two places.
Forgetting to simplify	Writing 40% as 40/100 and not reducing to 2/5. The correct answer choice may be 2/5.	Always simplify fractions by dividing numerator and denominator by their GCF.
Mishandling repeating decimals	Rounding 0.333… to 0.33, then getting 33% instead of 33⅓%.	Know the common repeating-decimal benchmarks (1/3, 2/3, 1/6, 5/6) and use exact fractions.
Confusing 'percent of' with 'percent'	Reading '0.5% of 200' as '50% of 200' and getting 100 instead of 1.	Convert the percent to a decimal first: 0.5% = 0.005. Then multiply: 0.005 × 200 = 1.
Percents greater than 100%	Thinking 150% can't be a valid answer because percents max out at 100.	Percents can exceed 100. 150% = 1.5 = 3/2. Think: '150% of 40 is 60.'

Five common conversion errors and their fixes

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KEY TAKEAWAY

Most conversion errors come from moving the decimal point the wrong direction or forgetting to simplify. A quick way to double-check: a percent should always be a bigger-looking number than its decimal equivalent (for example, 0.07 becomes 7%, not 0.07%). If your percent looks smaller than the decimal you started with, you moved the decimal the wrong way.

SECTION 8

Connections to Advanced ISEE Topics

Converting between fractions, decimals, and percents is not an isolated skill—it connects directly to several higher-level topics you will encounter on the ISEE. Understanding these connections shows you why fluency with conversions is so valuable.

Conversion Skill	Advanced ISEE Topic	How They Connect
Fraction → Decimal	Probability	Probability is usually expressed as a fraction, but you may need to compare it to a decimal threshold or percent.
Percent → Decimal	Percent Increase/Decrease	A 15% increase means multiplying by 1.15 (decimal form). Fast conversion is essential.
Decimal → Fraction	Ratios & Proportions	Cross-multiplication works best when values are in fraction form, so converting 0.6 to 3/5 speeds up solving.
All conversions	Data Analysis / Graphs	Pie charts use percents, tables use fractions, and calculations require decimals. You need all three.

How conversion skills feed into advanced ISEE topics

As you progress through your ISEE preparation, you will notice that conversion fluency saves you time on nearly every quantitative question. Percent increase and decrease problems, probability comparisons, and even some algebraic word problems all assume you can move freely between these three forms. Mastering this skill now creates a foundation for efficiency on every harder topic.

SECTION 9

Practice Problems

Try these five problems to solidify your conversion skills. Remember: on the ISEE there is no penalty for guessing, so always select an answer, even if you need to estimate. Use process of elimination to narrow your choices.

Question 1: CONCEPTUAL

Which of the following is equivalent to 0.125?

A. 1/12
B. 1/8
C. 1/5
D. 3/20

Show answer (B)

The correct answer is (B) 1/8. To convert 0.125 to a fraction, recognize that 0.125 = 125/1000. Simplify by dividing both numerator and denominator by 125: 125 ÷ 125 = 1 and 1000 ÷ 125 = 8, so 0.125 = 1/8. Choice (A) 1/12 = 0.0833…, which is smaller. Choice (C) 1/5 = 0.2, which is larger. Choice (D) 3/20 = 0.15, which is also larger. Knowing the benchmark 1/8 = 0.125 makes this problem nearly instant.

Question 2: BASIC CALCULATION

What is 7/8 expressed as a percent?

A. 78%
B. 85%
C. 87.5%
D. 90%

Show answer (C)

The correct answer is (C) 87.5%. First convert 7/8 to a decimal: 7 ÷ 8 = 0.875. Then multiply by 100 to get the percent: 0.875 × 100 = 87.5%. Choice (A) 78% would result from confusing the digits 7 and 8 with a percentage. Choice (B) 85% is close but incorrect—it does not correspond to any simple fraction with denominator 8. Choice (D) 90% = 9/10, which is a different fraction entirely.

Question 3: INTERMEDIATE

Column A: The decimal equivalent of 5/6. Column B: 0.84 Compare the two quantities.

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). To find the decimal equivalent of 5/6, divide 5 by 6: 5 ÷ 6 = 0.8333… (repeating). Column B is 0.84. Comparing the two: 0.8333… vs. 0.8400…. Looking at the hundredths digit, Column A has 3 and Column B has 4. Since 0.8333… < 0.84, Column B is greater. Choice (C) would require the values to be equal, which they are not. Choice (D) does not apply since both columns are fixed numbers with no variables.

Question 4: APPLIED

A store marks down a $240 jacket by 35%. The next week, the sale price is further reduced by 1/5. What is the final price of the jacket?

A. $108.00
B. $124.80
C. $156.00
D. $192.00

Show answer (B)

The correct answer is (B) $124.80. First, compute the price after the 35% markdown. 35% as a decimal is 0.35, and the discount amount is 0.35 × $240 = $84. The sale price is $240 − $84 = $156. Next, reduce this by 1/5. One-fifth of $156 is $156 ÷ 5 = $31.20. The final price is $156 − $31.20 = $124.80. Choice (A) $108.00 results from a common error: applying both discounts to the original price separately and subtracting both from $240. A student might compute 35% of $240 = $84 and 1/5 of $240 = $48, then subtract both from $240 to get $240 − $84 − $48 = $108.00, incorrectly treating the second discount as a percentage of the original rather than the already-reduced price. Choice (C) $156.00 is the price after only the first discount. Choice (D) $192.00 results from applying only the 1/5 reduction to the original price: $240 × 4/5 = $192.00.

Question 5: CRITICAL THINKING

x is a positive integer. Column A: The fraction x/(x + 1) expressed as a percent Column B: 90% Compare the two quantities.

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

Show answer (D)

The correct answer is (D). The value of x/(x + 1) expressed as a percent depends on the value of x. Test x = 1: 1/2 = 50%, which is less than 90%, so Column B is greater. Test x = 9: 9/10 = 90%, so the two quantities are equal. Test x = 100: 100/101 ≈ 99%, which is greater than 90%, so Column A is greater. Since different positive integer values of x produce different relationships between the columns, the relationship cannot be determined from the information given.

SUMMARY

Lesson Summary

Summary

Every rational number can be expressed as a fraction, a decimal, or a percent. To convert a fraction to a decimal, divide the numerator by the denominator. To convert a decimal to a percent, multiply by 100. To convert a percent to a fraction, write the percent over 100 and simplify. Memorize the benchmark conversions for halves, thirds, quarters, fifths, and eighths to save precious time on the ISEE.

When comparing values in mixed formats, convert everything to the same representation—decimals are usually the fastest choice. Watch out for common traps like moving the decimal point the wrong direction, forgetting to simplify fractions, or misreading small percents like 0.5%. On quantitative comparison questions, if the columns contain variables, test multiple values before choosing your answer. Remember: there is no guessing penalty on the ISEE, so always answer every question.