Rational Number Operations

A rational number is any number that can be written as a fraction of two integers. This includes integers, fractions, and terminating or repeating decimals. In this topic, you'll learn to add, subtract, multiply, and divide all types of rational numbers!

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What Are Rational Numbers?

Definition: A rational number can be written as a/b where a and b are integers and b ≠ 0.

Examples of Rational Numbers:

• Integers: 5 = 5/1, -3 = -3/1, 0 = 0/1
• Fractions: 2/3, -5/8, 7/4
• Decimals that terminate: 0.5 = 1/2, 0.75 = 3/4
• Decimals that repeat: 0.333... = 1/3, 0.666... = 2/3

NOT Rational:

• π (pi = 3.14159..., never ends, never repeats)
• √2 (square root of 2 = 1.41421..., never ends, never repeats)

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Adding and Subtracting Rational Numbers

The key is to work with common denominators!

Adding Fractions (Same Denominator)

When denominators are the same, just add the numerators!

Example: 2/7 + 3/7 = (2 + 3)/7 = 5/7

With negatives: -1/5 + 3/5 = (-1 + 3)/5 = 2/5

Adding Fractions (Different Denominators)

Find a common denominator first!

Example: 1/3 + 1/4

Step 1: Find LCD (Least Common Denominator) = 12

Step 2: Convert both fractions

• 1/3 = 4/12
• 1/4 = 3/12

Step 3: Add

• 4/12 + 3/12 = 7/12

Answer: 7/12

Subtracting Fractions

Same process as addition!

Example: 5/6 - 1/4

Step 1: LCD = 12

Step 2: Convert

• 5/6 = 10/12
• 1/4 = 3/12

Step 3: Subtract

• 10/12 - 3/12 = 7/12

Answer: 7/12

Adding and Subtracting with Negative Fractions

Use the integer rules you learned!

Example 1: -2/5 + (-1/5) = (-2 + -1)/5 = -3/5

Example 2: -3/4 - 1/4 = (-3 - 1)/4 = -4/4 = -1

Example 3: 2/3 - (-1/3) = 2/3 + 1/3 = 3/3 = 1

Remember: Subtracting a negative = adding a positive!

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Multiplying Rational Numbers

Multiplying fractions is easier than adding them - no common denominator needed!

Basic Fraction Multiplication

Rule: Multiply numerators, multiply denominators

Example: 2/3 × 3/5 = (2 × 3)/(3 × 5) = 6/15 = 2/5 (simplified)

Simplifying Before Multiplying

Cancel common factors first to make calculation easier!

Example: 4/9 × 3/8

Before canceling: (4 × 3)/(9 × 8) = 12/72 = 1/6

After canceling:

• 4 and 8 share factor 4: 4/8 = 1/2
• 3 and 9 share factor 3: 3/9 = 1/3
• (1 × 1)/(3 × 2) = 1/6 (much easier!)

Multiplying with Negative Fractions

Use the sign rules from integer multiplication!

Same signs → Positive:

• (2/3) × (1/4) = 2/12 = 1/6
• (-2/3) × (-1/4) = 2/12 = 1/6

Different signs → Negative:

• (2/3) × (-1/4) = -2/12 = -1/6
• (-2/3) × (1/4) = -2/12 = -1/6

Multiplying Mixed Numbers

Convert to improper fractions first!

Example: 2 1/3 × 1 1/2

Step 1: Convert

• 2 1/3 = 7/3
• 1 1/2 = 3/2

Step 2: Multiply

• (7 × 3)/(3 × 2) = 21/6

Step 3: Simplify

• 21/6 = 7/2 = 3 1/2

Answer: 3 1/2

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Dividing Rational Numbers

Remember: Dividing by a fraction means multiplying by its reciprocal!

Basic Fraction Division

Rule: Keep, Change, Flip (KCF)

1. Keep the first fraction
2. Change division to multiplication
3. Flip the second fraction (find reciprocal)

Example: 2/3 ÷ 4/5

Step 1: KCF

• 2/3 × 5/4

Step 2: Multiply

• (2 × 5)/(3 × 4) = 10/12 = 5/6

Answer: 5/6

Dividing with Negative Fractions

Use the division sign rules!

Same signs → Positive:

• (3/4) ÷ (1/2) = 3/4 × 2/1 = 6/4 = 3/2
• (-3/4) ÷ (-1/2) = 3/4 × 2/1 = 3/2

Different signs → Negative:

• (3/4) ÷ (-1/2) = 3/4 × (-2/1) = -6/4 = -3/2
• (-3/4) ÷ (1/2) = -3/4 × 2/1 = -3/2

Dividing Mixed Numbers

Convert to improper fractions, then use KCF!

Example: 3 1/4 ÷ 1 1/2

Step 1: Convert

• 3 1/4 = 13/4
• 1 1/2 = 3/2

Step 2: KCF

• 13/4 × 2/3

Step 3: Multiply

• (13 × 2)/(4 × 3) = 26/12 = 13/6

Step 4: Convert back

• 13/6 = 2 1/6

Answer: 2 1/6

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Working with Decimals

Decimals are rational numbers too!

Adding and Subtracting Decimals

Rule: Line up the decimal points!

Example: 3.45 + 12.8 - 5.23

Step 1: Line up 3.45 12.80

• -5.23

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10.02

Answer: 10.02

Multiplying Decimals

Multiply as if they were whole numbers, then place the decimal!

Example: 2.5 × 1.3

Step 1: Multiply 25 × 13 = 325

Step 2: Count decimal places (1 + 1 = 2)

Step 3: Place decimal: 3.25

Answer: 3.25

Dividing Decimals

Make the divisor a whole number!

Example: 7.2 ÷ 0.8

Step 1: Multiply both by 10

• 72 ÷ 8

Step 2: Divide

• 72 ÷ 8 = 9

Answer: 9

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Converting Between Forms

Decimal to Fraction

Terminating Decimal:

Example: 0.75

• Place over power of 10: 75/100
• Simplify: 3/4

Repeating Decimal:

Example: 0.333... = 1/3 (you'll learn the method for this later!)

Common ones to memorize:

• 0.333... = 1/3
• 0.666... = 2/3
• 0.111... = 1/9

Fraction to Decimal

Divide the numerator by the denominator!

Example: 3/8

• 3 ÷ 8 = 0.375

Example: 1/3

• 1 ÷ 3 = 0.333...

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Order of Operations with Rational Numbers

PEMDAS still applies!

Example: 1/2 + 3/4 × 2/3

Step 1: Multiply first

• 3/4 × 2/3 = 6/12 = 1/2

Step 2: Add

• 1/2 + 1/2 = 1

Answer: 1

Example with parentheses: (2/5 + 1/5) × 3

Step 1: Parentheses first

• 2/5 + 1/5 = 3/5

Step 2: Multiply

• 3/5 × 3 = 9/5 = 1 4/5

Answer: 1 4/5

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Real-World Applications

Cooking (Scaling Recipes)

Problem: A recipe calls for 2/3 cup of flour. You want to make 1.5 times the recipe. How much flour?

Solution:

• 2/3 × 1.5 = 2/3 × 3/2 = 6/6 = 1 cup

Answer: 1 cup

Shopping (Discounts)

Problem: A $45 shirt is on sale for 2/5 off. What's the discount amount?

Solution:

• 45 × 2/5 = 90/5 = $18 discount

Answer: $18 off (sale price =$27)

Construction (Board Cutting)

Problem: A 10.5-foot board is cut into pieces 1.75 feet long. How many pieces?

Solution:

• 10.5 ÷ 1.75 = 6

Answer: 6 pieces

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Common Mistakes to Avoid

❌ Mistake 1: Adding denominators

• Wrong: 1/4 + 1/4 = 2/8
• Right: 1/4 + 1/4 = 2/4 = 1/2

❌ Mistake 2: Not finding LCD

• Wrong: 1/3 + 1/4 = 2/7
• Right: 1/3 + 1/4 = 4/12 + 3/12 = 7/12

❌ Mistake 3: Forgetting to flip when dividing

• Wrong: 2/3 ÷ 1/2 = (2 × 1)/(3 × 2) = 2/6
• Right: 2/3 ÷ 1/2 = 2/3 × 2/1 = 4/3

❌ Mistake 4: Decimal placement errors

• Wrong: 2.5 × 1.3 = 32.5
• Right: 2.5 × 1.3 = 3.25

❌ Mistake 5: Sign errors with negatives

• Wrong: -1/2 + (-1/2) = 1
• Right: -1/2 + (-1/2) = -1

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Practice Strategy

For Fractions:

1. Check if you need common denominators (add/subtract = yes, multiply/divide = no)
2. Simplify before calculating when possible
3. Convert mixed numbers to improper fractions
4. Simplify your final answer

For Decimals:

1. Line up decimal points (add/subtract)
2. Count decimal places (multiply)
3. Move decimals to make whole numbers (divide)
4. Check that your answer makes sense

For Signs:

1. Determine sign of answer first (same → +, different → -)
2. Then do the operation
3. Apply the sign

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Summary

Rational numbers include all fractions, integers, and terminating/repeating decimals.

Operations:

• Add/Subtract: Need common denominators for fractions, line up decimals
• Multiply: Multiply across, no common denominator needed
• Divide: Use Keep-Change-Flip (multiply by reciprocal)

Sign Rules: Same as integers!

• Same signs → Positive
• Different signs → Negative

Master rational number operations and you're ready for solving equations, working with ratios, and tackling algebra!