Chapter 1: Number Concepts
What is a Whole Number?
A whole number is any of the numbers 0, 1, 2, 3, and so on. Whole numbers do not have fractions or decimals and are not negative.
What is Place Value?
Place value is the value of each digit in a number, based on its position. For example, in 3,245, the 3 is in the thousands place, so its value is 3,000.
What is Absolute Value?
The absolute value of a number is its distance from 0 on the number line, without considering direction. For example, the absolute value of -5 is 5, and the absolute value of 5 is also 5.
What is Distance?
In math, distance usually refers to how far apart two numbers are on a number line, which is the absolute value of their difference.
Example: The distance between 2 and -3 is |2 - (-3)| = |2 + 3| = 5.
Chapter 2: Fractions Operations
- Changing Improper Fractions to Mixed Numbers: Divide the numerator by the denominator. The quotient is the whole number, and the remainder over the denominator is the fraction part.
- Changing Mixed Numbers to Improper Fractions: Multiply the whole number by the denominator, add the numerator, and write the result over the denominator.
- Adding Fractions with Like Denominators: Add the numerators. Keep the denominator the same.
- Subtracting Fractions with Like Denominators: Subtract the numerators. Keep the denominator the same.
- Adding/Subtracting Fractions with Unlike Denominators: Find a common denominator, convert fractions, then add or subtract numerators.
- Adding/Subtracting Mixed Numbers with Unlike Denominators: Convert to improper fractions, find a common denominator, add/subtract, then simplify.
- Estimating Sums & Differences: Round fractions to the nearest half or whole number to estimate answers.
Multiplication & Division
- Multiplying Fractions & Whole Numbers: Multiply the numerator by the whole number, keep the denominator.
- Reciprocals: Two numbers are reciprocals if their product is 1 (e.g., 2/3 and 3/2).
- Multiplying Fractions and Mixed Numbers: Convert mixed numbers to improper fractions, multiply, then simplify.
- Dividing Fractions by Whole Numbers: Multiply by the reciprocal of the whole number (make it a fraction first).
- Dividing Whole Numbers by Fractions: Multiply the whole number by the reciprocal of the fraction.
- Dividing Fractions by Fractions: Multiply by the reciprocal of the divisor fraction.
- Dividing Mixed Numbers: Convert to improper fractions, then divide as fractions.
Chapter 3: Ratios & Proportions
- Ratios: Compare two quantities. Written as a:b, a/b, or "a to b".
- Proportions & Cross-Multiplying: An equation stating two ratios are equal; cross-multiply to solve.
- Ratio Tables: Tables showing pairs of numbers with the same ratio.
- Rates: A ratio comparing two different units (e.g., miles per hour).
- Problem-Solving with Proportions: Use proportions to solve for unknowns in real-world problems.
Chapter 4: Worked Examples
1. Improper Fraction to Mixed Number:
7/3 = 2 1⁄3 (3 goes into 7 two times with 1 left over)
7/3 = 2 1⁄3 (3 goes into 7 two times with 1 left over)
2. Mixed Number to Improper Fraction:
2 1⁄3 = (2×3+1)/3 = 7/3
2 1⁄3 = (2×3+1)/3 = 7/3
3. Add Fractions (Like Denominators):
2/7 + 3/7 = 5/7
2/7 + 3/7 = 5/7
4. Add Fractions (Unlike Denominators):
1/4 + 1/6 → common denominator is 12:
3/12 + 2/12 = 5/12
1/4 + 1/6 → common denominator is 12:
3/12 + 2/12 = 5/12
5. Multiply Fractions:
2/5 × 3/4 = 6/20 = 3/10
2/5 × 3/4 = 6/20 = 3/10
6. Divide Fractions:
2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6
2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6
7. Ratios:
6 apples to 9 oranges = 6:9 = 2:3
6 apples to 9 oranges = 6:9 = 2:3
8. Proportion Example:
If 2 pencils cost $1, how much do 10 pencils cost?
2/1 = 10/x ⇒ 2x = 10 ⇒ x = 5; Answer: $5
If 2 pencils cost $1, how much do 10 pencils cost?
2/1 = 10/x ⇒ 2x = 10 ⇒ x = 5; Answer: $5
Chapter 5: Pre-Test (Multiple Choice, 40 Questions)
Chapter 6: Questions & Answers
Q: How do you find a common denominator?
A: Find the least common multiple (LCM) of the denominators and rewrite each fraction with that denominator.
A: Find the least common multiple (LCM) of the denominators and rewrite each fraction with that denominator.
Q: What is a reciprocal?
A: The reciprocal of a fraction is flipped upside down. For example, the reciprocal of 3/4 is 4/3.
A: The reciprocal of a fraction is flipped upside down. For example, the reciprocal of 3/4 is 4/3.
Q: How do you estimate 5/8 + 1/3?
A: 5/8 ≈ 1/2, 1/3 ≈ 1/3, so estimated sum ≈ 1/2 + 1/3 ≈ 5/6.
A: 5/8 ≈ 1/2, 1/3 ≈ 1/3, so estimated sum ≈ 1/2 + 1/3 ≈ 5/6.
Q: How do you solve a proportion like 3/4 = x/12?
A: Cross-multiply: 3×12 = 4×x ⇒ 36 = 4x ⇒ x = 9.
A: Cross-multiply: 3×12 = 4×x ⇒ 36 = 4x ⇒ x = 9.