# Pre-MBA Quant Refresher: [Your name and programme start date] > A self-paced refresher of the mathematics the MBA core assumes on day one: algebra, functions, exponents and > logarithms, growth rates and compounding, percentages, probability basics and descriptive statistics. It is for > admitted students who have not used algebra or statistics for a few years, career changers from non-quantitative > backgrounds, and anyone preparing for a programme's quantitative waiver or placement exam. Each topic gives the > rules in plain language, KaTeX formulas, the matching Excel function, a fully solved Worked example A, a half-solved > Worked example B with gaps `[ ]` to fill, and an unsolved Exercise C, with every answer computed and collected in > the Answer key. A 12-question diagnostic at the start tells you which topics to skip. > Source basis: standard school and first-year university mathematics and statistics, the learning research of > Dunlosky et al. (2013) and Renkl and Atkinson (2003), and OpenStax open textbooks named as further reading; all > wording, problems and data are original. > Replace every `[bracketed]` value and every row marked *Example*. The business settings are fictional: Northfield > Outdoor Co., Corvane Robotics and Tallow Creek Medical Center. ## Document control | Field | Value | |---|---| | Document ID | [PRE-MBA-QR-001] | | Learner | [Your name] | | Programme and start date | [School A full-time MBA, YYYY-MM-DD] | | Waiver or placement exam (if any) | [Quant waiver exam on YYYY-MM-DD / none] | | Diagnostic 1 score and date | [ ] / 12, YYYY-MM-DD | | Diagnostic 2 score and date | [ ] / 12, YYYY-MM-DD | | Tools | [Excel version, calculator model] | | Reviewed by | [Study buddy or mentor] | | Last updated | YYYY-MM-DD | | Version | [0.1] | ## Contents 1. How this refresher works 2. Diagnostic and routing 3. Algebra 4. Functions 5. Exponents and logarithms 6. Growth rates and compounding 7. Percentages 8. Probability basics 9. Descriptive statistics 10. Common mistakes 11. Two-week practice plan 12. Self-test ## 1. How this refresher works The MBA core moves fast. In the first weeks, finance assumes you can rearrange a formula and solve for an exponent, microeconomics assumes you can find the maximum of a quadratic, and statistics assumes you know what a standard deviation measures. None of this is hard, but rusty skills cost hours in the first month that you would rather spend on cases, recruiting and your classmates. The refresher uses three methods with strong evidence behind them. **Retrieval practice**, answering questions from memory, beats re-reading for long-term retention (Roediger and Karpicke 2006), so every section ends in problems and the template ends in a self-test. **Worked examples with fading**, studying a solved problem, then completing a partly solved one, then solving one alone, cut the effort of learning a new procedure (Renkl and Atkinson 2003). **Spacing**, returning to the same skill after days rather than minutes, was one of the two strategies rated high utility by Dunlosky et al. (2013); the practice plan in section 11 builds it in. Every topic shows the Excel function next to the formula. Excel does not replace understanding: you need to know what the function computes to spot a wrong input. But it is the tool you will use in the core, and checking a hand answer in Excel trains both skills at once. ## 2. Diagnostic and routing Answer without notes. A basic calculator is fine. 1. Solve $3x - 7 = 2x + 5$. 2. Rearrange $\pi = Q(P - V) - F$ to give $Q$. 3. A line passes through $(2, 10)$ and $(6, 22)$. What is its slope? 4. Simplify $2^5 \times 2^3$ and give its value. 5. What are $\ln(e^3)$ and $\log_{10}(1000)$? 6. Revenue grew from 100 to 150 over 4 years. What is the compound annual growth rate? 7. A price rises 20% and then falls 20%. What is the net change? 8. An item costs 60 and sells for 80. What are the gross margin and the markup? 9. $P(A) = 0.3$, $P(B) = 0.5$ and $P(A \cap B) = 0.1$. What is $P(A \cup B)$? 10. A bet pays +500 with probability 0.2 and -100 otherwise. What is its expected value? 11. What are the mean and the median of 2, 4, 4, 5, 15? 12. A policy rate moves from 4% to 5%. Describe the change in percentage points and in percent. **Routing.** Use your marked answers. | Questions missed | Study section | Typical time | |---|---|---| | 1 or 2 | 3. Algebra | 2 to 3 hours | | 3 | 4. Functions | 2 to 3 hours | | 4 or 5 | 5. Exponents and logarithms | 1 to 2 hours | | 6 | 6. Growth rates and compounding | 2 hours | | 7, 8 or 12 | 7. Percentages | 1 to 2 hours | | 9 or 10 | 8. Probability basics | 2 to 3 hours | | 11 | 9. Descriptive statistics | 2 hours | | none | Self-test (section 12), then mba/pre-mba-excel-modeling-drills | 1 hour | ```mermaid flowchart TD A["Take the 12-question diagnostic"] --> B["Mark with the Answer key"] B --> C{"Score 11 or 12?"} C -->|"Yes"| D["Do the self-test, then move to Excel drills"] C -->|"No"| E["Study each section you missed, A then B then C"] E --> F["Log every miss in the error log"] F --> G["Re-take the diagnostic after two weeks"] G --> C D --> H["Next: accounting and finance primers"] ``` ## 3. Algebra **Rules you need.** Whatever you do to one side of an equation, do to the other. To isolate a variable, undo operations in reverse order (subtract before you divide). An inequality flips direction when you multiply or divide by a negative number. Two linear equations in two unknowns usually have one solution: substitute one into the other, or eliminate a variable. $$ ax + b = c \;\Rightarrow\; x = \frac{c - b}{a}, \qquad -2x > 6 \;\Rightarrow\; x < -3 $$ Excel: Goal Seek (Data, What-If Analysis) solves one equation numerically; `=MMULT(MINVERSE(A),b)` solves a linear system; Solver handles constraints. **Worked example A (solved).** Northfield sold 500 units of two products last weekend: jackets at 120 and tents at 300. Revenue was 96,000. How many of each? Let $J$ be jackets and $T$ tents. Then $J + T = 500$ and $120J + 300T = 96{,}000$. From the first, $J = 500 - T$. Substitute: $120(500 - T) + 300T = 96{,}000$, so $60{,}000 + 180T = 96{,}000$ and $T = 200$, $J = 300$. Check: $120 \times 300 + 300 \times 200 = 36{,}000 + 60{,}000 = 96{,}000$. Excel: with the coefficients in A1:B2 and the totals in D1:D2, `=MMULT(MINVERSE(A1:B2),D1:D2)` returns 300 and 200. **Worked example B (half solved).** Corvane Robotics splits a 30,000 marketing budget between search ads and social ads. Search yields 0.04 leads per unit spent, social 0.025. The team wants exactly 1,020 leads. How much goes to each? 1. Equations: $s + m = 30{,}000$ and $0.04s + 0.025m = [\ \ ]$. 2. Substitute $m = 30{,}000 - s$: $0.04s + 0.025(30{,}000 - s) = 1{,}020$, so $0.015s + [\ \ ] = 1{,}020$. 3. Solve: $s = [\ \ ]$ and $m = [\ \ ]$. 4. Check the leads total: $[\ \ ]$. **Exercise C (unsolved).** A product has variable cost 35 per unit and fixed costs of 60,000. Management expects to sell 4,000 units and wants a profit of 40,000. (a) Rearrange $\pi = Q(P - V) - F$ to solve for $P$ and find the required price. (b) At a price of 65, a variable cost of 25 and fixed costs of 40,000, for which quantities is profit at least zero? Write it as an inequality. ## 4. Functions **Rules you need.** A linear function $y = a + bx$ has intercept $a$ (the value at $x = 0$) and slope $b$ (the change in $y$ per one-unit change in $x$): $b = (y_2 - y_1)/(x_2 - x_1)$. A quadratic $y = ax^2 + bx + c$ with $a < 0$ has a maximum at $x = -b/(2a)$. Profit is revenue minus cost: $\pi(Q) = R(Q) - C(Q)$. With a linear demand curve $P = \alpha - \beta Q$, revenue $R = PQ = \alpha Q - \beta Q^2$ is a downward-opening parabola. $$ \text{slope } b = \frac{\Delta y}{\Delta x}, \qquad x^* = -\frac{b}{2a} \text{ for } y = ax^2 + bx + c $$ Excel: `=SLOPE(y_range, x_range)` and `=INTERCEPT(y_range, x_range)` fit a line through data; a scatter chart with a trendline shows it; Solver finds a maximum. **Worked example A (solved).** Northfield's private-label stove: cost $C(Q) = 40{,}000 + 25Q$, price 65, so revenue $R(Q) = 65Q$. Profit $\pi(Q) = 65Q - 40{,}000 - 25Q = 40Q - 40{,}000$, which is zero at $Q = 1{,}000$ (the break-even volume). At $Q = 1{,}500$, profit is $40 \times 1{,}500 - 40{,}000 = 20{,}000$. Second part: if demand is $P = 100 - 0.5Q$, revenue is $R = 100Q - 0.5Q^2$. Here $a = -0.5$ and $b = 100$, so revenue peaks at $Q^* = -100/(2 \times -0.5) = 100$ units, price $P = 100 - 50 = 50$ and revenue $5{,}000$. Excel check: build a column of $Q$ from 0 to 200 and `=Q*(100-0.5*Q)`; the largest value is at 100. **Worked example B (half solved).** Same demand $P = 100 - 0.5Q$, now with cost $C = 1{,}000 + 20Q$. Find the profit-maximising quantity, price and profit. 1. Profit: $\pi = 100Q - 0.5Q^2 - 1{,}000 - 20Q = [\ \ ]Q - 0.5Q^2 - 1{,}000$. 2. Maximum at $Q^* = -b/(2a) = [\ \ ]$. 3. Price: $P = 100 - 0.5Q^* = [\ \ ]$. 4. Profit: $\pi^* = [\ \ ]$. 5. Compare with Example A: why is the profit-maximising quantity lower than the revenue-maximising one? **Exercise C (unsolved).** Corvane sells service contracts with demand $P = 240 - 2Q$ and cost $C = 1{,}500 + 40Q$. Find $Q^*$, $P^*$ and maximum profit. ## 5. Exponents and logarithms **Rules you need.** $$ a^m a^n = a^{m+n}, \quad \frac{a^m}{a^n} = a^{m-n}, \quad (a^m)^n = a^{mn}, \quad a^{-n} = \frac{1}{a^n}, \quad a^{1/n} = \sqrt[n]{a} $$ $$ \ln(xy) = \ln x + \ln y, \quad \ln(x^n) = n\ln x, \quad \ln(e^x) = x, \quad b^n = c \;\Rightarrow\; n = \frac{\ln c}{\ln b} $$ Excel: `=LN(x)`, `=EXP(x)`, `=LOG10(x)`, `=LOG(x, base)`, powers with `^`. Excel's `=NPER(rate, 0, -pv, fv)` and `=RATE(nper, 0, -pv, fv)` solve the compounding equation for $n$ or $r$ directly. **Worked example A (solved).** How many years does 1,000 take to grow to 2,500 at 8% a year? $1{,}000 \times 1.08^n = 2{,}500$, so $1.08^n = 2.5$ and $n = \ln 2.5/\ln 1.08 = 0.9163/0.0770 = 11.906$ years. Check: $1{,}000 \times 1.08^{11.906} = 2{,}500.02$ (rounding). Excel: `=NPER(0.08,0,-1000,2500)` returns 11.906. **Worked example B (half solved).** The "rule of 72" says money doubles in about $72/r$ years when $r$ is in percent. Test it at 9%. 1. Rule of 72: $72/9 = [\ \ ]$ years. 2. Exact: $1.09^n = 2$, so $n = \ln 2 / \ln 1.09 = [\ \ ]$ years. 3. With continuous compounding at 9%: $e^{0.09n} = 2$, so $n = \ln 2/0.09 = [\ \ ]$ years. 4. Which of the three is the "rule of 69.3", and why does 72 work well for annual compounding at common rates? **Exercise C (unsolved).** (a) An investment grows from 5,000 to 8,000 in 6 years. What annual rate does that imply? Use $r = (FV/PV)^{1/n} - 1$ and check with `=RATE(6,0,-5000,8000)`. (b) What is $\log_{10}(0.001)$? ## 6. Growth rates and compounding **Rules you need.** The compound annual growth rate (CAGR) is the constant rate that takes a start value to an end value over $n$ years. The arithmetic mean of yearly growth rates overstates it whenever the rates vary. Compounding $m$ times a year at an annual rate $\text{APR}$ gives an effective annual rate above the APR; continuous compounding is the limit. Real growth removes inflation. $$ CAGR = \left(\frac{V_{\text{end}}}{V_{\text{start}}}\right)^{1/n} - 1, \qquad EAR = \left(1 + \frac{\text{APR}}{m}\right)^m - 1, \qquad 1 + g_{\text{real}} = \frac{1 + g_{\text{nominal}}}{1 + \pi} $$ Excel: `=RRI(n, start, end)` returns the CAGR; `=EFFECT(apr, m)` the effective annual rate; `=GEOMEAN(1+range)-1` the geometric mean growth of a series of rates (entered as an array formula in older Excel versions). **Worked example A (solved).** Corvane Robotics revenue (USD m) for six years: 12, 15, 21, 24, 27, 30. Yearly growth: 25.00%, 40.00%, 14.29%, 12.50%, 11.11%. The arithmetic mean is 20.58%. The CAGR is $(30/12)^{1/5} - 1 = 20.11\%$. If you project with the arithmetic mean you get $12 \times 1.2058^5 = 30.59$, not the actual 30: the arithmetic mean overstates compound growth. Excel: `=RRI(5,12,30)` returns 20.11%. **Worked example B (half solved).** A lender quotes 9% APR. Compute the effective annual rate for: 1. Quarterly compounding: $(1 + 0.09/4)^4 - 1 = [\ \ ]$. 2. Monthly compounding: $(1 + 0.09/12)^{12} - 1 = [\ \ ]$. 3. Continuous compounding: $e^{0.09} - 1 = [\ \ ]$. 4. Which number should you use to compare this loan with one quoted as an effective annual rate? **Exercise C (unsolved).** Northfield's nominal sales grew 7% last year and its prices rose 3%. (a) What was real (volume) growth, exactly and by the approximation $g_{\text{nominal}} - \pi$? (b) Why does the gap between the two widen when inflation is high? The chart compares 1,000 growing at 8% simple interest with 8% compound interest over 30 years. The gap is the "interest on interest" that makes compounding the central idea of finance. ```vega-lite { "$schema": "https://vega.github.io/schema/vega-lite/v5.json", "description": "Growth of 1,000 at 8 percent, simple versus compound (computed)", "width": 380, "height": 200, "data": {"values": [ {"year": 0, "type": "Simple", "value": 1000.0}, {"year": 5, "type": "Simple", "value": 1400.0}, {"year": 10, "type": "Simple", "value": 1800.0}, {"year": 15, "type": "Simple", "value": 2200.0}, {"year": 20, "type": "Simple", "value": 2600.0}, {"year": 25, "type": "Simple", "value": 3000.0}, {"year": 30, "type": "Simple", "value": 3400.0}, {"year": 0, "type": "Compound", "value": 1000.0}, {"year": 5, "type": "Compound", "value": 1469.33}, {"year": 10, "type": "Compound", "value": 2158.92}, {"year": 15, "type": "Compound", "value": 3172.17}, {"year": 20, "type": "Compound", "value": 4660.96}, {"year": 25, "type": "Compound", "value": 6848.48}, {"year": 30, "type": "Compound", "value": 10062.66} ]}, "mark": {"type": "line", "point": true}, "encoding": { "x": {"field": "year", "type": "quantitative", "title": "Year"}, "y": {"field": "value", "type": "quantitative", "title": "Value"}, "color": {"field": "type", "type": "nominal", "title": "Interest"} } } ``` ## 7. Percentages **Rules you need.** A percentage change is relative to the starting value. Successive changes multiply, they do not add. A change between two percentages is measured in percentage points; its relative change is in percent. Margin is profit over price; markup is profit over cost. A weighted average weights each item by its share. $$ \%\Delta = \frac{\text{new} - \text{old}}{\text{old}}, \quad (1 + a)(1 + b) - 1 \neq a + b, \quad \text{margin} = \frac{P - C}{P}, \quad \text{markup} = \frac{P - C}{C}, \quad \text{margin} = \frac{\text{markup}}{1 + \text{markup}} $$ Excel: `=(new-old)/old` formatted as a percentage; `=SUMPRODUCT(weights, values)` for a weighted average. **Worked example A (solved).** Northfield marks a 200 jacket down 30%, then takes a further 20% off the reduced price at the end of the season. Final price: $200 \times 0.7 \times 0.8 = 112$, a total discount of 44%, not 50%. In the same period the online conversion rate rose from 2.0% to 2.5%: that is 0.5 percentage points, or a 25% relative increase. Saying "conversion rose 0.5%" is wrong and in an interview it is the kind of slip that costs credibility. **Worked example B (half solved).** Northfield's sales mix and gross margins by category: apparel 50% of sales at 40% margin, equipment 30% at 25%, accessories 20% at 55%. 1. Blended margin: $0.5 \times 0.40 + 0.3 \times 0.25 + 0.2 \times 0.55 = [\ \ ]$. 2. Next year the mix shifts to apparel 30%, equipment 50%, accessories 20%, with the same category margins. New blended margin: $[\ \ ]$. 3. The change in blended margin is $[\ \ ]$ percentage points, caused entirely by mix, not by pricing. 4. Excel: write the `SUMPRODUCT` formula for step 1. **Exercise C (unsolved).** (a) After a 15% price increase, a product sells for 92. What was the original price? (b) A buyer uses a 40% markup on cost. What gross margin does that give? ## 8. Probability basics **Rules you need.** Probabilities lie between 0 and 1. The addition rule handles "or", the multiplication rule handles "and". Conditional probability restricts attention to the cases where the condition holds. Bayes' rule turns $P(B \mid A)$ into $P(A \mid B)$. Expected value weights each outcome by its probability. The binomial distribution counts successes in $n$ independent trials with the same probability $p$. $$ P(A \cup B) = P(A) + P(B) - P(A \cap B), \qquad P(A \mid B) = \frac{P(A \cap B)}{P(B)} $$ $$ P(A \mid B) = \frac{P(B \mid A)P(A)}{P(B \mid A)P(A) + P(B \mid A^c)P(A^c)}, \qquad E[X] = \sum x_i p_i, \qquad P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k} $$ Excel: `=SUMPRODUCT(values, probabilities)` for expected value; `=BINOM.DIST(k, n, p, FALSE)` for one value and `TRUE` for the cumulative probability; `=COMBIN(n, k)`. **Worked example A (solved).** Tallow Creek Medical Center screens patients for a condition with 2% prevalence. The test has 95% sensitivity (true positive rate) and 90% specificity (true negative rate). A patient tests positive. What is the probability they have the condition? Per 10,000 patients: 200 have the condition, of whom 190 test positive; 9,800 do not, of whom 980 test positive (10%). So $P(\text{condition} \mid +) = 190/(190 + 980) = 0.0190/(0.0190 + 0.0980) = 0.1624$, about 16%. Most positives are false positives because the condition is rare. This base-rate effect matters whenever a manager reads a fraud flag, a quality alarm or a hiring screen. **Worked example B (half solved).** Corvane can launch a new gripper now or run a pilot first. Full launch: 60% chance of success worth 400 (USD k), 40% chance of failure costing 250. Pilot: same 60% chance; success then worth 300 (later entry), failure costs only 60 (the pilot is stopped early). 1. $E[\text{launch}] = 0.6 \times 400 + 0.4 \times (-250) = [\ \ ]$. 2. $E[\text{pilot}] = 0.6 \times 300 + 0.4 \times (-60) = [\ \ ]$. 3. Which has the higher expected value, and by how much? 4. Name one reason a manager might still prefer the option with the lower expected value. **Exercise C (unsolved).** Each of 10 visitors to a Northfield store independently buys with probability 0.3. (a) What is the expected number of buyers? (b) What is the probability that at least 3 buy? Use $1 - P(X \le 2)$ and check with `=1-BINOM.DIST(2,10,0.3,TRUE)`. ## 9. Descriptive statistics **Rules you need.** The mean is the balance point; the median is the middle value. The sample standard deviation measures typical distance from the mean, with divisor $n - 1$. A z-score expresses a value in standard deviations from the mean. Quartiles split the sorted data into four parts; the IQR ($Q_3 - Q_1$) is the spread of the middle half, and values beyond $Q_3 + 1.5\,IQR$ or below $Q_1 - 1.5\,IQR$ are flagged as possible outliers (Tukey's rule). Correlation measures the strength of a linear relationship, from -1 to 1. $$ \bar{x} = \frac{1}{n}\sum x_i, \quad s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}}, \quad z = \frac{x - \bar{x}}{s}, \quad r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}} $$ Excel: `=AVERAGE`, `=MEDIAN`, `=STDEV.S` (sample; `STDEV.P` divides by $n$), `=STANDARDIZE(x, mean, sd)`, `=QUARTILE.INC(range, 1)`, `=CORREL(x, y)`. **Worked example A (solved).** Daily online orders at a Northfield regional site over 10 days: 42, 38, 51, 45, 47, 39, 44, 63, 41, 50. Sum 460, mean 46.00. Sorted: 38, 39, 41, 42, 44, 45, 47, 50, 51, 63, so the median is $(44 + 45)/2 = 44.50$. Sample SD 7.38 (`=STDEV.S(A1:A10)`). The 63-order day has $z = (63 - 46)/7.3786 = 2.30$: worth checking (a promotion, a data error) before planning capacity on the mean. **Worked example B (half solved).** Same 10 days. 1. $Q_1$ = 41.25 and $Q_3$ = [ ] (`=QUARTILE.INC(A1:A10,3)`). 2. IQR = [ ]. 3. Upper fence $Q_3 + 1.5 \times IQR$ = [ ]. 4. Is 63 flagged as a possible outlier by Tukey's rule? [ ]. 5. The mean is above the median. What does that tell you about the shape of the data? **Exercise C (unsolved).** Corvane tracks demo visits $x$ and qualified leads $y$ for five weeks: $x$ = 2, 4, 5, 7, 9; $y$ = 30, 44, 39, 55, 58. (a) Compute the means of $x$ and $y$. (b) Compute the correlation with `=CORREL` and interpret it in one sentence. (c) Does a strong correlation show that demo visits cause leads? ## 10. Common mistakes | Mistake | Example | Fix | |---|---|---| | Adding successive percentage changes | +20% then -20% "is zero" | Multiply factors: $1.2 \times 0.8 = 0.96$, a 4% fall | | Confusing percentage points with percent | Rate 4% to 5% called "up 1%" | Up 1 percentage point, or 25% in relative terms | | Confusing margin and markup | A 40% markup "is a 40% margin" | Margin = markup / (1 + markup) = 28.6% | | Averaging growth rates arithmetically | Mean of yearly growth used as CAGR | Use `=RRI` or the geometric mean | | Mixing nominal and real values | Comparing sales across years of high inflation | Deflate first, then compare | | Forgetting to flip an inequality | $-2x > 6$ written as $x > -3$ | Multiplying by a negative flips the sign | | Ignoring the base rate | Reading a positive test as "95% likely" | Draw the tree or table per 10,000 cases | | Using `STDEV.P` for a sample | Underestimates spread in small samples | Use `STDEV.S` for samples | | Reporting only the mean for skewed data | Order volume with one promotion day | Report median and IQR as well | | Reading correlation as causation | Demo visits "cause" leads | Look for a confounder; only an experiment shows cause | | Solving for a period count by trial and error | Guessing $n$ in $1.08^n = 2.5$ | Use logs: $n = \ln 2.5/\ln 1.08$ | ## 11. Two-week practice plan | Day | Session (about 60 to 90 minutes) | Spaced review | |---|---|---| | 1 | Diagnostic, routing, section 3 Algebra | None | | 2 | Section 4 Functions | Algebra Exercise C again from memory | | 3 | Section 5 Exponents and logarithms | Functions Example B | | 4 | Section 6 Growth rates and compounding | Algebra and logs, two problems each | | 5 | Section 7 Percentages | Growth Example B | | 6 | Rest or catch-up | Flashcards (mba/mba-concept-flashcards) | | 8 | Section 8 Probability basics | Percentages Exercise C | | 9 | Section 9 Descriptive statistics | Probability Example A from memory | | 10 | Mixed practice: one problem from each section in random order | Error log review | | 12 | Self-test (section 12) | All Exercise C problems you missed | | 14 | Diagnostic re-take; update Document control | Plan next primer | Interleaving (mixing problem types) in sessions 10 and 12 matters: in the core, no question tells you which topic it belongs to. ## 12. Self-test 1. Solve $2{,}000(1 + r)^5 = 3{,}000$ for $r$. 2. A customer satisfaction score rose from 12% "very satisfied" to 15%. State the change in percentage points and in percent. 3. How many years does it take for a value to triple at 5% a year? 4. Fixed costs are 5,000, variable cost 12 per unit and price 20. What is the break-even volume? 5. Demand is $P = 60 - 0.2Q$. At what quantity is revenue maximised, and what are the price and revenue there? 6. Sales rise 10%, then 10%, then fall 15%. What is the total change? 7. Each item from a supplier is defective with probability 0.05, independently. What is the probability that a box of 3 contains at least one defect? 8. Find the mean and median of 5, 7, 7, 9, 22. Which better describes a typical value, and why? 9. 30% of Corvane's leads come from trade events, and 20% of those convert; 5% of other leads convert. A lead has converted. What is the probability it came from an event? 10. Nominal revenue grew 8% while prices rose 5%. What was real growth? ## My notes and next steps | Item | My entry | |---|---| | Sections I needed (from the diagnostic) | [Sections 6 and 8] | | Concepts still shaky | [ ] | | Errors logged in mba/mba-error-log | [Date and short description] | | Next primer | [mba/pre-mba-excel-modeling-drills, then mba/pre-mba-accounting-primer] | | Diagnostic re-take date | YYYY-MM-DD | ## Sources and further reading - Dunlosky, J., Rawson, K. A., Marsh, E. J., Nathan, M. J., and Willingham, D. T. (2013). Improving students' learning with effective learning techniques. *Psychological Science in the Public Interest*, 14(1), 4-58. - Renkl, A., and Atkinson, R. K. (2003). Structuring the transition from example study to problem solving in cognitive skill acquisition: a cognitive load perspective. *Educational Psychologist*, 38(1), 15-22. - Roediger, H. L., and Karpicke, J. D. (2006). Test-enhanced learning: taking memory tests improves long-term retention. *Psychological Science*, 17(3), 249-255. - Rohrer, D., and Taylor, K. (2007). The shuffling of mathematics problems improves learning. *Instructional Science*, 35(6), 481-498. - Tukey, J. W. (1977). *Exploratory Data Analysis*. Addison-Wesley. - OpenStax (free; CC BY 4.0 or CC BY-NC-SA 4.0 by title), https://openstax.org/: *College Algebra 2e*; *Introductory Statistics 2e*; *Principles of Finance* (time value of money chapters). - Microsoft Support, Excel functions (by category), https://support.microsoft.com/en-us/excel/excel-functions-by-category (accessed 2026-10-28). - Further reading (by name only): a business mathematics text such as Haeussler, Paul and Wood, *Introductory Mathematical Analysis for Business, Economics, and the Life and Social Sciences*; Anderson, Sweeney and Williams, *Statistics for Business and Economics*; for test-style practice, the official GMAT Focus and GRE preparation materials from GMAC and ETS. ## Answer key **Diagnostic.** 1. $x = 12$. 2. $Q = (\pi + F)/(P - V)$. 3. Slope $= (22 - 10)/(6 - 2) = 3$. 4. $2^8 = 256$. 5. 3 and 3. 6. $1.5^{1/4} - 1 = 10.67\%$. 7. $1.2 \times 0.8 - 1 = -4\%$. 8. Margin $20/80 = 25\%$; markup $20/60 = 33.3\%$. 9. $0.3 + 0.5 - 0.1 = 0.7$. 10. $0.2 \times 500 + 0.8 \times (-100) = 20$. 11. Mean 6, median 4. 12. Up 1 percentage point, which is a 25% relative increase. **Algebra.** B: (1) 1,020; (2) $0.015s + 750 = 1{,}020$; (3) $s = 18{,}000$, $m = 12{,}000$; (4) $0.04 \times 18{,}000 + 0.025 \times 12{,}000 = 720 + 300 = 1{,}020$. C: (a) $P = (\pi + F)/Q + V = (40{,}000 + 60{,}000)/4{,}000 + 35 = 60$. (b) $65Q - 25Q - 40{,}000 \ge 0$, so $Q \ge 1{,}000$. **Functions.** B: (1) 80; (2) $Q^* = -80/(2 \times -0.5) = 80$; (3) $P = 60$; (4) $\pi^* = 80 \times 80 - 0.5 \times 6{,}400 - 1{,}000 = 2{,}200$; (5) each extra unit now carries a cost of 20, so the firm stops where marginal revenue equals 20 rather than 0. C: $\pi = 200Q - 2Q^2 - 1{,}500$, $Q^* = 50$, $P^* = 140$, maximum profit 3,500. **Exponents and logarithms.** B: (1) 8.0; (2) 8.043; (3) 7.702; (4) the continuous case, since $\ln 2 \times 100 = 69.3$; 72 sits close to the exact annual-compounding figure for rates of about 6% to 10% and has many divisors. C: (a) $1.6^{1/6} - 1 = 8.15\%$; (b) $-3$. **Growth rates and compounding.** B: (1) 9.31%; (2) 9.38%; (3) 9.42%; (4) the effective annual rate for the compounding the lender actually applies (9.38% if monthly). C: (a) exact $1.07/1.03 - 1 = 3.88\%$, approximation 4%; (b) the approximation drops the cross term $g_{\text{real}} \times \pi$, which grows with inflation. **Percentages.** B: (1) 38.5%; (2) $0.3 \times 0.40 + 0.5 \times 0.25 + 0.2 \times 0.55 = 35.5\%$; (3) 3.0 percentage points lower; (4) `=SUMPRODUCT(B2:B4,C2:C4)` with shares in B and margins in C. C: (a) $92/1.15 = 80$; (b) $0.4/1.4 = 28.6\%$. **Probability.** B: (1) 140; (2) 156; (3) the pilot, by 16 (USD k); (4) the pilot also limits the worst case from -250 to -60, which matters if the firm cannot absorb a large loss; a launch-now option might be preferred if early entry has strategic value not captured in the payoffs. C: (a) $10 \times 0.3 = 3$; (b) $P(X \le 2) = 0.3828$, so $P(X \ge 3) = 0.6172$. **Descriptive statistics.** B: (1) $Q_3 = 49.25$; (2) IQR $= 8.00$; (3) $49.25 + 12 = 61.25$; (4) yes, $63 > 61.25$; (5) the data are right-skewed: one high day pulls the mean up. C: (a) $\bar{x} = 5.4$, $\bar{y} = 45.2$; (b) $r = 0.94$, a strong positive linear association; (c) no: five weeks of observational data cannot rule out a common cause (for example a trade show week raising both). **Self-test.** 1. $r = 1.5^{1/5} - 1 = 8.45\%$ (`=RRI(5,2000,3000)`). 2. 3 percentage points; $3/12 = 25\%$ relative increase. 3. $n = \ln 3/\ln 1.05 = 22.5$ years. 4. $5{,}000/(20 - 12) = 625$ units. 5. $R = 60Q - 0.2Q^2$, maximum at $Q = 60/0.4 = 150$, $P = 30$, $R = 4{,}500$. 6. $1.1 \times 1.1 \times 0.85 - 1 = +2.85\%$. 7. $1 - 0.95^3 = 0.1426$. 8. Mean 10, median 7. The median: the single value 22 pulls the mean above four of the five values. 9. $0.06/(0.06 + 0.035) = 0.632$. 10. $1.08/1.05 - 1 = 2.86\%$. ## Revision history | Version | Date | Author | Change | |---|---|---|---| | 0.1 | YYYY-MM-DD | [Your name] | First version from the template, diagnostic 1 taken | | 0.2 | YYYY-MM-DD | [Your name] | *Example:* Diagnostic 2 taken, score 11/12, added own problems on logs | ## Standard and licence note Based on standard secondary-school and first-year university mathematics and statistics, as assumed by MBA core courses, and on published learning research (retrieval practice, worked examples, spacing). The explanations, problems, data and answers are original; no textbook text, test-preparation material or official GMAT, GRE or Executive Assessment question is reproduced. The template is not affiliated with or endorsed by any business school or test owner. Template licence: MIT.