Mathematical Reasoning
What is being measured
Arithmetic with whole numbers, fractions, decimals, and percents; ratios and proportions; unit rates; averages; elapsed time; and short applied word problems set in a facility context (counts, schedules, supply costs, staffing ratios).
Core concepts
Order of operations (parentheses, exponents, multiply/divide left to right, add/subtract left to right). Percent conversions - 15% = 0.15 = 15/100. Percent of a number: multiply by the decimal. Percent change: change ÷ original × 100. Successive discounts: apply each one in order; they do not add. Ratios: add the parts to get the total parts, then divide the total by that number. Unit rates: convert to a common unit before comparing.
Traps
- Percent increase versus percentage points - a rise from 20% to 25% is 5 percentage points but a 25% increase.
- Successive discounts - 25% off then 10% off is not 35% off.
- Mixed units - minutes versus hours, days versus weeks.
- Averages of averages - you cannot average two group averages unless the groups are the same size.
- Irrelevant numbers included to distract you, and how many more questions that require subtraction after you find both values.
Rule of thumb
Write the units next to each number and estimate the answer before computing. If your estimate and your calculation disagree by a lot, recheck the setup.
Worked example
A jacket costs $80. It is marked down 25%, then the sale price is reduced another 10%. Final price: 80 × 0.75 = $60; 60 × 0.90 = $54. A common wrong answer is 80 × 0.65 = $52, which incorrectly adds the discounts.
Ratio example
Staff and inmates split a group in a 3:5 ratio with 64 people total. Total parts = 8; one part = 8; the groups are 24 and 40.