Notes on Business Mathematics & General Mathematics

BUSINESS MATHEMATICS

LESSON 1: FUNDAMENTAL OPERATIONS ON: FRACTIONS DECIMALS AND PERCENTAGE

• Focuses on operations and conversions among fractions, decimals, and percentages.
• Key tasks include:
  • Performing operations on fractions (addition, subtraction, multiplication, division).
  • Performing operations on decimals.
  • Working with percentages (conversion and calculations).
  • Converting between fractions, decimals, and percentages.
• Core formulas and rules:
  • Fraction operations (with common denominator where needed):
  • Addition: \frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}
  • Subtraction: \frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd}
  • Multiplication: \frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}
  • Division: \frac{a}{b} \div \frac{c}{d} = \frac{ad}{bc}
  • Converting fractions to decimals: compute the division \frac{a}{b} = a \div b.
  • Decimal to fraction: write the decimal with a power of 10 in the denominator, then simplify to lowest terms.
  • Fraction/Decimal to Percent: multiply by 100%: \frac{a}{b} \times 100\%; Decimal to Percent: d \times 100\%; Percent to Decimal: p\% = \frac{p}{100}.
  • Converting between all three representations:
  • Fraction ⇄ Decimal ⇄ Percent are just different forms of the same quantity.
• Illustrative examples:
  • Addition: \frac{3}{5} + \frac{2}{3} = \frac{19}{15}\,.
  • Subtraction: \frac{4}{7} - \frac{1}{3} = \frac{5}{21}\,.
  • Multiplication: \frac{2}{3} \cdot \frac{3}{4} = \frac{1}{2}\,.
  • Division: \frac{2}{3} \div \frac{4}{5} = \frac{5}{6}\,.
  • Decimal ⇄ Fraction: \frac{3}{4} = 0.75 = 75\%.
  • Decimal to Fraction: 0.6 = \frac{3}{5}\,;\;0.125 = \frac{1}{8}.
  • Percent to Decimal: 25\% = 0.25\; ;\; 0.83 = 83\%.
• Conversions and relationships:
  • 1 = 100\%\;\Rightarrow\; x\% = \frac{x}{100}\text{ (as decimal)}
  • \text{Decimal } d = \frac{p}{100} \Rightarrow p\% = d\times 100\%
• Practical tips:
  • Align denominators before adding/subtracting fractions.
  • Check simplifications after every operation.
  • Use conversions to verify answers (e.g., convert to a common form).

LESSON 2: RATIO & PROPORTION

• Key concepts:
  • Ratio: a comparison expressed as a:b (or a/b).
  • Proportion: an equation that expresses two ratios as equal: a:b = c:d, with cross-multiplication ad = bc.
  • Base, Rate, and Percentage in ratio-context:
  • Base amount (the reference quantity).
  • Rate (the percentage or multiplier applied to the base).
  • Percentage (the rate expressed as a percent).
  • Practical applications include mixing ratios, scaling recipes, and financial calculations (percent changes).
• Proportion and solving use cross-multiplication:
  • If a:b = c:d, then ad = bc.
  • Example: If the ratio of apples to oranges is 3:2 and there are 15 apples, oranges = 15 × (2/3) = 10. Solve via cross-multiplication: (\frac{3}{2} = \frac{15}{x} \Rightarrow x = 10).
• Ratios and percent examples:
  • Percentage as a ratio: 5% corresponds to 5 parts per 100, i.e., 5/100.
  • Percentage of a total: if B is the base and r% is the rate, final value V = B(1 + r/100) for an increase, or V = B(1 - r/100) for a decrease.
  • Example: 5% of 200 = 200 \times \frac{5}{100} = 10.
  • Increasing a base by a percentage: V = B(1 + r/100).
• Worked cross-checks:
  • If you know a target for one part of a ratio, you can compute the other part using ad = bc.
• Summary equations:
  • Proportion: \frac{a}{b} = \frac{c}{d} \quad\Rightarrow\quad ad = bc
  • Base–Rate–Percentage relation: V = B\left(1 \pm \frac{r}{100}\right), \quad r\text{ in percent}

LESSON 3: BUYING AND SELLING

• Core concepts:
  • Markup: the amount added to cost to reach selling price.
  • Margin: profit as a portion of selling price or cost (two common definitions).
  • Markdown: reduction from original price to a lower price.
  • Trade discounts: reductions given by suppliers to retailers for bulk purchases.
• Formulas:
  • Cost (C), Selling Price (SP):
  • Markup (amount): M = SP - C
  • Markup percentage on cost: \text{Markup\% on cost} = \frac{SP - C}{C} \times 100\%
  • Margin (profit as a percentage):
  • Margin on selling price: \text{Margin (on SP)} = \frac{SP - C}{SP} \times 100\%
  • Margin on cost: \text{Margin (on cost)} = \frac{SP - C}{C} \times 100\%
  • Markdown: \text{Markdown} = \text{Original Price} - \text{New Price}
  • Markdown percentage: \text{Markdown\%} = \frac{\text{Markdown}}{\text{Original Price}} \times 100\%
  • Trade discounts (net price):
  • If List Price = $L$ and there is a discount $d\%$: \text{Net Price} = L\left(1 - \frac{d}{100}\right)
  • For multiple discounts, apply sequentially: \text{Net Price} = L\left(1 - \frac{d1}{100}\right)\left(1 - \frac{d2}{100}\right)\cdots
• Illustrative examples:
  • If cost C = 80 and selling price SP = 100, then markup M = 20; markup on cost = (\frac{20}{80} = 25\%).
  • Margin on selling price: (\frac{SP - C}{SP} = \frac{20}{100} = 20\%).
  • Markdown example: Original Price 100, New Price 85 → Markdown = 15; Markdown% = (\frac{15}{100} \times 100\% = 15\%).
  • Trade discount example: List Price 100, trade discount 15% → Net Price = 85.
• Practical relevance:
  • Accurate calculation of costs, pricing, and profitability.
  • Understanding how markups and margins affect business decisions.
  • Use of discounts and markdowns in competitive pricing while maintaining profitability.

GENERAL MATHEMATICS

LESSON 1: FUNCTIONS AND RELATIONS

• Distinction:
  • A relation is a set of ordered pairs (x, y) that defines how two sets relate.
  • A function is a special relation where every input x from the domain maps to exactly one output y.
• Key terms:
  • Domain: all possible input values x for which the relation is defined.
  • Range: all possible output values y produced by the relation.
  • Notation: f: X → Y denotes a function mapping from domain X to codomain Y.
  • Representations: explicit form f(x), mapping diagram, and graph.
• Examples:
  • Relation R: { (1,2), (2,3), (3,2) } is a function (each x has a single y).
  • Relation R: { (1,2), (1,3) } is not a function (x = 1 maps to two y-values).
• Foundational ideas:
  • Functions are the fundamental way to model rules that assign outputs to inputs.
  • Domain and range determine where a function is defined and what outputs it can produce.

LESSON 2: EVALUATION OF FUNCTIONS

• Purpose:
  • Evaluate a function at a specified input value: compute f(a).
• Typical notation:
  • Given f(x), evaluate at x = a: f(a).
• Example:
  • If f(x) = 3x - 7, then f(4) = 3(4) - 7 = 12 - 7 = 5.
• Considerations:
  • Some inputs may be outside the domain; ensure a is in the domain before evaluating.
  • If the function has piecewise definitions, choose the piece that applies to the input a.

LESSON 3: OPERATION OF FUNCTIONS

• Operations on functions:
  • Sum: (f + g)(x) = f(x) + g(x)
  • Difference: (f - g)(x) = f(x) - g(x)
  • Product: (f \cdot g)(x) = f(x) \cdot g(x)
  • Quotient: (f \div g)(x) = \frac{f(x)}{g(x)} with $g(x) \neq 0$.
• Examples:
  • If $f(x) = x^2$ and $g(x) = 2x$, then
  • $(f + g)(x) = x^2 + 2x$
  • $(f - g)(x) = x^2 - 2x$
  • $(f \cdot g)(x) = x^2 \cdot 2x = 2x^3$
  • $(f \div g)(x) = \frac{x^2}{2x} = \frac{x}{2}$ for $x \neq 0$.
• Domain considerations:
  • For division, avoid values of x where $g(x) = 0$.
  • Ensure that the resulting expression is defined for the chosen x.

LESSON 4: COMPOSITE FUNCTIONS

• Definition:
  • The composite function (f \circ g)(x) = f(g(x)).
• Domain considerations:
  • The domain of $f \circ g$ consists of all $x$ in the domain of $g$ such that $g(x)$ lies in the domain of $f$.
• Example:
  • If $f(x) = x^2$ and $g(x) = x + 1$, then $(f \circ g)(x) = (x + 1)^2$.
• Properties:
  • Associativity (where defined): (f \circ g) \circ h = f \circ (g \circ h) provided the expressions are defined.
• Visual intuition:
  • First apply the inner function to the input, then apply the outer function to the result.
• Practical note:
  • Composite functions are central to modeling multi-step processes and transformations in mathematics and applied contexts.