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Add, subtract, multiply, and divide rational numbers
Learn step-by-step with practice exercises built right in.
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!
Definition: A rational number can be written as a/b where a and b are integers and b ≠ 0.
Examples of Rational Numbers:
NOT Rational:
The key is to work with common denominators!
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
Find a common denominator first!
Example: 1/3 + 1/4
Step 1: Find LCD (Least Common Denominator) = 12
Step 2: Convert both fractions
Step 3: Add
Answer: 7/12
Same process as addition!
Example: 5/6 - 1/4
Step 1: LCD = 12
Step 2: Convert
Step 3: Subtract
Answer: 7/12
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!
Multiplying fractions is easier than adding them - no common denominator needed!
Rule: Multiply numerators, multiply denominators
Example: 2/3 × 3/5 = (2 × 3)/(3 × 5) = 6/15 = 2/5 (simplified)
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:
Use the sign rules from integer multiplication!
Same signs → Positive:
Different signs → Negative:
Convert to improper fractions first!
Example: 2 1/3 × 1 1/2
Step 1: Convert
Step 2: Multiply
Step 3: Simplify
Answer: 3 1/2
Remember: Dividing by a fraction means multiplying by its reciprocal!
Rule: Keep, Change, Flip (KCF)
Example: 2/3 ÷ 4/5
Step 1: KCF
Step 2: Multiply
Answer: 5/6
Use the division sign rules!
Same signs → Positive:
Different signs → Negative:
Convert to improper fractions, then use KCF!
Example: 3 1/4 ÷ 1 1/2
Step 1: Convert
Step 2: KCF
Step 3: Multiply
Step 4: Convert back
Answer: 2 1/6
Decimals are rational numbers too!
Rule: Line up the decimal points!
Example: 3.45 + 12.8 - 5.23
Step 1: Line up 3.45 12.80
10.02
Answer: 10.02
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
Make the divisor a whole number!
Example: 7.2 ÷ 0.8
Step 1: Multiply both by 10
Step 2: Divide
Answer: 9
Terminating Decimal:
Example: 0.75
Repeating Decimal:
Example: 0.333... = 1/3 (you'll learn the method for this later!)
Common ones to memorize:
Divide the numerator by the denominator!
Example: 3/8
Example: 1/3
PEMDAS still applies!
Example: 1/2 + 3/4 × 2/3
Step 1: Multiply first
Step 2: Add
Answer: 1
Example with parentheses: (2/5 + 1/5) × 3
Step 1: Parentheses first
Step 2: Multiply
Answer: 1 4/5
Problem: A recipe calls for 2/3 cup of flour. You want to make 1.5 times the recipe. How much flour?
Solution:
Answer: 1 cup
Problem: A $45 shirt is on sale for 2/5 off. What's the discount amount?
Solution:
Answer: $18 off (sale price = $27)
Problem: A 10.5-foot board is cut into pieces 1.75 feet long. How many pieces?
Solution:
Answer: 6 pieces
❌ Mistake 1: Adding denominators
❌ Mistake 2: Not finding LCD
❌ Mistake 3: Forgetting to flip when dividing
❌ Mistake 4: Decimal placement errors
❌ Mistake 5: Sign errors with negatives
For Fractions:
For Decimals:
For Signs:
Rational numbers include all fractions, integers, and terminating/repeating decimals.
Operations:
Sign Rules: Same as integers!
Master rational number operations and you're ready for solving equations, working with ratios, and tackling algebra!
Add: 2/5 + 1/5
The fractions have the same denominator, so add the numerators:
2/5 + 1/5 = (2 + 1)/5 = 3/5
Answer: 3/5
Subtract: 5/6 - 1/3
Find a common denominator. The LCD of 6 and 3 is 6.
Convert 1/3 to sixths: 1/3 = 2/6
Now subtract: 5/6 - 2/6 = 3/6 = 1/2
Answer: 1/2
Multiply: (-2/3) × (3/4)
Multiply numerators and denominators:
(-2/3) × (3/4) = (-2 × 3)/(3 × 4) = -6/12
Simplify: -6/12 = -1/2
Answer: -1/2
Divide: 3/4 ÷ 2/5
To divide fractions, multiply by the reciprocal:
3/4 ÷ 2/5 = 3/4 × 5/2
Multiply: (3 × 5)/(4 × 2) = 15/8
Convert to mixed number: 15/8 = 1 7/8
Answer: 15/8 or 1 7/8
Calculate: -1/2 + 3/4 - 1/3
Find the LCD of 2, 4, and 3. LCD = 12
Convert all fractions: -1/2 = -6/12 3/4 = 9/12 -1/3 = -4/12
Add/subtract from left to right: -6/12 + 9/12 - 4/12 = (-6 + 9 - 4)/12 = -1/12
Answer: -1/12