Fractions Addition Subtraction

Overview

Engineering reference data for Fractions Addition Subtraction in mathematics.

Key Formulas

Quadratic Formula

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Roots of ax² + bx + c = 0.

Pythagorean Theorem

$c^2 = a^2 + b^2$

Right triangle relationship.

Circle Area

$A = \pi r^2$

Area of a circle.

Logarithm

$\log_b(x) = \frac{\ln(x)}{\ln(b)}$

Change of base formula.

Variables

Fraction Operations Guide

Adding Fractions

To add fractions, first find a common denominator:

$\frac{a}{b} + \frac{c}{d} = \frac{a \cdot d + c \cdot b}{b \cdot d}$

For mixed numbers, convert to improper fractions first: $a \frac{b}{c} = \frac{a \cdot c + b}{c}$

Example from content: $2\frac{3}{4} + 4\frac{3}{8}$

1. Convert to common denominator (LCD=8): $2\frac{6}{8} + 4\frac{3}{8}$
2. Add whole numbers: $2 + 4 = 6$
3. Add fractions: $\frac{6}{8} + \frac{3}{8} = \frac{9}{8} = 1\frac{1}{8}$
4. Combine: $6 + 1\frac{1}{8} = 7\frac{1}{8}$

Subtracting Fractions

$\frac{a}{b} - \frac{c}{d} = \frac{a \cdot d - c \cdot b}{b \cdot d}$

When subtracting mixed numbers where the fractional part of the minuend is smaller:

1. Borrow 1 from the whole number: $1 = \frac{\text{denominator}}{\text{denominator}}$
2. Add to the fractional part
3. Subtract

Example from content: $6\frac{3}{4} - 4\frac{3}{8}$

1. Convert $6\frac{3}{4}$ to $6\frac{6}{8}$
2. Subtract: $6\frac{6}{8} - 4\frac{3}{8} = 2\frac{3}{8}$

Least Common Denominator (LCD)

The LCD is the smallest common multiple of denominators. For denominators 4 and 8:

• Multiples of 4: 4, 8, 12, 16, 20...
• Multiples of 8: 8, 16, 24...
• LCD = 8

Key Definitions

Numerator: Top number in a fraction (parts we have) Denominator: Bottom number (total equal parts) Proper Fraction: Numerator < Denominator ($\frac{3}{8}$) Improper Fraction: Numerator ≥ Denominator ($\frac{13}{8}$) Mixed Number: Whole number + proper fraction ($2\frac{3}{4}$)

Conversion Examples

From the extracted content:

• $9\frac{13}{8}$ → $\frac{9 \cdot 8 + 13}{8} = \frac{85}{8} = 10\frac{5}{8}$
• $31\frac{21}{16}$ → $\frac{31 \cdot 16 + 21}{16} = \frac{517}{16} = 32\frac{5}{16}$

References

• Original Source