Comprehensive Guide to Fractions, Decimals, and Percentages for Higher and Foundation Tiers

Core Conversions: Fractions, Decimals, and Percentages

• Conceptual Overview: Fractions, decimals, and percentages (FDP) are three distinct mathematical notations used to represent parts of a whole or proportions. For GCSE and university-level mathematics, fluidly moving between these forms is an essential calculation skill.

• Definitions and Etymology:     * Percentage: Derived from the Latin per centum, meaning "by the hundred." A percentage represents a fraction with a denominator of $100$.     * Decimal: A representation of a number based on powers of $10$, where place values to the right of the decimal point represent tenths ($0.1$), hundredths ($0.01$), and so on.     * Fraction: A numerical representation of a quotient where a numerator is divided by a denominator.

• Conversion Procedures:     * Fraction to Decimal: Divide the numerator ($\text{top number}$) by the denominator ($\text{bottom number}$). For example, to convert $\frac{5}{8}$, calculate 5  8 = 0.625.     * Decimal to Percentage: Multiply the decimal value by $100$. This is equivalent to shifting the decimal point two places to the right and appending the symbol $\%$. Example: $0.625 \times 100 = 62.5\%$.     * Percentage to Decimal: Divide the percentage value by $100$. This involves shifting the decimal point two places to the left. Example: 45\% = 45  100 = 0.45.     * Decimal to Fraction: Identify the place value of the final digit. If there are two decimal places, the denominator is $100$; if there are three, it is $1000$. For instance, $0.35$ becomes $\frac{35}{100}$, which simplifies to $\frac{7}{20}$ by dividing both parts by the highest common factor ($5$).     * Fraction to Percentage: First convert the fraction to a decimal, then multiply by $100$. Alternatively, if possible, find an equivalent fraction with a denominator of $100$. Example: $\frac{4}{25} = \frac{16}{100} = 16\%$.

Finding Percentages of Amounts

• The Non-Calculator Method (Unitary Strategy):     * Calculating $10\%$: Divide the total amount by $10$.     * Calculating $1\%$: Divide the total amount by $100$.     * Calculating $5\%$: Calculate $10\%$ and then divide that result by $2$.     * Calculating Compound Non-Standard Percentages: To find $35\%$, calculate $10\% \times 3$ and add $5\%$.

• The Multiplier Method (Calculator Strategy):     * Convert the required percentage into a decimal (the multiplier).     * Multiply the original amount by this decimal.     * Equation: $\text{Amount} \times \text{Multiplier} = \text{Result}$.     * Example: To find $23\%$ of \text{
  460}, use the calculation $460 \times 0.23 = 138$.

Percentage Increase and Decrease

• Standard Method (Two-Step):     1. Find the percentage of the amount.     2. For an increase, add the result to the original value. For a decrease, subtract the result from the original value.

• Multiplier Method (One-Step):     * Increase: Add the percentage to $100\%$ and convert to a decimal. To increase by $15\%$, the multiplier is $115\%$ or $1.15$.     * Decrease: Subtract the percentage from $100\%$ and convert to a decimal. To decrease by $15\%$, the multiplier is $85\%$ or $0.85$.

• Formulaic Representation:     * $\text{New Value} = \text{Original Value} \times (1 \pm i)$, where $i$ is the percentage change expressed as a decimal.

Reverse Percentages

• Definition: Calculating the original value before a percentage increase or decrease occurred, given the final value and the percentage change.

• Operational Procedure:     1. Determine the multiplier associated with the change. If a value increased by $20\%$, the multiplier is $1.2$. If it decreased by $20\%$, the multiplier is $0.8$.     2. Divide the final amount by the multiplier to return to the original amount ($100\%$).

• Key Warning: A common error is to calculate the percentage of the new value and add or subtract it. This is incorrect because the percentage change was applied to the original (unknown) value.

• Example Case Study:     * A coat is priced at \text{
  68} in a sale after a $15\%$ reduction. What was the original price?     * The sale price represents $85\%$ of the original ($100\% - 15\% = 85\%$).     * Multiplier = $0.85$.     * \text{Original Price} = 68  0.85 = 80.     * The original price was \text{
  80}.

Compound Percentages

• Core Logic: Compound percentages involve applying a percentage change repeatedly over multiple time periods. Unlike simple interest (where the percentage applies only to the original principal), compound interest applies the percentage to the new total at each interval.

• The Compound Interest Formula:     * $A = P \times (1 + r)^n$     * $A$: Final amount (Accumulated value).     * $P$: Principal (Initial amount).     * $r$: Interest rate per period (expressed as a decimal).     * $n$: Number of periods (e.g., years).

• Depreciation: When the value of an asset decreases over time by a percentage (e.g., car value). The formula uses a subtraction sign:     * $\text{Value} = \text{Initial Cost} \times (1 - r)^n$

• Comparison Example:     * \text{
  1000} invested at $5\%$ compound interest for $3$ years:     * $\text{Year 1}: 1000 \times 1.05 = 1050$     * $\text{Year 2}: 1050 \times 1.05 = 1102.50$     * $\text{Year 3}: 1102.50 \times 1.05 = 1157.63$     * Using the formula: $1000 \times (1.05)^3 = 1157.625$

Questions & Discussion

• Tier Differentiation:     * Foundation Tier: Focuses on basic conversions, non-calculator methods for finding percentages, and simple percentage increases/decreases.     * Higher Tier: Requires mastery of reverse percentages, compound interest/growth formulas, and more complex FDP algebraic problem-solving.