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.
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?
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
Fraction → Decimal
Percent → Decimal
Use Benchmarks
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.
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.
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.
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% |
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%?
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. |
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) |
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.
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!