MB Essential Math 40S cheat sheet

Simple interest: I = Prt, where r is the annual rate as a decimal and t is time in years; total amount A = P + I = P(1 + rt).

Compound interest: A = P(1 + i)^n, where i = annual rate ÷ number of compounding periods per year and n = periods per year × years.

Rule of 72: doubling time in years ≈ 72 ÷ annual percentage rate; accurate only for rates roughly between 4% and 15%.

Present value from a future amount: P = A ÷ (1 + i)^n; use it to compare lump-sum offers against payment streams.

Loan payment on an amortized loan: PMT = P·i ÷ (1 − (1 + i)^(−n)), with i the periodic rate and n the total number of payments.

A vehicle or home purchase budget must include taxes, fees, insurance, and maintenance, not just the sticker price; affordability is judged on total monthly outflow.

Currency exchange: amount in target currency = amount in source currency × exchange rate; a buy rate and a sell rate differ, so round-trip conversions lose money.

Percent change = (new − old) ÷ old × 100%; a percent increase followed by the same percent decrease never returns to the original value.

Unit price comparison: divide total cost by quantity and convert to a common unit before comparing two package sizes.

Mean = sum of values ÷ count; median = middle value of an ordered list (average the two middle values if the count is even); mode = most frequent value.

Weighted mean = Σ(value × weight) ÷ Σ(weights); use it when categories such as assignments carry different weights.

Range = maximum − minimum; it is sensitive to outliers, so report it alongside the median rather than the mean when data are skewed.

Standard deviation measures typical distance from the mean; roughly 68% of data lie within one standard deviation of the mean in a roughly symmetric distribution.

A z-score = (value − mean) ÷ standard deviation; a positive z-score is above the mean, and |z| > 2 flags an unusually extreme value.

A box-and-whisker plot shows minimum, first quartile, median, third quartile, and maximum; the interquartile range IQR = Q3 − Q1.

An outlier by the 1.5 rule lies below Q1 − 1.5·IQR or above Q3 + 1.5·IQR.

Correlation describes the strength and direction of a linear association; correlation does not establish causation.

Sampling bias arises when the sample is not representative; a voluntary-response or convenience sample usually over-represents strong opinions.

Probability of an event = number of favourable outcomes ÷ total number of equally likely outcomes; 0 ≤ P ≤ 1.

Complement rule: P(not A) = 1 − P(A); use it whenever 'at least one' appears in a question.

Addition rule: P(A or B) = P(A) + P(B) − P(A and B); subtract the overlap so it is not counted twice.

Independent events: P(A and B) = P(A) × P(B); for dependent events multiply by the conditional probability of the second given the first.

Expected value = Σ(outcome value × its probability); a positive expected value means the bet or venture is favourable on average over many trials.

Odds in favour of A = P(A) : P(not A); convert odds a:b to probability a ÷ (a + b).

Experimental probability comes from observed frequencies and approaches theoretical probability as the number of trials grows.

Mutually exclusive events cannot occur together, so P(A and B) = 0 and the addition rule reduces to P(A) + P(B).

Linear relation: y = mx + b, with slope m = (y₂ − y₁) ÷ (x₂ − x₁) and b the y-intercept; slope is the constant rate of change.

To build a model from a table, check whether first differences are constant (linear) or whether successive ratios are constant (exponential).

Exponential growth or decay: y = a·b^x, where a is the initial value and b > 1 grows while 0 < b < 1 decays.

Graph-reading rule: the slope of a graph is a rate, so the units of the answer are y-units per x-unit.

Break-even point is where revenue equals cost; profit = revenue − cost, and profit is positive only beyond the break-even quantity.

Interpolation estimates a value between known data points; extrapolation estimates beyond them and is less reliable.

A line of best fit minimizes the total squared vertical distances from the data points and is used to predict one variable from another.

When solving a system of two linear relations, the solution is the intersection point; parallel lines with different intercepts have no solution.

Perimeter and area: rectangle A = lw, triangle A = ½bh, circle A = πr² and C = 2πr.

Volume: rectangular prism V = lwh, cylinder V = πr²h, sphere V = (4/3)πr³, cone V = (1/3)πr²h.

Surface area of a cylinder = 2πr² + 2πrh; of a rectangular prism = 2(lw + lh + wh).

Pythagorean theorem a² + b² = c² applies only to right triangles; c is always the hypotenuse.

Scale factor k on a 3-D object multiplies lengths by k, surface area by k², and volume by k³.

1 m = 100 cm, 1 m² = 10 000 cm², 1 m³ = 1 000 000 cm³; area and volume conversions use the square and cube of the length factor.

1 L = 1000 mL = 1000 cm³, so a container of volume 5000 cm³ holds 5 L.

Density = mass ÷ volume; rearrange to mass = density × volume or volume = mass ÷ density.

A composite figure's area or volume is found by splitting it into standard shapes, computing each part, then adding or subtracting.

Trigonometric ratios in a right triangle: sin θ = opposite ÷ hypotenuse, cos θ = adjacent ÷ hypotenuse, tan θ = opposite ÷ adjacent.

To find a missing side, choose the ratio that links the known side and the unknown side, then solve; to find a missing angle, use the inverse function.

Angles of elevation and depression are measured from the horizontal; the angle of elevation from A to B equals the angle of depression from B to A.

The sine law a ÷ sin A = b ÷ sin B = c ÷ sin C applies to any triangle; use it when you know a side and its opposite angle plus one more part.

The cosine law c² = a² + b² − 2ab·cos C applies to any triangle; use it with two sides and the included angle or with all three sides.

Angle sum of any triangle is 180°; in a right triangle the two acute angles are complementary.

Bearing is measured clockwise from north and written with three digits, so due east is 090° and due south is 180°.

In a non-right triangle, the largest angle faces the longest side; check whether the given information allows two possible triangles (the ambiguous case).