Fractions, Ratios, Proportions, and Numbers: Interactive Study Guide

About Numbers

What is a Whole Number?

A whole number is a number without fractions or decimals. Whole numbers include 0, 1, 2, 3, 4, 5, etc. (no negatives, no fractions).

What is Place Value?

Place value is the value of a digit depending on its position in a number. For example, in 345, the 3 is in the "hundreds" place (so it's worth 300), the 4 is in the "tens" place (worth 40), and the 5 is in the "ones" place (worth 5).

What is Absolute Value?

The absolute value of a number is its distance from zero, ignoring sign. For example, the absolute value of -7 is 7; the absolute value of 5 is 5. It's always positive or zero.

What is Distance (on a Number Line)?

The distance between two numbers is how far apart they are, measured as the absolute value of their difference. For example, the distance between 3 and -4 is |3 - (-4)| = |3 + 4| = 7.

Fractions & Mixed Numbers

Changing Improper Fractions to Mixed Numbers: Divide the numerator by the denominator. The quotient is the whole number part; the remainder is the new numerator.
Changing Mixed Numbers to Improper Fractions: Multiply the whole number by the denominator, add the numerator, and place over the original denominator.

Example:
Convert 11/4 to a mixed number: 11 ÷ 4 = 2 remainder 3, so 2 3/4.
Convert 3 2/5 to an improper fraction: 3 × 5 + 2 = 17, so 17/5.

Operations with Fractions

Adding Fractions with Like Denominators: Add numerators, keep denominator.
Subtracting Fractions with Like Denominators: Subtract numerators, keep denominator.
Adding or Subtracting Fractions with Unlike Denominators: Find the least common denominator (LCD), make equivalent fractions, then add or subtract numerators.
Adding Mixed Numbers with Unlike Denominators: Convert fractions to common denominators, add fractions, add whole numbers, simplify if needed.
Subtracting Mixed Numbers with Unlike Denominators: Find LCD, subtract fractions and whole numbers; you may need to borrow.
Estimating Sums and Differences: Round each mixed number or fraction to nearest whole or "easy" fraction, then add or subtract.

Example:

2/5 + 1/5 = (2+1)/5 = 3/5
3/4 - 1/4 = (3-1)/4 = 2/4 = 1/2
1/2 + 1/3 → LCD is 6: (3/6) + (2/6) = 5/6

Multiplying Fractions

Multiplying Fractions and Whole Numbers: Multiply the whole number by the numerator, keep the denominator.
Multiplying Fractions: Reciprocals: The reciprocal of a/b is b/a. Multiplying a number by its reciprocal gives 1.
Multiplying Fractions and Mixed Numbers: Reducing: Convert mixed numbers to improper fractions first. Cancel (reduce) any common factors before multiplying. Simplify the answer.

Example:

3 × 2/7 = (3×2)/7 = 6/7
Reciprocal of 3/5 is 5/3
1 1/4 × 2/3 → 1 1/4 = 5/4, so 5/4 × 2/3 = 10/12 = 5/6

Dividing Fractions

Dividing Fractions by Whole Numbers: Multiply the fraction by the reciprocal of the whole number.
Dividing Whole Numbers by Fractions: Multiply the whole number by the reciprocal of the fraction.
Dividing Fractions by Fractions: Multiply the first fraction by the reciprocal of the second.
Dividing Mixed Numbers: Change mixed numbers to improper fractions, then proceed as above.

Example:

3/4 ÷ 2 = 3/4 × 1/2 = 3/8
5 ÷ 1/4 = 5 × 4/1 = 20
2/3 ÷ 5/7 = 2/3 × 7/5 = 14/15

Ratios, Proportions & Rates

Ratios: A comparison of two quantities, expressed as a:b or a/b.
Proportions and Cross-Multiplying: A proportion is an equation showing two ratios are equal. Cross-multiply to solve for unknowns.
Ratio Tables: Tables showing how quantities increase or decrease in the same ratio.
Rates: A ratio comparing two quantities with different units (e.g., miles/hour).
Problem-Solving with Proportions: Use proportions and cross-multiplying to solve real-world problems.

Example:

Ratio: 3 apples to 5 oranges = 3:5
Proportion: 2/3 = 4/6 (cross-multiplied: 2×6 = 3×4)
Rate: 120 miles in 2 hours = 60 miles/hour
Ratio Table:

Apples 2 4 6

Oranges 3 6 9

Apples	2	4	6
Oranges	3	6	9

Examples of the Topics

Improper Fraction to Mixed: 15/4 = 3 3/4
Mixed to Improper: 2 5/6 = (2×6 + 5)/6 = 17/6
Add like denominators: 1/7 + 3/7 = 4/7
Subtract unlike denominators: 5/8 - 1/4 = 5/8 - 2/8 = 3/8
Add mixed numbers (unlike): 1 2/3 + 2 1/4 = (1+2) + (8/12+3/12) = 3 + 11/12
Estimate: 4 2/3 + 1 3/10 ≈ 5 + 1 = 6
Multiply: 2/5 × 5/6 = 10/30 = 1/3
Divide: 3/5 ÷ 2/3 = 3/5 × 3/2 = 9/10
Ratio: 12 girls : 9 boys = 4:3
Proportion: If x/12 = 3/4, then x = 12×3/4 = 9

Fractions, Ratios, Proportions, and Numbers: Study Guide

About Numbers

What is a Whole Number?

What is Place Value?

What is Absolute Value?

What is Distance (on a Number Line)?

Fractions & Mixed Numbers

Operations with Fractions

Multiplying Fractions

Dividing Fractions

Ratios, Proportions & Rates

Examples of the Topics

Pre-Test (40 Questions)

Questions & Answers

Post-Test (40 Questions)