Essential Mathematical Calculations for Economics and Management

NoteLearning objectives

By the end of this module, you should be able to:

• Apply the correct order of operations in numerical and business calculations.
• Use absolute value, powers, and square roots and know their properties.
• Manipulate fractions, percentages, and index numbers.
• Solve everyday problems with the rule of proportionality.

1 Diagnostic Activity

Attempt these questions without consulting the lesson.

1. Evaluate \(4 + 3 \times 5\).
2. Simplify \(\dfrac{18}{24}\).
3. A price rises from €80 to €92. Calculate the percentage increase.

NoteCheck your answers

1. \(4 + 3 \times 5 = 4 + 15 = 19\) (multiplication before addition).
2. \(\dfrac{18}{24} = \dfrac{3}{4}\) (divide numerator and denominator by 6).
3. \(\dfrac{92 - 80}{80} \times 100 = \dfrac{12}{80} \times 100 = 15\%\).

Use your results to identify the topics that require particular attention before reading the Core Concepts section.

2 Opening Problem

A clothing retailer increases the price of a jacket by 20% for the new season. At the end of the season, the retailer applies a 20% discount to the new price.

Does the jacket return to its original price?

Write down your prediction before continuing. In particular, is the overall change 0%, positive, or negative?

3 Core Concepts

Mathematics is an indispensable universal language in economics and management. Behind every decision about production, pricing, investment, or financing lie calculations — sometimes simple, sometimes more involved, but always based on the fundamental rules of real numbers. Mastering these basics is therefore not merely an academic exercise: it is a daily tool for reasoning and professional effectiveness.

1. Order of Operations

The order of operations is a fundamental rule of mathematics that lets us evaluate a numerical expression correctly. The order in which we perform the calculations can completely change the result: we cannot add, multiply, or raise to a power in just any order.

TipKey rule — PEMDAS

1. Parentheses: \(( )\), \([ ]\), \(\{ \}\)
2. Exponents (powers and roots)
3. Multiplication and Division (left to right)
4. Addition and Subtraction (left to right)

Warning

A calculator does not always apply priorities correctly, especially without parentheses. When in doubt, add parentheses.

CautionExercise 1 — evaluate the following (click for solutions)

1. \(8 \div 2 \times (2 + 2)\) \(\rightarrow\) \(16\)
2. \(4 \times (5 - (9 - 2))\) \(\rightarrow\) \(-8\)
3. \(10 + 2 \times (-1) \times 2^{3}\) \(\rightarrow\) \(-6\)
4. \(\big(10 + 2 \times (-1) \times 2^{3} - 4 \times (5 - (9 - 2))\big)\) \(\rightarrow\) \(2\)
5. \((10 + 2 \times (-1) \times 2^{3} - 4 \times (5 - (9 - 2))) \div 8\) \(= \dfrac{1}{4}\)
6. \(\big[12 - (3 + 2^{3} \times (4 - 1)) \div 3\big] \times 2 - 5^{2}\) \(\rightarrow\) \(-19\)

CautionExercise 2 — place parentheses to make each equality true

1. \(3 + 5 \times 2 = 16\) \(\rightarrow\) \((3 + 5) \times 2 = 16\)
2. \(10 - 2 + 4 = 4\) \(\rightarrow\) \(10 - (2 + 4) = 4\)

2. Absolute Value

The absolute value of a real number \(a\), written \(|a|\), is its distance from zero:

• \(|a| = a\) if \(a \geq 0\);
• \(|a| = -a\) if \(a < 0\).

TipUseful properties

• \(|a| \geq 0\)
• \(|a| = |{-a}|\)
• \(|a \times b| = |a| \times |b|\)
• \(\left|\dfrac{a}{b}\right| = \dfrac{|a|}{|b|}\) if \(b \neq 0\)
• \(|a| = 0 \iff a = 0\)
• \(|a| = |b| \iff a = b \text{ or } a = -b\)

Warning

In general \(|a + b| \neq |a| + |b|\) (this is the triangle inequality: \(|a + b| \leq |a| + |b|\)).

Applications:

• Thermal comfort — gap between actual and target temperature: \(|T^{\text{actual}} - T^{\text{target}}|\)
• Quality control — with measured length \(L_r\) and target \(L_c\): overall defect \(= |L_r - L_c| + |l_r - l_c|\)

CautionSynthesis exercise (click for solution)

\[ -2 \times \big|-4 - |1 - 5|\big| - \big|(-2)^{2}\big| \times |4 - 7| + |-3|^{2} \]

Step by step:

• \(|1 - 5| = 4\), so \(\big|-4 - 4\big| = 8\) and \(-2 \times 8 = -16\);
• \(|(-2)^2| = 4\) and \(|4 - 7| = 3\), so \(-4 \times 3 = -12\);
• \(|-3|^2 = 9\).

Total: \(-16 - 12 + 9 = \mathbf{-19}\).

3. Powers

A power \(a^{n}\) is the product of \(a\) by itself \(n\) times.

TipKey properties

• \(a^{n} \times a^{m} = a^{n+m}\)
• \(\dfrac{a^{n}}{a^{m}} = a^{n-m}\)
• \(a^{0} = 1\)
• \(a^{-n} = \dfrac{1}{a^{n}}\)
• \((a^{n})^{m} = a^{nm}\)
• \(a^{1/2} = \sqrt{a}\)

Warning

\(a^{n} + a^{m} \neq a^{n+m}\).

Economic application — compound (exponential) growth: each increase applies to an already-increased base, e.g. compound interest and capitalisation (developed in Modules 4 and 5).

CautionExercise — evaluate (click for solution)

\[ \left( \frac{2^{3} \times 4^{-2} \times 8}{2^{-1} \times \sqrt{64} \times 8} \right)^{2} \]

Rewrite in base \(2\): numerator \(2^3 \times 2^{-4} \times 2^3 = 2^{2}\); denominator \(2^{-1} \times 2^{3} \times 2^{3} = 2^{5}\). Ratio \(2^{2-5} = 2^{-3}\), squared \(2^{-6} = \tfrac{1}{64}\).

Result: \(\dfrac{1}{64}\).

4. Square Root

The square root of \(a \geq 0\) is the positive number \(b\) such that \(b^{2} = a\).

TipUseful properties

• \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\)
• \(\sqrt{\dfrac{a}{b}} = \dfrac{\sqrt{a}}{\sqrt{b}}\) (for \(b > 0\))

Warning

\(\sqrt{a + b} \neq \sqrt{a} + \sqrt{b}\).

5. Fractions

A fraction is a division of two real numbers, written \(\dfrac{a}{b}\) with \(b \neq 0\).

TipFundamental rules

• Addition / subtraction: with the same denominator, add the numerators; otherwise reduce to a common denominator (the LCM is convenient).
• Multiplication: multiply numerators together and denominators together.
• Division: multiply by the reciprocal of the second fraction.

CautionExercise — evaluate (click for solution)

\[ \left( \frac{3}{4} + \frac{2}{3} \times \left(1 - \frac{5}{6}\right) \right) \div \left( \frac{7}{8} - \frac{1}{2} \right) \]

Inner bracket \(1 - \tfrac56 = \tfrac16\), so \(\tfrac23 \times \tfrac16 = \tfrac{2}{18} = \tfrac19\). Then \(\tfrac34 + \tfrac19 = \tfrac{27}{36} + \tfrac{4}{36} = \tfrac{31}{36}\). Divisor \(\tfrac78 - \tfrac12 = \tfrac38\). Finally \(\tfrac{31}{36} \div \tfrac38 = \tfrac{31}{36} \times \tfrac83 = \tfrac{248}{108} = \tfrac{62}{27}\).

Result: \(\dfrac{62}{27}\).

6. Percentages and Index Numbers

A percentage is a fraction with denominator \(100\). For example, \(25\%\) of \(80\) is \(\dfrac{25}{100} \times 80 = 20\).

TipPercentage change formula

\[ \Delta\% = \frac{V_{\text{final}} - V_{\text{initial}}} {V_{\text{initial}}} \times 100 \]

Applying a change of rate \(t\) multiplies the value by the factor \((1 + t)\) (where \(t\) is expressed as a decimal). Two successive changes \(t_1\) then \(t_2\) multiply by \((1 + t_1)(1 + t_2)\) — they do not simply add.

CautionDerivation — why successive discounts multiply (click to expand)

Applying a discount of rate \(t\) multiplies the price by the factor \((1 - t)\). Applying two successive discounts \(t_1\) then \(t_2\) multiplies by \((1 - t_1)\) then by \((1 - t_2)\), hence by the product \((1 - t_1)(1 - t_2)\). The overall discount rate \(T\) satisfies \(1 - T = (1 - t_1)(1 - t_2)\), so

\[ T = 1 - (1 - t_1)(1 - t_2). \]

This is not \(t_1 + t_2\): successive discounts do not simply add, because each rate applies to a base already reduced by the previous one.

Index numbers:

\[ \text{Index} = \left( \frac{\text{current value}}{\text{reference value}} \right) \times 100 \]

• Index \(= 100\): stable; Index \(> 100\): increase; Index \(< 100\): decrease.

7. Proportionality Rule

If two ratios are equal, the quantities are proportional:

\[ \frac{a}{b} = \frac{c}{x} \;\Rightarrow\; x = \frac{b \times c}{a}. \]

This “rule of three” is the everyday tool for scaling recipes, converting prices, comparing unit costs, or splitting quantities proportionally.

CautionExercise — unit price (click for solution)

If \(3\) kg of coffee cost €\(27\), what do \(5\) kg cost at the same rate?

\[ x = \frac{27 \times 5}{3} = 45 \text{ €}. \]

4 Worked Example

Successive discounts — a coat sale

A coat costs €\(200\). It receives a \(10\%\) sale discount, a \(5\%\) loyalty discount, and a \(15\%\) birthday discount.

Step 1 — Compute the final price.

Each discount applies to the already-reduced price, so we multiply by the successive factors:

\[ 200 \times 0.90 \times 0.95 \times 0.85 = 145.35 \text{ €}. \]

Step 2 — Compute the overall discount rate.

\[ T = 1 - 0.90 \times 0.95 \times 0.85 = 1 - 0.72675 = 0.27325, \]

i.e. about \(\mathbf{27.33\%}\) — not \(30\%\) (which would be the naive sum \(10 + 5 + 15\)).

Step 3 — Express as an index number.

\[ \text{Index} = \frac{145.35}{200} \times 100 = 72.675. \]

The coat now costs \(72.7\%\) of its original price — the index fell below \(100\), confirming a decrease.

Result: The three discounts together reduce the price by \(27.33\%\), not \(30\%\), because each percentage applies to a smaller base.

5 Guided Practice

A company’s revenue rises from €\(400{,}000\) to €\(460{,}000\).

1. Calculate the absolute change.
2. Calculate the relative change.
3. Express the relative change as a percentage.
4. Interpret the result in one sentence.

TipHint

The percentage change formula is:

\[ \Delta\% = \frac{\text{new value} - \text{initial value}} {\text{initial value}} \times 100. \]

Make sure you divide by the initial value (€\(400{,}000\)), not the final value.

CautionShow the solution

1. Absolute change: \(460{,}000 - 400{,}000 = 60{,}000\) €.
2. Relative change: \(\dfrac{60{,}000}{400{,}000} = 0.15\).
3. Percentage: \(0.15 \times 100 = 15\%\).
4. Revenue increased by \(15\%\) relative to the previous period.

6 Interactive Exploration

Use the sliders to explore what happens when you apply an increase and then a decrease of the same percentage to a starting value.

viewof initialValue = Inputs.range([10, 1000], {
  value: 100,
  step: 10,
  label: "Initial value (€)"
})

viewof increaseRate = Inputs.range([0, 100], {
  value: 20,
  step: 1,
  label: "Increase (%)"
})

viewof decreaseRate = Inputs.range([0, 100], {
  value: 20,
  step: 1,
  label: "Decrease (%)"
})

valueAfterIncrease = initialValue * (1 + increaseRate / 100)
finalValue = valueAfterIncrease * (1 - decreaseRate / 100)
overallChange = 100 * (finalValue - initialValue) / initialValue

html`
<div class="metric-grid">
  <div class="metric-card">
    <span class="metric-label">Initial value</span>
    <strong>${initialValue.toFixed(2)} €</strong>
  </div>
  <div class="metric-card">
    <span class="metric-label">After the increase</span>
    <strong>${valueAfterIncrease.toFixed(2)} €</strong>
  </div>
  <div class="metric-card">
    <span class="metric-label">Final value</span>
    <strong>${finalValue.toFixed(2)} €</strong>
  </div>
  <div class="metric-card">
    <span class="metric-label">Overall change</span>
    <strong>${overallChange.toFixed(2)}%</strong>
  </div>
</div>
`

Investigation

1. Set both rates to \(20\%\). What is the overall change? Is it \(0\%\)?
2. Repeat with \(10\%\), \(30\%\), and \(50\%\). Does the pattern hold?
3. Does the initial value affect the overall percentage change? Explain.
4. Find a decrease rate that exactly reverses an increase of \(25\%\). Why is this rate not \(25\%\)?

7 Applied Problem

A retail chain monitors the performance of three product categories over a quarter. The data are as follows:

Category	Revenue at start	Revenue at end
Electronics	€\(120{,}000\)	€\(138{,}000\)
Clothing	€\(85{,}000\)	€\(79{,}050\)
Food	€\(200{,}000\)	€\(210{,}000\)

1. For each category, calculate the percentage change in revenue. State whether it is an increase or a decrease.
2. Which category has the largest absolute change? Which has the largest relative change?
3. The chain targets an overall revenue index of \(105\) (base = start of quarter). Did it reach this target?

8 Common Errors

WarningCommon error — dividing by the final value

A student calculates the percentage change from €\(80\) to €\(100\) as:

\[ \frac{100 - 80}{100} \times 100 = 20\%. \]

The correct calculation uses the initial value as the reference:

\[ \frac{100 - 80}{80} \times 100 = 25\%. \]

Dividing by the final value produces a smaller percentage and is mathematically incorrect. The initial value is always the reference base for a percentage change.

9 Check Your Understanding

1. Calculation: Evaluate \(\big[3 + 2^{2} \times (8 - 5)\big] \div 5\).
2. Conceptual: Why do two successive discounts of \(10\%\) and \(10\%\) not produce a total discount of \(20\%\)?
3. Applied: A firm’s costs rise from €\(50{,}000\) to €\(53{,}500\). Express this as both a percentage change and an index number (base = initial cost).
4. Error diagnosis: A student claims that a \(+30\%\) increase followed by a \(-30\%\) decrease returns to the original value. What is the actual overall change? What did the student overlook?
5. Find the value: What single percentage discount is equivalent to successive discounts of \(15\%\) and \(10\%\)?

10 Summary

1. Order of operations governs every calculation: always resolve parentheses first, then exponents, then multiplication/division, then addition/subtraction. Errors in order of operations contaminate every subsequent step.
2. Successive percentage changes multiply, they do not add: applying a +\(t_1\%\) increase then a −\(t_2\%\) decrease gives an overall factor of \((1 + t_1)(1 - t_2)\), which is almost never equal to \(1 + t_1 - t_2\).
3. Always divide by the initial value for a percentage change: the reference base is where you started, not where you ended up.

11 Independent Work

• Complete exercises 1 to 10 in the exercise bank.
• Correct your answers using the detailed solutions.
• Record any errors in your error log (error / cause / corrected method / prevention rule).
• Review the glossary entries for: percentage, index number, proportionality, absolute value, common factor (LCM/GCF).

12 Continue

Continue to Module 2 →