MATH114: Quantitative Reasoning - Test 4 Formula Sheet

Growth Models

Linear Growth (Basic)

New Value = Initial Value + (Growth Rate × Time)

Conversions

Dimensional analysis: Multiply by conversion factors so units cancel: $\text{result} = \text{given} \times \frac{\text{wanted unit}}{\text{current unit}}$. result = given * (wanted unit/current unit)
Quick examples: monthly $\to$ annual (×12), annual $\to$ monthly (÷12).

Exponential Growth

Formula (rate 'r'): $ Q = Q_0 \times (1 + r)^t $ Q = Q_0 * (1 + r)^t
Approx. Doubling Time ($T_{dbl}$): $ T_{dbl} \approx \frac{70}{P} $ (where P is % growth rate) T_dbl ≈ 70 / P
Exact Doubling Time ($T_{dbl}$): $ T_{dbl} = \frac{\log(2)}{\log(1 + r)} $ T_dbl = log(2) / log(1 + r)
Formula (doubling time): $ Q = Q_0 \times 2^{(t / T_{dbl})} $ Q = Q_0 * 2^(t / T_dbl)

Exponential Decay (Half-Life)

Formula (half-life $T_{half}$): $ Q = Q_0 \times (0.5)^{(t / T_{half})} $ (equivalently $\left(\frac{1}{2}\right)^{(t / T_{half})}$) Q = Q_0 * (0.5)^(t / T_half) (equivalently (1/2)^(t / T_half))

Linear Modeling

Slope (Rate of Change)

Rate of Change = Slope (m) = $\frac{\text{Change in Dependent Variable}}{\text{Change in Independent Variable}}$ (Change in Dependent Variable) / (Change in Independent Variable)
$ m = \frac{y_2 - y_1}{x_2 - x_1} $ m = (y2 - y1) / (x2 - x1)

Equation of a Line

General Form: $\text{Dependent Variable} = \text{Initial Value} + (\text{Rate of Change} \times \text{Independent Variable})$ Dependent Var = Initial Value + (Rate of Change * Independent Var)
Slope-Intercept Form: $ y = mx + b $ y = mx + b
y: Dependent Variable, x: Independent Variable, m: Slope, b: y-intercept (Initial Value)

Rate of Change Rule

$\text{Change in Dependent Var} = \text{Rate of Change} \times \text{Change in Independent Var}$ Change in Dependent Var = Rate of Change * Change in Independent Var
$ \Delta y = m \times \Delta x $ Δy = m * Δx

Financial Formulas

Simple Interest

Interest = $ P \times r \times t $ Interest = P * r * t
Amount $ A = P \times (1 + r \times t) $ A = P * (1 + r * t)
P: Principal, r: annual rate (decimal), t: time (years), A: Amount

Compound Interest

Compounded Annually: $ A = P \times (1 + APR)^Y $ A = P * (1 + APR)^Y
Compounded n times/year: $ A = P \times \left(1 + \frac{APR}{n}\right)^{(nY)} $ A = P * (1 + APR/n)^(nY)
Compounded Continuously: $ A = P \times e^{(APR \times Y)} $ A = P * e^(APR * Y)
A: Accumulated Balance (FV), P: Principal (PV), APR: Annual Rate, Y: Years, n: compounding periods/year, $e \approx 2.71828$
APY (Annual Percentage Yield): Effective annual rate
$ APY = \left(1 + \frac{APR}{n}\right)^n - 1 $ (for n compoundings) APY = (1 + APR/n)^n - 1
$ APY = e^{APR} - 1 $ (for continuous compounding) APY = e^APR - 1

Savings/Loan Payments (Ordinary Annuity)

Payment (PMT) based on desired Future Value (FV): $$ PMT = \frac{FV \times \left(\frac{APR}{n}\right)}{\left[\left(1 + \frac{APR}{n}\right)^{(nY)} - 1\right]} $$ PMT = (FV * (APR/n)) / (((1 + APR/n)^(nY)) - 1)
Payment (PMT) based on Loan Principal (PV): $$ PMT = \frac{PV \times \left(\frac{APR}{n}\right)}{\left[1 - \left(1 + \frac{APR}{n}\right)^{(-nY)}\right]} $$ PMT = (PV * (APR/n)) / (1 - (1 + APR/n)^(-nY))
Future Value (FV) from regular payment (PMT): $$ FV = PMT \cdot \frac{\left(1+\frac{APR}{n}\right)^{nY}-1}{\frac{APR}{n}} $$ FV = PMT * (((1 + APR/n)^(nY)) - 1) / (APR/n)
PMT: Regular Payment, PV: Principal/Loan Amount, FV: Future Value, APR: Annual Percentage Rate (decimal), n: Payments/compounding periods per year, Y: Years

Investment Returns

Total Return: $ \frac{(A - P)}{P} \times 100\% $ (A - P) / P * 100%
Annual Return: $ \left(\frac{A}{P}\right)^{(1/Y)} - 1 $ (decimal, multiply by 100 for %) (A / P)^(1/Y) - 1

Descriptive Statistics & Change

Measures of Center

Mean ($\mu$ or $\bar{x}$): $\frac{\sum x}{n} = \frac{\text{Sum of values}}{\text{Count of values}}$ Mean (μ or x̄): Sum of values / Count of values
Median: Middle value of sorted data.
Mode: Most frequent value(s).

Measures of Variation

Range: $\text{Highest Value} - \text{Lowest Value}$ Range: Highest Value - Lowest Value
Standard Deviation ($\sigma$ or s): Measures data spread.
Range Rule of Thumb: $ \text{Std Dev} \approx \frac{\text{Range}}{4} $ Std Dev ≈ Range / 4

Change Metrics

Absolute Change: $\text{New Value} - \text{Reference Value}$ Absolute Change: New Value - Reference Value
Relative Change: $ \frac{\text{Absolute Change}}{\text{Reference Value}} \times 100\% $ Relative Change: (Absolute Change / Reference Value) * 100%
Or: $ \frac{(\text{New Value} - \text{Reference Value})}{\text{Reference Value}} \times 100\% $ Or: ((New Value - Reference Value) / Reference Value) * 100%

Distributions

Standard Score (z-score)

$ z = \frac{(x - \mu)}{\sigma} $ z = (x - μ) / σ
x: data value
μ: population mean
σ: population std dev

Empirical Rule (68-95-99.7)

Approx. 68% of data within 1 std dev of the mean (μ ± σ).
Approx. 95% of data within 2 std dev of the mean (μ ± 2σ).
Approx. 99.7% of data within 3 std dev of the mean (μ ± 3σ).

Relevant Excel Formulas

Note: Replace range with cells (e.g., A1:A10), n with number, p with power, x with value, μ with mean, σ with std dev, y's/x's with known data ranges, rate with periodic rate, nper with total periods, pmt with periodic payment, pv with present value, fv with future value, nominal_rate with APR, npery with compoundings per year.

Basic Calculations

Power: =POWER(n, p) or =n^p
Log (Base 10): =LOG10(n)
Log (Natural): =LN(n)
Exponential ($e^n$): =EXP(n)

Descriptive Statistics

Mean: =AVERAGE(range)
Median: =MEDIAN(range)
Mode (Single): =MODE.SNGL(range)
Mode (Multi): =MODE.MULT(range)
Min: =MIN(range)
Max: =MAX(range)
Range: =MAX(range)-MIN(range)
Std Dev (Sample): =STDEV.S(range)
Std Dev (Pop.): =STDEV.P(range)
Quartiles: =QUARTILE.INC(range, q) (q=0,1,2,3,4)

Normal Distribution

Z-Score: =STANDARDIZE(x, μ, σ)
Area left of Z: =NORM.S.DIST(z, TRUE)
Z from Area: =NORM.S.INV(area)
Area left of x: =NORM.DIST(x, μ, σ, TRUE)
x from Area: =NORM.INV(area, μ, σ)

Growth/Trend

Linear Forecast: =FORECAST.LINEAR(x, known_y's, known_x's)
Exponential Forecast: =GROWTH(known_y's, [known_x's], [new_x's], [const])

Financial Functions

Present Value: =PV(rate, nper, pmt, [fv], [type])
Future Value: =FV(rate, nper, pmt, [pv], [type])
Payment: =PMT(rate, nper, pv, [fv], [type])
Number of Periods: =NPER(rate, pmt, pv, [fv], [type])
Rate: =RATE(nper, pmt, pv, [fv], [type], [guess])
Effective Rate (APY): =EFFECT(nominal_rate, npery)

Standard Normal (Z) Table - Percentiles

Table shows percentile (area to LEFT × 100) for Z-score.

Z	Pct(%)	Z	Pct(%)	Z	Pct(%)	Z	Pct(%)
-3.0	0.13	-1.4	8.08	0.2	57.93	1.8	96.41
-2.9	0.19	-1.3	9.68	0.3	61.79	1.9	97.13
-2.8	0.26	-1.2	11.51	0.4	65.54	2.0	97.72
-2.7	0.35	-1.1	13.57	0.5	69.15	2.1	98.21
-2.6	0.47	-1.0	15.87	0.6	72.57	2.2	98.61
-2.5	0.62	-0.9	18.41	0.7	75.80	2.3	98.93
-2.4	0.82	-0.8	21.19	0.8	78.81	2.4	99.18
-2.3	1.07	-0.7	24.20	0.9	81.59	2.5	99.38
-2.2	1.39	-0.6	27.43	1.0	84.13	2.6	99.53
-2.1	1.79	-0.5	30.85	1.1	86.43	2.7	99.65
-2.0	2.28	-0.4	34.46	1.2	88.49	2.8	99.74
-1.9	2.87	-0.3	38.21	1.3	90.32	2.9	99.81
-1.8	3.59	-0.2	42.07	1.4	91.92	3.0	99.87
-1.7	4.46	-0.1	46.02	1.5	93.32
-1.6	5.48	0.0	50.00	1.6	94.52
-1.5	6.68	0.1	53.98	1.7	95.54