Skip to content

Formulas: finance and economics, grade 12 (3 units)

Revenue and profit: a quadratic model

  • The revenue – p\displaystyle p the price of a unit, q\displaystyle q the number of units sold
    R=p⋅q\displaystyle R=p\cdot q
  • The profit – the revenue minus the cost
    P=R−C\displaystyle P=R-C
  • A parabola – the revenue or the profit as a function of the price
    y=ax2+bx+c\displaystyle y=ax^{2}+bx+c
  • The vertex’s x\displaystyle x – where the revenue or the profit is greatest (when a<0\displaystyle a<0)
    x=−b2a\displaystyle x=-\frac{b}{2a}a≠0\displaystyle a\neq0
  • The quadratic formula – the prices where the revenue or the profit is 0
    x1,2=−b±b2−4ac2a\displaystyle x_{1,2}=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}

Linear programming

  • A constraint – an inequality; the points that satisfy it are a half-plane
    ax+by≤c\displaystyle ax+by\le c
  • The objective function – the greatest profit is at one of the feasible region’s corner points
    P=ax+by\displaystyle P=ax+by

Analytic geometry

  • The distance between the points (x1,y1)\displaystyle (x_1,y_1) and (x2,y2)\displaystyle (x_2,y_2)
    d=(x2−x1)2+(y2−y1)2\displaystyle d=\sqrt{(x_2-x_1)^{2}+(y_2-y_1)^{2}}
  • The midpoint of the segment between the two points
    M=(x1+x22,y1+y22)\displaystyle M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)
  • The slope of the line through the two points
    m=y2−y1x2−x1\displaystyle m=\frac{y_2-y_1}{x_2-x_1}
  • The equation of the line through the point (x1,y1)\displaystyle (x_1,y_1) with slope m\displaystyle m
    y−y1=m(x−x1)\displaystyle y-y_1=m(x-x_1)
  • Parallel lines – the same slope
    m1\displaystyle m_1 =\displaystyle = m2\displaystyle m_2

The normal distribution

  • In a normal distribution – μ\displaystyle \mu the mean, σ\displaystyle \sigma the standard deviation: about 68% of the data within one standard deviation of the mean (34% on each side)
    P(μ−σ<x<μ+σ)≈68%\displaystyle P(\mu-\sigma<x<\mu+\sigma)\approx68\%
  • About 95% within two standard deviations of the mean (another 13.5% on each side)
    P(μ−2σ<x<μ+2σ)≈95%\displaystyle P(\mu-2\sigma<x<\mu+2\sigma)\approx95\%
  • About 2.5% above μ+2σ\displaystyle \mu+2\sigma, and about 2.5% below μ−2σ\displaystyle \mu-2\sigma
    P(x>μ+2σ)≈2.5%\displaystyle P(x>\mu+2\sigma)\approx2.5\%
  • The standard score of the value x\displaystyle x – how many standard deviations it is above the mean (or below it, when z<0\displaystyle z<0)
    z=x−μσ\displaystyle z=\frac{x-\mu}{\sigma}
  • The value whose standard score is z\displaystyle z
    x=μ+z⋅σ\displaystyle x=\mu+z\cdot\sigma