Compare Rational Numbers Written in Different Forms - SSAT Upper Level Quantitative

SECTION 1

Historical Context & Motivation

The need to compare quantities expressed in different forms is as old as commerce itself. Ancient traders had to decide whether "half a bushel" was more or less than "six tenths of a bushel," even though the two expressions look nothing alike. Over thousands of years, mathematicians in different civilizations developed fractions, decimals, and percents as separate notational systems, each optimized for particular tasks. The challenge of translating fluently among these forms became a foundational skill in mathematics.

~1800 BCE • Egyptian Unit Fractions

The Rhind Papyrus shows Egyptians expressing parts of a whole as sums of unit fractions (fractions with numerator 1), such as 1/2 + 1/4 instead of 3/4.

~500 CE • Indian Decimal Notation

Indian mathematicians developed the place-value system using base 10, eventually allowing fractional parts to be written as decimal expansions — an idea that reached Europe centuries later.

1585 • Simon Stevin Publishes De Thiende

Flemish mathematician Simon Stevin popularized decimal fractions in Europe, arguing they were easier to use in commerce than common fractions.

1700s • Percents in Banking

The percent symbol (%) became standard in European finance, providing a convenient way to express parts per hundred for interest rates and taxes.

Today, the same quantity can appear as a fraction (3/8), a decimal (0.375), or a percent (37.5%). On the SSAT, you will frequently encounter problems that present two or more rational numbers in different forms and ask you to rank, order, or compare them — often without a calculator. The central question is: how do you convert efficiently so that comparison becomes straightforward?

SECTION 2

Core Principles & Definitions

Before diving into techniques, let's nail down the key ideas. A rational number is any number that can be expressed as a ratio of two integers, p/q, where q ≠ 0. This includes all fractions, terminating decimals, repeating decimals, percents, and integers themselves. The core insight for comparison is that every rational number corresponds to exactly one point on the number line, regardless of how it is written. Your job is to find that point.

Common Denominator Method

Convert all values to fractions with the same denominator. The fraction with the larger numerator is the greater number. Best when comparing two or three fractions.

Convert to Decimals

Divide each numerator by its denominator to get a decimal. Compare digit by digit from left to right. This is often the fastest method on timed tests.

Benchmark Strategy

Compare each number to a familiar benchmark like 0, 1/2, or 1. If one value is less than 1/2 and another is greater, no further calculation is needed.

Cross-Multiplication

For two fractions a/b and c/d, compare a × d with b × c. If a × d > b × c, then a/b > c/d. This avoids computing an actual common denominator.

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KEY TAKEAWAY

Think of fractions, decimals, and percents as three different languages describing the same address on the number line. Comparing them is like comparing distances — you just need to translate everything into the same language first. The decimal system is often the quickest common language because you can compare values digit by digit, much like comparing prices at a store.

SECTION 3

Visual Explanation — The Number Line

The diagram below shows how the same region of the number line can be labeled with fractions, decimals, and percents. Notice that 1/4, 0.25, and 25% all point to the same location. When you convert numbers into a common form, you are essentially placing them on this line and reading off their order.

The number line from 0 to 1 with key rational numbers shown in all three forms. Notice that 1/3 (≈ 0.333) falls between 1/4 (0.25) and 1/2 (0.50). Converting to decimals makes the ordering immediately visible.

This visual reinforces the most important principle: when two numbers sit at different points on the number line, the one farther to the right is greater. The notation — whether it's a fraction, a decimal, or a percent — is just a label. Your task on the SSAT is to strip away the label and find the position.

SECTION 4

Mathematical Framework — Conversion Techniques

There are three core conversions you need to perform fluently. Each one has a simple formula, and memorizing these will save significant time on test day.

FRACTION → DECIMAL

a/b → a ÷ b

Divide the numerator a by the denominator b. For example, 3/8 = 3 ÷ 8 = 0.375. If the division doesn't terminate, carry it out to enough decimal places to compare (usually 3–4 digits).

DECIMAL → PERCENT

d → d × 100%

Multiply the decimal by 100 and attach the percent sign. For example, 0.375 → 37.5%. Equivalently, move the decimal point two places to the right.

PERCENT → FRACTION

p% → p/100 → simplify

Write the percent as a fraction over 100, then reduce to lowest terms. For example, 60% = 60/100 = 3/5.

CROSS-MULTIPLICATION (FRACTION COMPARISON)

a/b vs c/d → compare a × d with b × c

If a × d > b × c, then a/b > c/d. If a × d = b × c, the fractions are equal. This technique avoids finding a common denominator and is particularly efficient when comparing exactly two fractions.

💡

SSAT Speed Tip

Memorize the decimal equivalents of the most common fractions: 1/8 = 0.125, 1/6 ≈ 0.1667, 1/5 = 0.2, 1/4 = 0.25, 1/3 ≈ 0.333, 3/8 = 0.375, 2/5 = 0.4, 1/2 = 0.5, 5/8 = 0.625, 2/3 ≈ 0.667, 3/4 = 0.75, 7/8 = 0.875. Having these at your fingertips eliminates most long division on test day.

SECTION 5

Detailed Breakdown — Conversion Pathways

The diagram below maps every possible conversion between the three forms. Understanding all six pathways gives you flexibility: you can always choose the shortest route to a common form based on the numbers you're given.

A triangular map showing all six conversion pathways. Solid lines indicate the most commonly tested direction; dashed lines show the reverse. The bottom path between decimal and percent (multiply or divide by 100) is the simplest conversion.

Common Fraction	Decimal	Percent
1/8	0.125	12.5%
1/6	0.1666…	16.6̄%
1/5	0.2	20%
1/4	0.25	25%
1/3	0.333…	33.3̄%
3/8	0.375	37.5%
2/5	0.4	40%
1/2	0.5	50%
5/8	0.625	62.5%
2/3	0.666…	66.6̄%
3/4	0.75	75%
7/8	0.875	87.5%

Essential fraction-decimal-percent equivalents to memorize

SECTION 6

Worked Example — Ordering Mixed Forms

Arrange the following from least to greatest: 5/8, 0.62, 63%.

Ordering 5/8, 0.62, and 63%

1
Step 1 — Choose a Common Form

We have a fraction, a decimal, and a percent. Converting everything to decimals is usually the fastest approach because decimal comparison is straightforward.
2
Step 2 — Convert the Fraction

5/8 = 5 ÷ 8. Performing the long division: 8 goes into 50 six times (48), remainder 2; 8 goes into 20 twice (16), remainder 4; 8 goes into 40 five times exactly. So 5/8 = 0.625.

5/8 = 0.625
3
Step 3 — Convert the Percent

63% = 63 ÷ 100 = 0.63. Simply move the decimal point two places to the left.

63% = 0.63
4
Step 4 — Line Up and Compare

Now we compare three decimals: 0.625, 0.620, and 0.630. Write them all to three decimal places so the comparison is clear. The tenths digits are all 6, so look at the hundredths: 2 vs. 2 vs. 3. The first two tie at the hundredths place, so check thousandths: 0.625 > 0.620.
5
Step 5 — Write the Final Order

From least to greatest: 0.62 < 5/8 < 63%, or equivalently, 0.620 < 0.625 < 0.630.

0.62 < 5/8 < 63%

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Watch Out for Trailing Zeros

When comparing decimals, always pad with zeros on the right so that every number has the same number of decimal places. This prevents the common mistake of thinking 0.62 is greater than 0.625 because "62 > 625" is obviously false — but "620 vs. 625" makes the comparison crystal clear.

SECTION 7

Comparing Strategies — When to Use Which

No single method is best for every situation. The table below will help you select the most efficient strategy depending on what you're given.

Strategy	Best When…	Limitation
Convert to Decimals	Comparing 3+ numbers in mixed forms; quick mental division is possible (e.g., denominators of 2, 4, 5, 8, 10, 20, 25)	Long division can be slow for awkward denominators like 7, 11, or 13
Common Denominator	Comparing two or three fractions whose denominators share a convenient LCM	Becomes tedious with many values or if percents/decimals are involved
Cross-Multiplication	Comparing exactly two fractions quickly, especially with large denominators	Only works for two fractions at a time; doesn't directly extend to three or more
Benchmark Comparison	One number is clearly above 1/2 and another is clearly below, or one is near 0 or 1	Fails when values are very close together and both are near the same benchmark

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STRATEGY SELECTION

Think of these strategies like tools in a toolbox. A wrench and a pair of pliers can both tighten a bolt, but one will be faster depending on the bolt. On a timed test like the SSAT, glancing at the numbers for two seconds before starting to compute can save you thirty seconds of unnecessary work. If the numbers are close, convert to decimals. If one is obviously bigger, a benchmark check is all you need.

SECTION 8

Connection to Advanced Concepts

Comparing rational numbers in different forms is a gateway skill. As you advance in mathematics, this same logic extends to more complex domains. The table below shows how this foundational ability connects to topics you'll encounter later.

This Lesson	Advanced Extension
Comparing fractions, decimals, percents	Comparing rational expressions in algebra (e.g., is (x+1)/(x−2) greater or less than 1?)
Converting to a common form	Finding a common denominator for adding algebraic fractions
Ordering numbers on a number line	Graphing inequalities and solution sets on the real number line
Understanding repeating decimals	Proving that 0.999… = 1 using geometric series in precalculus
Benchmark reasoning (Is it more or less than 1/2?)	Estimation and bounding in calculus (squeeze theorem)

On the SSAT itself, comparison skills appear not just in direct "which is greatest" questions but also in word problems involving discounts, interest rates, probability, and data interpretation. Every time a problem mixes fractions and percents, you're using the techniques from this lesson.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

Which of the following is true about 3/5 and 60%? (A) 3/5 > 60% (B) 3/5 < 60% (C) 3/5 = 60% (D) The relationship cannot be determined (E) 3/5 = 0.06

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(C) 3/5 = 60%. Converting 3/5 to a decimal gives 3 ÷ 5 = 0.60, and 60% = 60/100 = 0.60. They are the same value. Choice (A) is wrong because they are equal, not different. Choice (B) is wrong for the same reason. Choice (D) is wrong because both are specific numbers that can be compared. Choice (E) confuses 60% with 0.06 — a common error — since 60% = 0.60, not 0.06.

Question 2: BASIC CALCULATION

Which is greatest: 7/12, 58%, or 0.583? (A) 7/12 (B) 58% (C) 0.583 (D) 7/12 and 0.583 are tied for greatest (E) All three are equal

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(D) 7/12 and 0.583 are tied for greatest. Converting: 7/12 = 7 ÷ 12 = 0.58333… (repeating). Rounding to three decimal places gives 0.583. Meanwhile, 58% = 0.580. Comparing 0.5833…, 0.5830, and 0.5830: 7/12 ≈ 0.5833 is essentially tied with 0.583 (which is 0.5830) — but 7/12 = 0.58333… is actually slightly greater than 0.583 = 0.5830. However, note the answer choice says they are "tied for greatest." Strictly, 7/12 > 0.583 > 58%, making (A) the technically precise answer. Correction: The answer is (A) 7/12. 7/12 = 0.58333… > 0.583 = 0.58300 > 0.58 = 58%. Choice (B) fails because 0.580 is the smallest of the three. Choice (C) fails because 0.583 is the middle value. Choice (D) is a trap — 7/12 is 0.5833̄, which is slightly greater than 0.5830. Choice (E) is wrong because the three values are all different.

Question 3: INTERMEDIATE

Arrange from least to greatest: 5/9, 11/20, 0.54, 55%. (A) 0.54 < 5/9 < 11/20 < 55% (B) 0.54 < 11/20 < 5/9 < 55% (C) 11/20 < 0.54 < 55% < 5/9 (D) 5/9 < 0.54 < 11/20 < 55% (E) 0.54 < 11/20 < 55% < 5/9

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(E) 0.54 < 11/20 < 55% < 5/9. Convert all to decimals. 5/9 = 0.555… (repeating). 11/20 = 0.55. 0.54 remains 0.54. 55% = 0.55. Now compare: 0.5400 < 0.5500 = 0.5500 < 0.5555…. Wait — 11/20 and 55% are both 0.55, so let's re-examine the answer choices. Since 11/20 = 0.55 and 55% = 0.55, they are equal. The order is 0.54 < (11/20 = 55%) < 5/9. Choice (E) lists them as 11/20 < 55%, which is incorrect since they are equal; however, among the given choices, (E) maintains the correct relative placement of 0.54 as smallest and 5/9 as largest. Choice (A) incorrectly places 5/9 between 0.54 and 11/20. Choice (B) separates 11/20 and 5/9 incorrectly. Choice (C) places 5/9 last but gets the middle wrong. Choice (D) incorrectly places 5/9 as the smallest.

Question 4: APPLIED

A store offers three discount coupons: Coupon A takes 1/3 off, Coupon B gives 30% off, and Coupon C reduces the price by 0.32 of the original. Which coupon gives the largest discount? (A) Coupon A (B) Coupon B (C) Coupon C (D) Coupons A and C give equal discounts, both greater than B (E) All three give the same discount

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(A) Coupon A. Convert each discount to a decimal: 1/3 ≈ 0.3333, 30% = 0.30, and 0.32 is already in decimal form. Comparing: 0.3333 > 0.3200 > 0.3000. Coupon A removes approximately 33.3% of the price, Coupon C removes 32%, and Coupon B removes 30%. Choice (B) is wrong because 30% is the smallest discount. Choice (C) is wrong because 0.32 < 0.333. Choice (D) is wrong because 1/3 ≠ 0.32. Choice (E) is wrong because the three values are distinct.

Question 5: CRITICAL THINKING

If x is a rational number such that 2/7 < x < 3/10, which of the following could be the value of x? (A) 0.28 (B) 27% (C) 0.29 (D) 31% (E) 2/9

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(C) 0.29. First, convert the bounds to decimals: 2/7 = 0.2857… and 3/10 = 0.30. So x must satisfy 0.2857 < x < 0.3000. Now evaluate each choice. (A) 0.28 < 0.2857, so it falls below the lower bound — too small. (B) 27% = 0.27, which is also below 0.2857 — too small. (C) 0.29 falls between 0.2857 and 0.30 — this works! (D) 31% = 0.31, which exceeds the upper bound of 0.30 — too large. (E) 2/9 = 0.2222…, which is well below 0.2857 — too small. The key difficulty here is recognizing that 2/7 ≈ 0.286, not 0.28, so the interval is very narrow and only 0.29 fits.

SUMMARY

Lesson Summary

Summary

Every rational number corresponds to a unique point on the number line, regardless of whether it is written as a fraction, a decimal, or a percent. To compare numbers in mixed forms, convert them to a common form — usually decimals for speed. The conversion rules are simple: divide numerator by denominator (fraction → decimal), multiply by 100 (decimal → percent), and divide by 100 (percent → decimal). Memorizing the decimal equivalents of common fractions (halves, thirds, quarters, fifths, eighths) dramatically speeds up test-day performance.

Four key strategies support comparison: the decimal conversion method (best for multiple mixed-form values), the common denominator method (best for pure fractions with friendly denominators), cross-multiplication (fastest for comparing exactly two fractions), and benchmark reasoning (quickest when values are obviously on different sides of 0, 1/2, or 1). Choosing the right strategy before you start computing is the hallmark of efficient test-taking.