Fractions and Decimals Mastery

1. The Basics: Types of Fractions

Conversion Tip: To turn \(2\frac{3}{5}\) into an improper fraction, calculate \((2 \times 5) + 3 = 13\). The result is \(\frac{13}{5}\).

2. Arithmetic with Fractions

Addition & Subtraction

You must have a Common Denominator.

Simple Example: \(\frac{1}{4} + \frac{2}{3}\)
Common Denominator is 12:
\(\frac{3}{12} + \frac{8}{12} = \frac{11}{12}\)
Mixed Number Subtraction: \(3\frac{1}{2} - 1\frac{4}{5}\)
1. Convert to improper: \(\frac{7}{2} - \frac{9}{5}\)
2. Common denominator 10: \(\frac{35}{10} - \frac{18}{10}\)
3. Solve: \(\frac{17}{10} = 1\frac{7}{10}\)

Multiplication & Division

Multiplication: \(\frac{2}{5} \times \frac{3}{7} = \frac{2 \times 3}{5 \times 7} = \frac{6}{35}\)
Division (Keep, Change, Flip): \(\frac{3}{4} \div \frac{2}{5}\)
\(\frac{3}{4} \times \frac{5}{2} = \frac{15}{8} = 1\frac{7}{8}\)

3. Conversions: Fractions & Decimals

Fraction to Decimal: Divide the top by the bottom. \(\frac{3}{8} = 3 \div 8 = 0.375\).
Decimal to Fraction: Use place value. \(0.45 = \frac{45}{100} = \frac{9}{20}\).

4. Recurring Decimals to Fractions

This is a high-level skill. We use algebra to cancel out the repeating part. More details and examples here.

Example 1: Single repeating digit (\(0.\dot{7}\))
Let \(x = 0.777...\)
\(10x = 7.777...\)
Subtract: \(9x = 7 \Rightarrow x = \frac{7}{9}\)
Example 2: Two repeating digits (\(0.\dot{4}\dot{1}\))
Let \(x = 0.4141...\)
\(100x = 41.4141...\)
Subtract: \(99x = 41 \Rightarrow x = \frac{41}{99}\)
Example 3: Partial repeat (\(0.2\dot{5}\))
Let \(x = 0.2555...\)
\(10x = 2.555...\)
\(100x = 25.555...\)
Subtract: \(90x = 23 \Rightarrow x = \frac{23}{90}\)
Example 4: Three digits (\(0.\dot{1}2\dot{3}\))
Let \(x = 0.123123...\)
\(1000x = 123.123...\)
Subtract: \(999x = 123 \Rightarrow x = \frac{123}{999} = \frac{41}{333}\)
Example 5: Mixed whole numbers (\(1.2\dot{3}\))
Let \(x = 1.2333...\)
\(10x = 12.333...\)
\(100x = 123.333...\)
Subtract: \(90x = 111 \Rightarrow x = \frac{111}{90} = 1\frac{21}{90} = 1\frac{7}{30}\)

5. Fractions to Recurring Decimals

Use long division until a pattern emerges.

Example 1: \(\frac{2}{3} = 2 \div 3 = 0.666... = 0.\dot{6}\)
Example 2: \(\frac{5}{11} = 5 \div 11 = 0.4545... = 0.\dot{4}\dot{5}\)
Example 3: \(\frac{1}{6} = 1 \div 6 = 0.1666... = 0.1\dot{6}\)