GED Prep: Number Sense and Operations (Fractions, Decimals, Percentages, Order of Operations)

GED – Number Sense and Operations (Fractions, Decimals, Percentages, Order of Operations)

GED Number Sense and Operations Study Guide

Topic: Fractions, Decimals, Percentages, and Order of Operations

What This Is

Number sense and operations are the foundation of the GED Math test. You’ll need to convert between fractions, decimals, and percentages, solve real-world problems (like discounts, taxes, or measurements), and apply the order of operations (PEMDAS/BODMAS) to simplify expressions. For example, a typical test question might ask: "A shirt costs $40 and is on sale for 25% off. What is the sale price?" Mastering these skills ensures you can handle budgeting, recipes, construction measurements, and more—both on the test and in daily life.

Key Terms & Rules

• Fraction: A part of a whole, written as numerator/denominator (e.g., 3/4). To simplify, divide numerator and denominator by their greatest common divisor (GCD).
• Decimal: A fraction written in base 10 (e.g., 0.75 = 75/100). To convert a fraction to a decimal, divide the numerator by the denominator.
• Percentage: A fraction out of 100 (e.g., 25% = 25/100 = 0.25). To convert a decimal to a percentage, multiply by 100.
• Order of Operations (PEMDAS/BODMAS):
  Parentheses → Exponents → Multiplication/Division (left to right) → Addition/Subtraction (left to right).
  Example: 3 + 4 × 2 = 3 + 8 = 11 (not 14, because multiplication comes first).
• Improper Fraction: A fraction where the numerator is larger than the denominator (e.g., 7/4). Convert to a mixed number by dividing (7 ÷ 4 = 1 3/4).
• Mixed Number: A whole number + a fraction (e.g., 2 1/2). Convert to an improper fraction by multiplying the whole number by the denominator and adding the numerator (2 × 2 + 1 = 5/2).
• Equivalent Fractions: Fractions that represent the same value (e.g., 1/2 = 2/4 = 50%). Multiply or divide numerator and denominator by the same number to find equivalents.
• Decimal to Fraction: Write the decimal as a fraction over 10, 100, or 1,000 (e.g., 0.6 = 6/10 = 3/5). Simplify if possible.
• Percentage Formula:
• Part = Percentage × Whole (e.g., 20% of 50 = 0.20 × 50 = 10).
• Percentage = (Part ÷ Whole) × 100 (e.g., What percent is 15 of 60? (15 ÷ 60) × 100 = 25%).
• Adding/Subtracting Fractions: Find a common denominator, then add/subtract numerators. Example: 1/4 + 1/6 = 3/12 + 2/12 = 5/12.
• Multiplying Fractions: Multiply numerators and denominators straight across. Example: 2/3 × 4/5 = 8/15.
• Dividing Fractions: Multiply by the reciprocal (flip the second fraction). Example: 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8.

Step-by-Step / Process Flow

1. Solving a Percentage Problem (e.g., Discounts, Tax, Tips)

Question: A laptop costs $800. It’s on sale for 15% off, and there’s a 7% sales tax. What’s the final price? 1. Find the discount amount: 15% of $800 = 0.15 × 800 = $120.
2. Subtract the discount: $800 – $120 = $680 (sale price).
3. Calculate tax: 7% of $680 = 0.07 × 680 = $47.60.
4. Add tax to sale price: $680 + $47.60 = $727.60.

2. Converting Between Fractions, Decimals, and Percentages

Question: Convert 3/8 to a decimal and a percentage.
1. Fraction to Decimal: Divide 3 ÷ 8 = 0.375.
2. Decimal to Percentage: 0.375 × 100 = 37.5%.

3. Order of Operations (PEMDAS)

Question: Simplify 6 + (3 × 2)² ÷ 4 – 1.
1. Parentheses: 3 × 2 = 6.
2. Exponents: 6² = 36.
3. Division: 36 ÷ 4 = 9.
4. Addition/Subtraction (left to right): 6 + 9 – 1 = 14.

4. Adding/Subtracting Fractions with Unlike Denominators

Question: 5/6 – 1/4 = ? 1. Find the least common denominator (LCD): LCD of 6 and 4 is 12.
2. Convert fractions: 5/6 = 10/12, 1/4 = 3/12.
3. Subtract numerators: 10/12 – 3/12 = 7/12.

Common Mistakes

• Mistake: Forgetting to simplify fractions.
  Correction: Always reduce fractions to lowest terms (e.g., 4/8 = 1/2). Why? The GED often includes simplified answers as distractors.

• Mistake: Misapplying order of operations (e.g., adding before multiplying).
  Correction: Use PEMDAS/BODMAS and work left to right for multiplication/division and addition/subtraction. Why? The test includes traps like 8 ÷ 2(2 + 2) to trick you into ignoring parentheses.

• Mistake: Converting percentages incorrectly (e.g., 5% = 0.5 instead of 0.05).
  Correction: Move the decimal two places left for percentages (5% = 0.05). Why? A common distractor is 5% = 5.0 or 0.5.

• Mistake: Adding denominators when adding fractions (e.g., 1/2 + 1/3 = 2/5).
  Correction: Find a common denominator first. Why? The GED includes this as a wrong answer choice.

• Mistake: Ignoring units (e.g., mixing dollars and cents).
  Correction: Convert all values to the same unit (e.g., $1.50 = 150 cents). Why? Real-world problems often test unit consistency.

Exam Insights

1. Most-Tested Concepts:
2. Converting between fractions, decimals, and percentages (especially in word problems).
3. Applying percentages to discounts, taxes, and tips.

4. Order of operations (PEMDAS) with exponents and parentheses.

5. Tricky Distinctions:

6. "Of" in word problems usually means multiplication (e.g., "20% of 50" = 0.20 × 50).
7. Reciprocals in division (e.g., 1/2 ÷ 3/4 = 1/2 × 4/3).

8. Mixed numbers vs. improper fractions (e.g., 1 3/4 vs. 7/4).

9. Common Distractors:

10. Answers that ignore order of operations (e.g., 3 + 4 × 2 = 14 instead of 11).
11. Percentage errors (e.g., 25% of 80 = 200 instead of 20).

12. Fraction simplification traps (e.g., 6/9 = 2/3 is correct, but 6/9 = 1/3 is a distractor).

13. Calculator Tips:

14. Use the % key for percentages (e.g., 20% of 50 = 50 × 20% = 10).
15. Convert fractions to decimals for easier calculations (e.g., 3/8 = 0.375).
16. For order of operations, enter the expression exactly as written (e.g., 6 + (3 × 2)² ÷ 4 – 1).

Quick Check Questions

1. What is 3/5 as a percentage?
   A) 30% B) 50% C) 60% D) 75%
   Answer: C) 60%. Convert 3/5 to a decimal (0.6) and multiply by 100.

2. Simplify: 12 ÷ (3 + 1) × 2 – 4
   A) 0 B) 2 C) 4 D) 8
   Answer: B) 2. Parentheses first (3 + 1 = 4), then division (12 ÷ 4 = 3), then multiplication (3 × 2 = 6), then subtraction (6 – 4 = 2).

3. A $60 shirt is on sale for 30% off. What is the sale price?
   A) $18 B) $30 C) $42 D) $48
   Answer: C) $42. 30% of $60 = $18 discount; $60 – $18 = $42.

Last-Minute Cram Sheet

1. PEMDAS/BODMAS: Parentheses → Exponents → Multiply/Divide (left to right) → Add/Subtract (left to right).
2. Percentage to Decimal: Move decimal 2 places left (e.g., 45% = 0.45).
3. Decimal to Percentage: Move decimal 2 places right (e.g., 0.7 = 70%).
4. Fraction to Decimal: Divide numerator by denominator (e.g., 3/4 = 0.75).
5. Adding Fractions: Find a common denominator first.
6. Dividing Fractions: Multiply by the reciprocal (flip the second fraction).
7. ⚠️ Order of Operations Trap: 8 ÷ 2(2 + 2) = 16 (not 1) because parentheses come first.
8. ⚠️ Percentage Trap: "20% off $50" = $10 discount, not $40.
9. Mixed Numbers: Convert to improper fractions for calculations (e.g., 2 1/2 = 5/2).
10. Real-World Keywords:
11. "Of" = multiply (e.g., 10% of 200 = 0.10 × 200).
12. "Per" = divide (e.g., miles per hour = miles ÷ hours).