Compare rational numbers in different forms. - ISEE Middle Level Quantitative Reasoning

SECTION 1

Why Do We Write Numbers in Different Forms?

Have you ever noticed that a store might advertise "50% off" while your math teacher writes the same idea as "½"? People have been using different ways to express the same number for thousands of years. Understanding how fractions, decimals, and percents connect to each other is one of the most useful skills in all of math.

Long ago, different cultures invented different systems for writing parts of whole numbers. Each form—fractions, decimals, and percents—was created to solve specific, real-world problems. Let's take a quick trip through history to see how these forms developed.

1800 BCE • Ancient Egyptian Fractions

Egyptians used unit fractions (fractions with 1 on top, like ⅓ and ¼) to divide bread and land fairly among workers.

500 CE • Indian Decimal System

Mathematicians in India developed the base-10 place-value system we use today, which eventually led to decimal numbers like 0.75.

1585 • Simon Stevin Publishes 'The Tenth'

A Flemish mathematician showed Europeans how to use decimals for everyday calculations, making math faster for merchants and scientists.

1700s • Percents in Business

The word 'percent' comes from the Latin 'per centum' meaning 'per hundred.' Bankers and traders adopted percents to describe interest rates and profits.

Today, the ISEE loves to test whether you can recognize that numbers written in different forms can actually be equal—or figure out which one is larger. The big question this lesson answers: How do you compare numbers when one is a fraction, another is a decimal, and another is a percent?

SECTION 2

Core Principles: Fractions, Decimals & Percents

A rational number is any number that can be written as a fraction where both the top number (numerator) and the bottom number (denominator) are integers, and the denominator is not zero. Fractions, terminating decimals, repeating decimals, and percents are all rational numbers. The key idea is that the same value can look completely different depending on which form you use.

Convert to a Common Form

To compare numbers in different forms, change them all into the same form—usually decimals. This makes it easy to line up place values and compare.

Fraction → Decimal

Divide the numerator by the denominator. For example, ¾ means 3 ÷ 4 = 0.75. This is the most important conversion to master.

Percent → Decimal

Move the decimal point two places to the left and drop the % sign. For example, 45% becomes 0.45. Think of it as dividing by 100.

Use Benchmarks

Memorize common equivalents like ½ = 0.5 = 50%. Use these 'benchmark' values to estimate and compare quickly without doing long division.

SECTION 3

Seeing the Connection: A Number Line View

One of the best ways to compare rational numbers is to picture them on a number line. When you place fractions, decimals, and percents on the same number line, you can instantly see which value is larger. The farther to the right a number sits, the greater it is.

Each dot shows the same value written as a fraction, a decimal, and a percent. Notice how ⅕ (violet) is to the left of ½ (amber), confirming that 0.20 is less than 0.50. The farther right, the larger the number.

The number line makes it clear: once every value is in the same form, comparing is just like reading a ruler. On the ISEE, you might not draw a number line, but you can always picture one in your head to check your work.

SECTION 4

Conversion Methods You Need to Know

Here are the key conversion formulas. Master these and you'll be able to compare any pair of rational numbers on the ISEE.

FRACTION TO DECIMAL

Decimal = Numerator ÷ Denominator

Example: ⅜ → 3 ÷ 8 = 0.375. Just divide the top by the bottom.

DECIMAL TO PERCENT

Percent = Decimal × 100

Example: 0.375 × 100 = 37.5%. Move the decimal point two places to the right.

PERCENT TO DECIMAL

Decimal = Percent ÷ 100

Example: 65% ÷ 100 = 0.65. Move the decimal point two places to the left.

COMPARING FRACTIONS WITH COMMON DENOMINATORS

a/c vs. b/c → just compare a vs. b

When two fractions share the same denominator, the one with the larger numerator is greater. Example: 5/8 > 3/8 because 5 > 3.

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ISEE Test Tip

SECTION 5

Benchmark Equivalents You Should Memorize

Knowing common equivalents by heart is like having a cheat sheet in your brain. These benchmark values show up again and again on the ISEE. If you memorize the table below, many comparison problems become almost instant.

Fraction	Decimal	Percent
1/10	0.1	10%
1/5	0.2	20%
1/4	0.25	25%
1/3	0.333…	33⅓%
2/5	0.4	40%
1/2	0.5	50%
3/5	0.6	60%
2/3	0.666…	66⅔%
3/4	0.75	75%
4/5	0.8	80%
1	1.0	100%

Common Fraction–Decimal–Percent Equivalents

This flowchart shows how to convert between fractions, decimals, and percents. The two worked examples at the bottom demonstrate the steps for ⅜ and 85%.

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Key takeaway

MEMORIZATION TIP

SECTION 6

Worked Example: Comparing Step by Step

Let's walk through a full comparison problem, just like you'd see on the ISEE. Suppose the question asks: Which is greatest: ⅝, 0.59, or 62%?

Worked example

1
Step 1 — Choose a Common Form

We have a fraction (⅝), a decimal (0.59), and a percent (62%). Let's convert everything to decimals. Decimals are usually the easiest form to compare because you can line up place values.
2
Step 2 — Convert the Fraction

⅝ means 5 ÷ 8. Let's do that division: 8 goes into 5.000 → 0.625.

⅝ = 0.625
3
Step 3 — Convert the Percent

62% means 62 ÷ 100. Move the decimal two places left: 0.62.

62% = 0.62
4
Step 4 — Line Up and Compare

Now we compare 0.625, 0.59, and 0.62. Add trailing zeros so they all have three decimal places: 0.625, 0.590, 0.620. Compare from left to right: 0.625 > 0.620 > 0.590.

⅝ is the greatest!
5
Step 5 — Final Order

From least to greatest: 0.59 < 62% < ⅝. The answer is ⅝.

0.59 < 0.62 < 0.625

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ISEE Strategy: Add Trailing Zeros

SECTION 7

Comparison Strategies: Strengths & Pitfalls

There are several ways to compare rational numbers. Each method has strengths and potential traps. Here's a side-by-side look at the main strategies.

Strategy	How It Works	Watch Out For…
Convert all to decimals	Divide numerator by denominator for fractions; divide by 100 for percents. Then compare digit by digit.	Long division can be slow without a calculator. Use benchmarks to avoid dividing when possible.
Common denominator	Find a common denominator for two fractions and then compare numerators.	Only works for fractions. Large denominators make this slow.
Cross-multiply	For two fractions a/b and c/d, compare a × d vs. c × b. The larger product tells you which fraction is bigger.	Keep track of which product goes with which fraction. Only use this for comparing two fractions.
Benchmark estimation	Compare each number to a known benchmark like ½ (0.5 or 50%). If one is above ½ and one is below, you're done!	This doesn't always work when both numbers are close to the same benchmark.

Four Strategies for Comparing Rational Numbers

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Key takeaway

WHEN IN DOUBT, CONVERT TO DECIMALS

SECTION 8

Looking Ahead: Rational Numbers in Harder Math

The skills you're building now—converting and comparing rational numbers—are the foundation for many topics you'll encounter in higher-level math. Here's a quick preview.

What You're Learning Now	Where It Leads
Comparing fractions and decimals	Ordering rational numbers on a number line, including negatives (7th-8th grade)
Converting percents	Percent increase/decrease, tax, tip, and discount problems (pre-algebra)
Finding common denominators	Adding and subtracting algebraic fractions in Algebra 1
Understanding repeating decimals	Distinguishing rational from irrational numbers (like √2 and π)

How These Skills Connect to Future Math

The ISEE is testing more than just this one skill. It's checking whether you have the number sense needed to succeed in more advanced courses. Every time you practice converting and comparing, you're strengthening that number sense for the future.

SECTION 9

Practice Problems

Try these five problems. Three are standard multiple-choice questions and two are quantitative comparison questions—both types appear on the ISEE. Remember: there is no penalty for guessing, so always pick an answer!

Question 1: CONCEPTUAL

Which of the following is equal to 0.4?

A. 1/4
B. 2/5
C. 1/2
D. 3/4

Show answer

The correct answer is (B) 2/5. Convert each fraction to a decimal: 1/4 = 0.25, 2/5 = 0.4, 1/2 = 0.5, and 3/4 = 0.75. Only 2/5 equals 0.4. Choice A (0.25) is too small, choice C (0.5) is too large, and choice D (0.75) is much too large.

Question 2: BASIC CALCULATION

Which shows these numbers in order from least to greatest? ⅜, 40%, 0.35

A. 0.35, ⅜, 40%
B. ⅜, 0.35, 40%
C. 40%, ⅜, 0.35
D. 0.35, 40%, ⅜

Show answer

The correct answer is (A). Convert all three numbers to decimals to compare them easily. ⅜ = 3 ÷ 8 = 0.375. 40% = 40 ÷ 100 = 0.40. 0.35 is already a decimal. Now compare: 0.35 < 0.375 < 0.40. Replacing the decimals back with the original forms gives 0.35 < ⅜ < 40%, so the correct order from least to greatest is 0.35, ⅜, 40%. Choice B incorrectly places ⅜ before 0.35. Choice C reverses the order entirely. Choice D incorrectly places 40% between 0.35 and ⅜.

Question 3: INTERMEDIATE

Quantitative Comparison: Column A: 7/12 Column B: 58%

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

Show answer

The correct answer is (A). Convert each value to a decimal for easy comparison. Divide 7 by 12: 7 ÷ 12 = 0.5833… Convert 58% to a decimal: 58% = 0.58. Now compare: 0.5833… vs. 0.5800. Since 0.5833… > 0.5800, Column A is greater. Choice B is incorrect because 0.58 is the smaller value. Choice C is incorrect because the two quantities are not equal. Choice D does not apply because both columns contain fixed numbers with no variables.

Question 4: APPLIED

Maria scored 17 out of 20 on her science quiz. Jake scored 82% on the same quiz. Who scored higher?

A. Maria, because 17/20 > 82%
B. Jake, because 82% > 17/20
C. They scored the same.
D. Maria, because 17 > 8.2

Show answer

The correct answer is (A). Convert Maria's score to a percentage: 17 ÷ 20 = 0.85, which equals 85%. Compare: 85% > 82%, so Maria scored higher. Choice B is incorrect because 82% is less than 85%. Choice C is incorrect because the scores are not equal. Choice D names the right person but uses flawed reasoning — you cannot compare 17 to 8.2 directly because they represent completely different quantities.

Question 5: CRITICAL THINKING

Quantitative Comparison: A positive fraction has a numerator of n and a denominator of 2n.

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

Show answer

The correct answer is (A). Step 1: Simplify Column A. The fraction n/(2n) has n in both the numerator and denominator. Since n is a positive number, cancel it: n/(2n) = 1/2. Step 2: Convert 1/2 to a percentage. 1/2 = 50%. Step 3: Compare. Column A = 50% and Column B = 45%. Since 50% > 45%, Column A is greater. Note: Even though n is a variable, n/(2n) always simplifies to exactly 1/2 no matter what positive value n takes, so the relationship is always definite. Choice (D) is tempting because of the variable, but the fraction reduces to a constant, making (D) incorrect.