Fractions, decimals and percentages Revision Notes

Fractions represent parts of a whole and come in three main types: proper fractions, improper fractions, and mixed numbers. A proper fraction has a numerator smaller than its denominator, for example, \( \frac{3}{5} \). An improper fraction has a numerator equal to or larger than its denominator, such as \( \frac{7}{4} \). Mixed numbers combine a whole number with a proper fraction, for example, \( 1 \frac{3}{4} \).

Simplifying a fraction means expressing it in its simplest form where the numerator and denominator share no common factors other than 1. To simplify, find the greatest common divisor (GCD) of numerator and denominator and divide both by it.

Example: Simplify \( \frac{18}{24} \). The GCD of 18 and 24 is 6. Dividing numerator and denominator by 6 gives \( \frac{18 \div 6}{24 \div 6} = \frac{3}{4} \). Thus, \( \frac{18}{24} \) simplifies to \( \frac{3}{4} \).

• Proper fraction: numerator < denominator (e.g., \( \frac{3}{5} \))
• Improper fraction: numerator ≥ denominator (e.g., \( \frac{7}{4} \))
• Mixed number: whole number + proper fraction (e.g., \( 1 \frac{3}{4} \))
• Simplify by dividing numerator and denominator by their GCD

Improper fractions can be converted to mixed numbers by dividing the numerator by the denominator. The quotient is the whole number part, and the remainder over the denominator forms the fractional part.

Example: Convert \( \frac{11}{4} \) to a mixed number.

Divide 11 by 4: 11 ÷ 4 = 2 remainder 3. So, \( \frac{11}{4} = 2 \frac{3}{4} \). Conversely, to convert a mixed number to an improper fraction, multiply the whole number by the denominator and add the numerator, placing the result over the original denominator. For \( 2 \frac{3}{4} \), calculate \( (2 \times 4) + 3 = 11 \), so the improper fraction is \( \frac{11}{4} \).

• Divide numerator by denominator for improper to mixed number
• Multiply whole number by denominator and add numerator for mixed to improper

To convert a fraction to a decimal, divide the numerator by the denominator using long division or a calculator.

Example: Convert \( \frac{3}{8} \) to a decimal. 3 ÷ 8 = 0.375.

To convert a decimal to a percentage, multiply by 100 and add the % sign. For example, 0.375 × 100 = 37.5%. Conversely, to convert a percentage to a decimal, divide by 100. For example, 45% = 0.45. To convert a decimal to a fraction, write the decimal over the appropriate power of 10 and simplify if possible. For example, 0.75 = \( \frac{75}{100} = \frac{3}{4} \).

• Fraction to decimal: numerator ÷ denominator
• Decimal to percentage: decimal × 100%
• Percentage to decimal: percentage ÷ 100
• Decimal to fraction: write over power of 10 and simplify

Equivalent fractions represent the same value even if numerators and denominators differ. For example, \( \frac{1}{2} = \frac{2}{4} = \frac{50}{100} \). Similarly, decimals and percentages can represent the same quantity: 0.5 = 50%. Recognising equivalence is essential for comparing and converting between forms.

Example: Show that 0.25, \( \frac{1}{4} \), and 25% are equivalent.

Convert \( \frac{1}{4} \) to decimal: 1 ÷ 4 = 0.25. Convert 0.25 to percentage: 0.25 × 100 = 25%. Thus, all three represent the same value.

• Equivalent fractions have equal values despite different numerators and denominators
• Decimals and percentages can represent the same value
• Check equivalence by converting to a common form

Recurring decimals have digits that repeat infinitely, for example, 0.17 recurring (written as 0.17̅). To convert a recurring decimal to a fraction, use algebraic methods.

Example: Convert 0.17̅ to a fraction. Let \( x = 0.171717... \). Multiply both sides by 100 (since two digits repeat): \( 100x = 17.171717... \). Subtract the original equation: \( 100x - x = 17.171717... - 0.171717... \) gives \( 99x = 17 \). Therefore, \( x = \frac{17}{99} \). This fraction is already in simplest form.

To convert a fraction to a recurring decimal, divide numerator by denominator. If the division does not terminate, the decimal repeats. For example, \( \frac{1}{3} = 0.333... \) with 3 recurring.

• Use algebraic subtraction to convert recurring decimals to fractions
• Multiply by power of 10 matching repeat length
• Subtract original decimal to eliminate repeating part
• Divide numerator by denominator to find recurring decimal from fraction