Operations with Rational Numbers- Lesson 2.5 Guide

What Are Rational Numbers? (Quick Refresher)

Rational numbers are any numbers you can write as a fraction. That includes integers, fractions, and decimals that either terminate or repeat. If it can be expressed as a/b where b ≠ 0, it's rational.

Examples:

• 1/2 (fraction)
• 0.75 (terminating decimal)
• -3/4 (negative fraction)
• 0.333... (repeating decimal)
• 5 (all integers are rational: 5 = 5/1)

The Four Operations

Lesson 2.5 covers the basics: adding, subtracting, multiplying, and dividing rational numbers. Each operation follows specific rules. Most students stumble on the sign rules and finding common denominators. That's where I'll focus.

Adding Rational Numbers

You need a common denominator. Period. No way around it.

Same denominators? Add the numerators, keep the denominator.

Example: 1/5 + 3/5 = 4/5

Different denominators? Find the LCD (Least Common Denominator) first.

Example: 1/4 + 1/6

• LCD of 4 and 6 is 12
• Convert: 1/4 = 3/12, 1/6 = 2/12
• Add: 3/12 + 2/12 = 5/12

For negative numbers, treat the signs as you go. A positive plus a negative means you subtract and keep the sign of the larger absolute value.

Subtracting Rational Numbers

Subtraction is just addition with a twist. Keep, change, flip.

When subtracting a rational number, add its opposite:

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

• Change subtraction to addition
• Flip the second fraction's sign: -1/4 becomes +1/4
• Now add: 2/3 + 1/4
• LCD is 12: 8/12 + 3/12 = 11/12

Two negatives in a row? They cancel out. That's it.

Multiplying Rational Numbers

Multiplying fractions is straightforward: numerator × numerator, denominator × denominator.

Example: 2/3 × 4/5 = (2×4)/(3×5) = 8/15

Sign rule: Positive × positive = positive. Negative × negative = positive. Positive × negative = negative.

Shortcut for mixed numbers: Convert to improper fractions first. Then multiply. Don't try to multiply mixed numbers directly—it doesn't work.

Dividing Rational Numbers

Division flips the second fraction (the divisor), then multiplies. This is called "multiply by the reciprocal."

Example: 2/3 ÷ 1/6

• Flip the divisor: 1/6 becomes 6/1
• Multiply: 2/3 × 6/1
• Simplify before multiplying: 2/3 × 6/1 = 2/1 × 2/1 = 4

Or multiply across: (2×6)/(3×1) = 12/3 = 4

Quick Reference Table

Simplifying Fractions

Always simplify your answer. Divide numerator and denominator by their GCF (Greatest Common Factor).

8/12 → GCF is 4 → 8÷4 = 2, 12÷4 = 3 → 2/3

If you leave answers unsimplified, expect points to be deducted on tests.

Common Mistakes to Avoid

• Forgetting to find the LCD when adding or subtracting. You can't just add across.
• Screwing up the sign rules with negatives. Draw a number line if you need to.
• Not converting mixed numbers before multiplying or dividing. This is a major error source.
• Skipping simplification. Your answer isn't finished until it's reduced.
• Flipping the wrong fraction when dividing. Flip the divisor, not the dividend.

Getting Started: Practice Problems

Work through these. No calculators until you've tried by hand.

1. 1/2 + 1/4 = ?
2. 3/5 - 1/3 = ?
3. 2/7 × 3/4 = ?
4. 5/6 ÷ 2/3 = ?
5. -3/4 + 1/2 = ?

Answers:

1. 3/4
2. 4/15
3. 6/28 = 3/14
4. 5/6 × 3/2 = 15/12 = 5/4
5. -1/4

If you missed any, go back and identify where you went wrong. Probably one of the five mistakes above.

Final Notes

Rational number operations aren't complicated. The rules are consistent. The only way to get faster is practice. There's no secret method. Students who struggle usually haven't put in the reps.

Focus on sign rules and finding LCDs. Those two skills trip up most people. Master those, and the rest falls into place.