Percentages, Ratios, and Proportions
Percentage questions hinge on one relationship: part = whole × rate. Be able to find a percentage of a quantity, express one quantity as a percentage of another, and reverse the process to find the original amount. Memorise key conversions (1/8 = 12.5%, 1/5 = 20%, 3/4 = 75%) and use multipliers for increases and decreases: a 20% increase is ×1.20, a 15% decrease is ×0.85.
Ratios describe relative shares. To divide a quantity in a given ratio, add the ratio parts to get the total number of shares, divide the quantity by that total, then multiply by each part. For proportions, set up equivalent fractions and cross-multiply.
Traps: percentage increase then decrease of the same rate does not return to the original value (a 20% rise then 20% fall leaves 96%, not 100%); percentages of different bases cannot simply be added; and ratio questions sometimes give a difference between shares rather than the total. Always identify the base before computing.
Worked example: share $240 in the ratio 3:5. Total shares = 8, so one share = $30; shares are $90 and $150. Rule of thumb: write the base down explicitly before touching the calculator.