DSST Business Mathematics cheat sheet

Rounding rule: identify the place value to keep, then inspect the single digit immediately to its right; 5 or more rounds up, 4 or less leaves the kept digit unchanged.

Adding or subtracting fractions requires a common denominator; multiply numerator and denominator of each fraction by the same factor, then combine numerators only.

Dividing by a fraction equals multiplying by its reciprocal: (a/b) ÷ (c/d) = (a/b) × (d/c).

Convert a decimal to a fraction by placing digits over the matching power of ten and reducing: 0.375 = 375/1000 = 3/8.

Ratio a:b compares parts; the total equals a + b parts, so each part = total ÷ (a + b). Use this to split an amount in a given ratio.

Mean = sum of values ÷ count of values. The mean is pulled toward outliers; the median (middle value of an ordered list) is not.

Weighted average = Σ(value × weight) ÷ Σ(weights); use when items carry different frequencies or importance.

Percent change = (new − old) ÷ old × 100; a percent increase followed by an equal percent decrease does not return to the original value.

Order of operations: parentheses, exponents, multiplication and division left to right, then addition and subtraction left to right.

A line graph shows trends over time; slope of the segment equals the rate of change per period (rise ÷ run).

A bar graph compares discrete categories; bar height or length is read against the vertical scale, so verify whether the axis starts at zero.

A pie chart shows parts of a whole; each sector's share = category value ÷ total, and sector angle = share × 360° (1% = 3.6°).

A broken or truncated axis exaggerates differences; check the scale labels before drawing conclusions from a chart.

A stacked bar shows both total per category (full bar height) and composition (segment heights).

A scatter plot displays correlation between two variables; a trend line through the points supports estimation but does not prove causation.

To read a value between gridlines, interpolate linearly: fraction of the gap × gap size, added to the lower gridline value.

Double-check units (thousands, millions) and time intervals printed on the axis; misreading them is the most common graph error.

Simple interest formula: I = P × r × t, where P is principal, r is the annual rate as a decimal, and t is time in years.

Amount (maturity value) under simple interest: A = P(1 + rt).

Convert time to years: months ÷ 12, days ÷ 360 for ordinary interest (banker's rule) or ÷ 365 for exact interest.

Solve for any unknown by rearranging: P = I ÷ (rt); r = I ÷ (Pt); t = I ÷ (Pr).

Ordinary interest uses a 360-day year and yields slightly more interest than exact interest on the same day count.

Simple interest accrues only on principal; interest is never added to the base for later periods.

To find the rate that doubles principal under simple interest: set I = P, then r × t = 1, so t = 1/r years.

When a partial payment is made, compute interest on the outstanding principal up to the payment date, then reduce principal.

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

Compound interest earned = A − P.

More frequent compounding raises the effective yield: quarterly (4), monthly (12), daily (360 or 365) periods per year.

Effective annual rate (EAR) = (1 + r/m)^m − 1, where m is compounding periods per year; compare rates using EAR, not nominal rate.

Rule of 72: approximate doubling time ≈ 72 ÷ annual percentage rate; accurate for rates roughly between 5% and 12%.

Present value (discounting): P = A ÷ (1 + i)^n; this reverses the compound amount formula.

Continuous compounding: A = P × e^(rt), the limit of more frequent compounding.

Compound interest grows principal faster than simple interest because each period's interest is added to the base.

Annuity: a series of equal payments made at equal time intervals; the payment interval and interest period usually match.

Future value of an ordinary annuity (payments at period end): FV = PMT × [((1 + i)^n − 1) ÷ i].

Present value of an ordinary annuity: PV = PMT × [(1 − (1 + i)^(−n)) ÷ i]; used for loan and lease valuations.

Annuity due (payments at period start) = ordinary annuity value × (1 + i).

Sinking fund payment: PMT = FV × i ÷ ((1 + i)^n − 1), the deposit needed each period to reach a target sum.

Amortized loan payment uses the present value of an annuity formula solved for PMT.

In an amortization schedule, each payment covers interest on the remaining balance first; the remainder reduces principal, so principal repayment accelerates over time.

Total interest on an annuity = (PMT × n) − amount borrowed or deposited.

Gross pay = hours worked × hourly rate; overtime typically pays 1.5× the regular rate for hours beyond 40 in a week.

Net pay = gross pay − total deductions (mandatory plus voluntary).

Mandatory deductions commonly include federal income tax withholding, state tax, Social Security (FICA) and Medicare.

FICA: Social Security is 6.2% of gross up to an annual wage base; Medicare is 1.45% of gross with no wage cap (employee share).

Gross pay for salaried employees = annual salary ÷ number of pay periods per year (26 biweekly, 24 semimonthly, 12 monthly).

Commission pay = commission rate × sales; a draw against commission is subtracted later from earned commission.

Percentage method for withholding: taxable wages × withholding rate from the applicable table bracket, plus the bracket base amount.

Deduction as a percent of gross = deduction ÷ gross pay × 100; useful for comparing benefit costs.

Trade (single) discount: net price = list price × (1 − discount rate); discount amount = list × rate.

Series (chain) discounts multiply: net price = list × (1 − d1) × (1 − d2) × …; they are never added together.

Single equivalent discount = 1 − [(1 − d1)(1 − d2)…]; e.g., 20% and 10% equal about 28%, not 30%.

Markup on cost: selling price = cost × (1 + markup rate on cost).

Markup on selling price: cost = selling price × (1 − markup rate on selling price).

Convert markup on cost to markup on selling price: rate_sp = rate_cost ÷ (1 + rate_cost).

Gross profit = selling price − cost; gross margin % = gross profit ÷ selling price × 100.

Cash discount terms 2/10, n/30 mean 2% off if paid within 10 days, net due in 30; net price = invoice × (1 − 0.02).

Markdown % = markdown amount ÷ original selling price × 100.

Straight-line depreciation: annual depreciation = (cost − salvage value) ÷ useful life in years.

Book value at any time = cost − accumulated depreciation.

Declining-balance depreciation: annual depreciation = book value at start of year × rate, where rate = (1 ÷ life) × multiplier (often 2 for double-declining).

Double-declining balance ignores salvage value in the rate but never depreciates book value below salvage value.

Units-of-production depreciation: (cost − salvage) ÷ total estimated units × units used in the period.

Sum-of-the-years'-digits: depreciation = (cost − salvage) × (remaining life ÷ sum of years' digits).

Net worth = total assets − total liabilities; a personal or business balance sheet reports it at a point in time.

Working capital = current assets − current liabilities; it measures short-term liquidity.

Common stockholders own the residual: they vote for directors and receive dividends only after preferred claims.

Preferred stock typically pays a fixed dividend rate on par value and usually carries no voting rights.

Cumulative preferred stock accumulates unpaid dividends (dividends in arrears) before common dividends are paid.

Par value is the stated value per share used for dividend and legal calculations; market value is the trading price.

Dividend per share = total dividend declared ÷ shares outstanding.

Bonds are debt: the issuer owes periodic interest (coupon = face value × coupon rate) and repayment of face value at maturity.

Bondholders are creditors paid before stockholders in liquidation; stockholders bear greater risk and greater upside.

Earnings per share (EPS) = net income ÷ shares outstanding.

Price-earnings ratio = market price per share ÷ earnings per share.

Inventory turnover = cost of goods sold ÷ average inventory; average inventory = (beginning + ending) ÷ 2.

Days' sales in inventory = 365 ÷ inventory turnover.

Accounts receivable turnover = net credit sales ÷ average accounts receivable.

Average collection period = 365 ÷ receivables turnover.

Total asset turnover = net sales ÷ average total assets; it measures how efficiently assets generate revenue.

Higher turnover generally signals efficient use of the asset, but very high turnover can indicate understocking or overly tight credit.

Return on assets = net profit margin × total asset turnover (DuPont relationship).

Inventory turnover uses cost of goods sold in the numerator, not sales, to keep units consistent with inventory at cost.

Current ratio = current assets ÷ current liabilities; a common benchmark is around 2:1.

Quick (acid-test) ratio = (cash + marketable securities + receivables) ÷ current liabilities; excludes inventory.

Debt-to-equity ratio = total liabilities ÷ total stockholders' equity.

Return on equity = net income ÷ average stockholders' equity.

Net profit margin = net income ÷ net sales × 100.

Gross profit margin = (net sales − cost of goods sold) ÷ net sales × 100.

Average daily sales = annual net sales ÷ 365; used for turnover and collection calculations.

A ratio alone is not meaningful; it requires comparison with prior periods, industry norms, or budget.

Convert a fraction to a decimal by dividing numerator by denominator; convert a decimal to a percent by multiplying by 100.

Percent means per hundred: 7.5% = 0.075 as a decimal multiplier.

Multiply a dollar amount by a percent by converting the percent to a decimal first: 8% of $450 = 0.08 × 450 = $36.

Round money to the nearest cent (two decimals) only at the final step; rounding intermediate steps compounds error.

When a problem says round to the nearest dollar, examine the cents digit: 50 cents or more rounds up.

To express one number as a percent of another: part ÷ whole × 100.

A ratio with mixed units must be converted to a common unit before simplifying.

Use simple interest for single short-term notes without compounding; use compound interest when interest is credited to principal.

Match the rate period to the payment period: monthly compounding requires rate ÷ 12 and months × 1 for n.

For a single lump sum use the compound amount formula; for a stream of equal deposits use the annuity formula.

An ordinary annuity values payments at period end; annuity due shifts every payment one period earlier and multiplies by (1 + i).

Loans use present value of an annuity; savings goals use future value of an annuity.

If compounding frequency increases while the nominal rate is fixed, both the effective rate and the future value increase.

The Rule of 72 estimates doubling time; use exact logarithms when precision matters: n = ln(2) ÷ ln(1 + i).

Overtime earnings = overtime hours × regular rate × 1.5; total gross = regular earnings + overtime earnings.

Take-home pay is always less than gross pay; deductions are subtracted, never added.

A series discount always yields a lower net price than the largest single discount applied alone.

Markup based on selling price and markup based on cost give different rates for the same dollar markup.

Depreciation expense reduces book value but is a non-cash charge; net worth on a balance sheet reflects accumulated depreciation.

Salvage value is the floor for declining-balance depreciation and is subtracted in straight-line calculations.

A capital loss or gain affects net worth only when the asset is sold or revalued, not when market price fluctuates daily.

Shares outstanding × market price per share = market capitalization.

Book value per share = (total stockholders' equity − preferred equity) ÷ common shares outstanding.

Dividend yield = annual dividend per share ÷ market price per share × 100.

Payout ratio = dividends per share ÷ earnings per share.

Asset turnover combined with profit margin determines return on assets; a low-margin firm can still earn well with high turnover.

A stock split increases shares outstanding and lowers par value proportionally; total equity is unchanged.

Turnover ratios use average balances when beginning and ending figures are both available; using ending balances alone can distort results.

Proportion setup: part/whole = percent/100; cross-multiply to solve for the unknown.

To divide a sum in a ratio a:b:c, total parts = a + b + c; each share = sum × (its part ÷ total parts).

Average of a data set changes by the same amount as every value if a constant is added to each value.

Median is found by ordering values and taking the middle; with an even count, average the two middle values.

Mode is the most frequent value; it is unaffected by extreme values.

Range = highest value − lowest value; it measures spread, not center.

Arithmetic mean of rates is wrong for averaging speeds or growth rates; use the harmonic or geometric mean respectively.

In a line graph, a steeper slope indicates a faster rate of change; a flat segment indicates no change.

In a pie chart, the largest sector corresponds to the largest share of the total, not necessarily the largest dollar amount if categories differ in base.

In a bar graph, compare bar lengths only when the axis scale is identical across bars.

A histogram shows frequency distribution of continuous data; bars touch because intervals are contiguous.

To find a total from a pie chart, divide a known sector value by its share: total = part ÷ share.

A trend line's slope estimates average change per period; extrapolation beyond the data is an assumption, not a fact.

Label units and periods on every axis; a graph without a stated scale cannot support a quantitative answer.

Assets are what is owned (cash, inventory, equipment, receivables); liabilities are what is owed (loans, payables, taxes).

Net worth (owner's equity) = assets − liabilities; it can be negative if liabilities exceed assets.

Straight-line depreciation spreads an equal expense each year over the asset's life.

Accelerated depreciation (declining balance, SYD) front-loads expense, lowering early taxable income.

Book value is an accounting figure, not necessarily the resale or market value of an asset.

A balance sheet is a snapshot at a date; an income statement covers a period.

Increased liabilities without matching asset growth reduce net worth dollar for dollar.

Cost + markup = selling price; selling price − markup = cost.

Markup amount is the same dollar figure whether expressed on cost or on selling price; only the base changes.

If markup on cost is 25%, markup on selling price is 20% (0.25 ÷ 1.25).

A markdown is computed on the original selling price unless the problem states otherwise.

Maintained markup = (selling price − cost) ÷ selling price, using the actual price after markdowns.

Break-even selling price equals cost plus all fixed and variable costs allocated per unit.

A cash discount is taken by the buyer; a trade discount is deducted by the seller before invoicing.

Retirement savings: use future value of an annuity to project accumulated deposits plus interest.

Mortgage payment: use present value of an annuity solved for PMT with monthly rate and total months.

Loan balance after k payments = present value of the remaining payments at the original rate.

Sinking fund deposits earn interest, so total deposits are less than the target fund amount.

Effective rate comparison is the correct way to choose between two nominal rates with different compounding.

Doubling time exact formula: n = log(2) ÷ log(1 + i); Rule of 72 is only an approximation.

Increasing the payment frequency at the same nominal rate raises the effective yield.