Understanding percentages, growth and financial applications
Ratio, proportion and rates • Ratio and proportion
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Key concepts
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Definition of percentage
Percentage is the number of parts per hundred and provides a scale for comparing quantities of different sizes. The notation p% means p parts out of 100, which equals the fraction p/100 and the decimal p ÷ 100. Limitations include that percentages describe relative amounts but require a reference quantity. The reference quantity must be stated when expressing one number as a percentage of another.
Conversion between percent, fraction and decimal
Conversion from percent to decimal uses division by 100: p% → p ÷ 100. Conversion from percent to fraction uses p% → p/100, followed by simplification when possible. Conversion back to percent from a decimal or fraction uses multiplication by 100. Accurate conversion ensures correct use of multipliers in subsequent calculations. Decimal and fraction forms support algebraic manipulation and comparison. Decimal form is convenient for multipliers and calculators; fraction form can simplify exact reasoning and cancellation.
Multiplicative interpretation and percentage multipliers
Percentage change acts multiplicatively: an increase of p% multiplies a quantity by (1 + p/100). A decrease of p% multiplies a quantity by (1 - p/100). Percentages greater than 100% use the same multiplier principle and produce values larger than the original. Repeated percentage changes use successive multiplication by the respective multipliers. Order of successive percentage changes matters when different rates apply sequentially.
Expressing and comparing percentages
One quantity as a percentage of another uses the ratio (part ÷ whole) × 100%. Comparison of two quantities using percentages converts both values to the same base and compares relative size, or computes percentage difference: (difference ÷ reference) × 100%. Careful choice of reference avoids ambiguity. Percentage point differences describe absolute differences between percentage values; percent differences describe relative change.
Percentage change and reverse percentage
Percentage increase or decrease uses the multiplier method to find a new value. Reverse percentage problems solve for the original value by dividing the final value by the multiplier: original = final ÷ (1 ± p/100). Reverse problems require identification of whether the stated percentage is applied to the original or the final amount. Rounding can affect reverse percentage results. Exact algebraic manipulation gives precise original values when percentages are rational numbers.
Simple interest
Simple interest calculates linear growth on the initial principal only. The formula uses I = P × r × t, where I is interest, P is principal, r is annual rate as a decimal, and t is time in years. The total amount after t years is A = P(1 + rt). Simple interest does not compound; interest earned in each period does not earn further interest. Simple interest suits short-term loans and some straightforward financial calculations.
Compound interest and exponential growth
Compound interest calculates growth where interest is added to the principal and then earns interest in later periods. The standard formula for annual compounding is A = P(1 + r)^n, where r is the annual rate as a decimal and n is the number of years. For m compounding periods per year, the formula is A = P(1 + r/m)^{mn}. Compound growth is exponential and can produce values that grow faster than simple interest for the same nominal rate. Compound decay uses the same form with a negative effective rate.
Iterative processes and growth/decay models
Iterative growth or decay uses a recurrence relation of the form A_{n+1} = k × A_n, where k is the growth/decay multiplier. The nth term equals A_0 × k^n. Iterative models include repeated investments, population models, and depreciation processes. Interpretation requires identification of k and initial value A_0. Algebraic solution gives exact values; calculators and spreadsheets support many-step iteration and compound schedules.
Key notes
Important points to keep in mind