Comparing and Ordering Rational Numbers

⚖️ Comparing & Ordering Rational Numbers

Part 1 of 5 — What Counts as a Rational Number?

Topics in This Part

🔑 Key Concept: A rational number is any number you can write as a fraction $\dfrac{a}{b}$ of two integers, where $b \ne 0$. Whole numbers, negatives, terminating decimals, and repeating decimals are all rational. Before we can compare them, we need to see how they all live together on one number line.

What Is a Rational Number?

A number is rational when it can be written as $\dfrac{a}{b}$ with $a$ and $b$ integers and $b \ne 0$.

So every kind of number you'll meet in this lesson — integers, fractions, mixed numbers, and decimals — is just a different costume for a point on the number line.

💡 The word rational comes from ratio. A rational number is a ratio of two whole numbers.

Concept Check 🎯

The Number Line: Bigger = Farther Right

Every rational number is a single point on the number line. The rule for comparing is simple:

🔑 The Golden Rule: The number farther to the right is greater. The number farther to the left is smaller — even when it looks "bigger."

$\cdots\; -2 \;<\; -1 \;<\; -\tfrac{1}{2} \;<\; 0 \;<\; \tfrac{1}{2} \;<\; 1 \;<\; 2 \;\cdots$

A few quick reads off the line:

⚠️ Watch the trap: With negatives, the number with the larger digits is actually the smaller value. $-9$ is less than $-2$.

Read the Number Line 🔽

For each pair, pick the correct symbol so the statement is true.

Same Value, Different Costume

The hardest part of comparing rational numbers is that the same value shows up in many forms. You must be able to switch between them.

$\tfrac{1}{2} \;=\; 0.5 \;=\; 50\%, \qquad \tfrac{3}{4} \;=\; 0.75, \qquad 1\tfrac{1}{4} \;=\; \tfrac{5}{4} \;=\; 1.25$

To compare two rational numbers fairly, you first put them in the same form — either both as fractions with a common denominator, or both as decimals. Those are exactly the two methods we build next.

💡 In Part 2 we compare with common denominators; in Part 3 we compare with decimals. Both always agree — pick whichever is easier for the numbers in front of you.

Switch the Costume 🧮

Rewrite each value in the requested form.

1) Write $\dfrac{1}{2}$ as a decimal: $\,?$ 2) Write $0.25$ as a fraction in lowest terms (type 1/4): $\,?$ 3) Write the mixed number $1\tfrac{1}{2}$ as a decimal: $\,?$

⚖️ Comparing & Ordering Rational Numbers

Part 2 of 5 — Comparing with Common Denominators

🔑 The Idea: When two fractions have the same denominator, the one with the larger numerator is larger. So to compare any two fractions, first rewrite them over a common denominator.

The Common-Denominator Method

To compare $\dfrac{a}{b}$ and $\dfrac{c}{d}$:

1. Find a common denominator (the LCD, or just $b \times d$).
2. Rewrite both fractions over that denominator.
3. Compare the numerators.

Worked Example: $\dfrac{3}{4}$ vs $\dfrac{5}{8}$

The LCD of $4$ and $8$ is $8$. Rewrite $\dfrac{3}{4}$:

$\frac{3}{4} = \frac{3 \cdot 2}{4 \cdot 2} = \frac{6}{8}$

Now compare numerators over the same denominator $8$:

$\frac{6}{8} \;>\; \frac{5}{8} \quad\Longrightarrow\quad \frac{3}{4} > \frac{5}{8}$

✅ Check with decimals: $\frac{3}{4} = 0.75$ and $\frac{5}{8} = 0.625$, and $0.75 > 0.625$ ✓

Concept Check 🎯

When the LCD Isn't Obvious

If the denominators share no easy factor, use the LCD — the smallest number both divide into.

Worked Example: $\dfrac{7}{12}$ vs $\dfrac{5}{8}$

The LCD of $12$ and $8$ is $24$.

$\frac{7}{12} = \frac{7 \cdot 2}{12 \cdot 2} = \frac{14}{24}, \qquad \frac{5}{8} = \frac{5 \cdot 3}{8 \cdot 3} = \frac{15}{24}$

Compare numerators: $14 < 15$, so

$\frac{7}{12} \;<\; \frac{5}{8}$

💡 Shortcut — cross multiply. To compare $\dfrac{a}{b}$ and $\dfrac{c}{d}$ (positive), compare $a \cdot d$ with $c \cdot b$. Here $7 \cdot 8 = 56$ and $5 \cdot 12 = 60$; since $56 < 60$, the first fraction is smaller. Same answer, less writing.

Cross-Multiply to Compare 🧮

For each pair of positive fractions, compute the two cross products $a\cdot d$ and $c\cdot b$ to decide which is bigger.

1) Compare $\dfrac{3}{4}$ and $\dfrac{5}{7}$. Enter $3 \times 7 = \,?$ 2) Same pair: enter $5 \times 4 = \,?$ 3) Which fraction is larger — type 3/4 or 5/7.

Rewrite Over a Common Denominator 🧮

1) Rewrite $\dfrac{2}{3}$ over the denominator $12$. Enter the new numerator. 2) Rewrite $\dfrac{3}{4}$ over the denominator $12$. Enter the new numerator. 3) Using your answers, which is larger — type 2/3 or 3/4.

⚖️ Comparing & Ordering Rational Numbers

Part 3 of 5 — Comparing with Decimals

🔑 Why decimals help: Once two numbers are written as decimals, you compare them place by place — like reading a price tag. No common denominators required.

Turning a Fraction into a Decimal

A fraction is just a division problem: $\dfrac{a}{b} = a \div b$.

Some fractions give a terminating decimal (it stops); others give a repeating decimal (a digit pattern repeats forever, marked with a bar). Both are still rational.

💡 To compare a fraction and a decimal, it is usually easiest to turn the fraction into a decimal and line up the place values.

Comparing Decimals: Line Up the Places

Write the decimals with the same number of places (pad with zeros), then compare left to right.

Worked Example: Compare $0.6$ and $0.58$

Pad $0.6$ to two places: $0.60$.

$0.60 \quad\text{vs}\quad 0.58$

Tenths place: $6 > 5$, so $0.6 > 0.58$.

⚠️ Common mistake: thinking $0.58 > 0.6$ "because it has more digits." Length does not mean larger. Pad with zeros and compare place by place.

Worked Example: Compare $\dfrac{2}{5}$ and $\dfrac{3}{7}$

$\frac{2}{5} = 0.40, \qquad \frac{3}{7} = 0.428\ldots$

Tenths are equal ($4 = 4$); hundredths: $2 > 0$, so $\dfrac{3}{7} > \dfrac{2}{5}$.

Concept Check 🎯

Convert to Decimals 🧮

Write each fraction as a decimal. (Each of these terminates.)

1) $\dfrac{1}{4} = \,?$ 2) $\dfrac{7}{10} = \,?$ 3) $\dfrac{5}{8} = \,?$

Decimal Showdown 🔽

Convert where needed, then choose the correct symbol.

⚖️ Comparing & Ordering Rational Numbers

Part 4 of 5 — Negatives & Ordering a Whole List

🔑 The Big Skill: Real problems mix positives, negatives, fractions, and decimals all at once. The trick is to put everything in one common form (usually decimals), then walk the number line from left to right.

Comparing Negative Rational Numbers

The number-line rule still holds: farther right is greater. Since negatives sit left of $0$, the one closer to zero is the larger value.

Worked Example: Compare $-\dfrac{2}{3}$ and $-\dfrac{3}{4}$

Convert to decimals (keep the sign):

$-\frac{2}{3} = -0.66\overline{6}, \qquad -\frac{3}{4} = -0.75$

On the number line, $-0.666\ldots$ is to the right of $-0.75$ (it is closer to $0$), so

$-\frac{2}{3} \;>\; -\frac{3}{4}$

⚠️ The negative flip: Among positives, $\frac{3}{4} > \frac{2}{3}$. Add the minus signs and the order reverses: $-\frac{3}{4} < -\frac{2}{3}$. Bigger size = more negative = smaller value.

Concept Check 🎯

Ordering a Mixed List

To order a list of rational numbers from least to greatest:

1. Convert every number to the same form (decimals are easiest).
2. Negatives come first (most negative is least), then $0$, then positives.
3. Read off the order, then translate back to the original forms.

Worked Example: Order $-1.5,\; \dfrac{2}{5},\; -\dfrac{3}{4},\; 0,\; \dfrac{2}{3}$

Convert to decimals:

$-1.5,\quad \frac{2}{5}=0.4,\quad -\frac{3}{4}=-0.75,\quad 0,\quad \frac{2}{3}=0.66\overline{6}$

Place on the line, least to greatest:

$-1.5 \;<\; -0.75 \;<\; 0 \;<\; 0.4 \;<\; 0.666\ldots$

So the order is:

$-1.5 \;<\; -\frac{3}{4} \;<\; 0 \;<\; \frac{2}{5} \;<\; \frac{2}{3}$

Build the Order 🔽

You are ordering $\;\dfrac{1}{2},\; -0.3,\; \dfrac{3}{4},\; -1,\; 0\;$ from least to greatest. Fill each slot.

Order It Yourself 🧮

Order $\;\dfrac{1}{4},\; -\dfrac{1}{2},\; 0.6,\; -0.1\;$ from least to greatest.

Type each value (as written above) in order. Use 1/4 for the fraction and -1/2 for the negative fraction.

1) Least $= \,?$ 2) Next $= \,?$ 3) Next $= \,?$ 4) Greatest $= \,?$

⚖️ Comparing & Ordering Rational Numbers

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) recognize rational numbers, (2) compare with common denominators, (3) compare with decimals, and (4) order a mixed list including negatives. Let's put it all together.

Quick Reference

⚠️ Two traps to remember: a longer decimal is not automatically larger ($0.5 > 0.45$), and a "bigger looking" negative is smaller ($-9 < -2$).

Mixed Practice 🧮

1) Compare $\dfrac{4}{5}$ and $\dfrac{7}{9}$ by cross multiplying: enter $4 \times 9 = \,?$ 2) Same pair: enter $7 \times 5 = \,?$ 3) Which is larger — type 4/5 or 7/9.

Mixed Practice 🎯

Exit Quiz ✅

Answer all three to finish the lesson.