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Adding and Subtracting Fractions

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Adding and Subtracting Fractions

When adding and subtracting fractions, the main idea is to have the same denominator. If the denominators are different, find the least common denominator (LCD) before adding or subtracting.

  • Adding fractions:

\frac{a}{d} + \frac{b}{d} = \frac{a + b}{d}

If the denominators are not the same, find the LCD and rewrite each fraction:

\frac{a}{x} + \frac{b}{y} = \frac{a \cdot y}{x \cdot y} + \frac{b \cdot x}{y \cdot x}

Subtracting fractions:
Similar to addition, but subtract the numerators.

Diagram:

Example: Adding \frac{3}{4} + \frac{2}{5}

\begin{array}{c} \begin{aligned} &\frac{3}{4} = \frac{15}{20} \ &\frac{2}{5} = \frac{8}{20} \ &\frac{15}{20} + \frac{8}{20} = \frac{23}{20} = 1 \frac{3}{20} \end{aligned} \end{array}

Hard Questions:

Q1: Add \frac{5}{6} + \frac{3}{4} .

Step-by-Step Solution:

  1. Find the LCD of 6 and 4, which is 12.
  2. Rewrite the fractions:

\frac{5}{6} = \frac{10}{12}, \quad \frac{3}{4} = \frac{9}{12}

Add the fractions:

\frac{10}{12} + \frac{9}{12} = \frac{19}{12} = 1 \frac{7}{12}

Answer:

  • The sum is 1 \frac{7}{12} .

Q2: Subtract \frac{7}{9} - \frac{5}{12} .

Step-by-Step Solution:

  1. Find the LCD of 9 and 12, which is 36.
  2. Rewrite the fractions:

\frac{7}{9} = \frac{28}{36}, \quad \frac{5}{12} = \frac{15}{36}

Subtract the fractions:

\frac{28}{36} - \frac{15}{36} = \frac{13}{36}

Answer:

  • The difference is \frac{13}{36} .

Q3: Add \frac{3}{5} + \frac{4}{7} .

Step-by-Step Solution:

  1. Find the LCD of 5 and 7, which is 35.
  2. Rewrite the fractions:

\frac{3}{5} = \frac{21}{35}, \quad \frac{4}{7} = \frac{20}{35}

Add the fractions:

\frac{21}{35} + \frac{20}{35} = \frac{41}{35} = 1 \frac{6}{35}

Answer:

  • The sum is 1 \frac{6}{35} .

Adding and Subtracting Fractions with Like Denominators

Concept Explanation:

When adding or subtracting fractions with like denominators, the process is simpler because the denominators are already the same. You simply add or subtract the numerators and keep the same denominator.

For example:

\frac{a}{d} + \frac{b}{d} = \frac{a + b}{d}

\frac{a}{d} - \frac{b}{d} = \frac{a - b}{d}

Hard Questions:

Q1: Add \frac{7}{11} + \frac{5}{11} .

Step-by-Step Solution:

  1. The denominators are the same, so add the numerators:

\frac{7 + 5}{11} = \frac{12}{11} = 1 \frac{1}{11}

Answer:

  • The sum is 1 \frac{1}{11} .

Q2: Subtract \frac{9}{13} - \frac{4}{13} .

Step-by-Step Solution:

  1. The denominators are the same, so subtract the numerators:

\frac{9 - 4}{13} = \frac{5}{13}

Answer:

  • The difference is \frac{5}{13} .

Q3: Add \frac{8}{10} + \frac{2}{10} .

Step-by-Step Solution:

  1. The denominators are the same, so add the numerators:

\frac{8 + 2}{10} = \frac{10}{10} = 1

Answer:

  • The sum is 1.

Adding and Subtracting Fractions with Negatives

Concept Explanation:

When adding or subtracting fractions involving negative numbers, follow the same rules for fractions, but pay attention to the signs.

For example:

\frac{-a}{b} + \frac{c}{b} = \frac{-a + c}{b}

\frac{-a}{b} - \frac{c}{b} = \frac{-a - c}{b}

Hard Questions:

Q1: Add \frac{-3}{7} + \frac{5}{7} .

Step-by-Step Solution:

  1. The denominators are the same, so add the numerators:

\frac{-3 + 5}{7} = \frac{2}{7}

Answer:

  • The sum is \frac{2}{7} .

Q2: Subtract \frac{-4}{9} - \frac{2}{9} .

Step-by-Step Solution:

  1. The denominators are the same, so subtract the numerators:

\frac{-4 - 2}{9} = \frac{-6}{9} = \frac{-2}{3}

Answer:

  • The difference is \frac{-2}{3} .

Q3: Add \frac{-7}{12} + \frac{-5}{12} .

Step

-by-Step Solution:

  1. The denominators are the same, so add the numerators:

\frac{-7 + (-5)}{12} = \frac{-12}{12} = -1

Answer:

  • The sum is -1 .

Adding and Subtracting Fractions with Unlike Denominators

Concept Explanation:

When adding or subtracting fractions with unlike denominators, you must first find the least common denominator (LCD), convert each fraction, and then proceed with addition or subtraction.

For example:

\frac{a}{x} + \frac{b}{y} = \frac{a \cdot y}{x \cdot y} + \frac{b \cdot x}{y \cdot x}

Hard Questions:

Q1: Add \frac{5}{8} + \frac{2}{3} .

Step-by-Step Solution:

  1. Find the LCD of 8 and 3, which is 24.
  2. Rewrite the fractions:

\frac{5}{8} = \frac{15}{24}, \quad \frac{2}{3} = \frac{16}{24}

Add the fractions:

\frac{15}{24} + \frac{16}{24} = \frac{31}{24} = 1 \frac{7}{24}

Answer:

  • The sum is 1 \frac{7}{24} .

Q2: Subtract \frac{7}{10} - \frac{3}{4} .

Step-by-Step Solution:

  1. Find the LCD of 10 and 4, which is 20.
  2. Rewrite the fractions:

\frac{7}{10} = \frac{14}{20}, \quad \frac{3}{4} = \frac{15}{20}

Subtract the fractions:

\frac{14}{20} - \frac{15}{20} = \frac{-1}{20}

Answer:

  • The difference is \frac{-1}{20} .

Q3: Add \frac{3}{5} + \frac{7}{8} .

Step-by-Step Solution:

  1. Find the LCD of 5 and 8, which is 40.
  2. Rewrite the fractions:

\frac{3}{5} = \frac{24}{40}, \quad \frac{7}{8} = \frac{35}{40}

Add the fractions:

\frac{24}{40} + \frac{35}{40} = \frac{59}{40} = 1 \frac{19}{40}

Answer:

  • The sum is 1 \frac{19}{40} .

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