Solving Business Problems with Linear Equations

What's the Big Idea?

Imagine you decide to start a small business selling freezits in your neighbourhood. You have to buy the plastic sleeves, the ingredients for the juice, and you need to pay for the electricity to run the freezer. You also need to decide on a selling price for each freezit. The big question is: how many freezits do you need to sell just to cover all your costs? If you sell fewer, you make a loss. If you sell more, you start making a profit.

This is a classic business problem that can be solved easily with a simple but powerful mathematical tool: a linear equation. At its heart, a linear equation helps us find an unknown value when we know the relationship between other values. It’s like a balanced scale, where both sides must always be equal. In business, we use it to find the "break-even point," calculate profit, predict costs, and make smart pricing decisions.

This section is your first step into the world of business mathematics. We will learn how to read a business problem, translate it into the language of algebra, and solve it step-by-step. Mastering this skill will give you a clear and simple way to make confident financial decisions, whether you are running a tuckshop, a poultry project, or planning your own future business.

Key Vocabulary

• Variable: A symbol, usually a letter like x, that represents an unknown number we are trying to find.
• Constant: A fixed number in an equation that does not change.
• Equation: A mathematical statement showing that two expressions are equal, connected by an equals sign (=).
• Linear Equation: An equation that describes a straight-line relationship between variables, where the variable is not raised to a power higher than one.
• Break-even Point: The exact point where a business's total income equals its total costs, resulting in zero profit and zero loss.

Simple Linear Equation

A linear equation is a straightforward way to express a relationship. The most basic form looks like this: ax + b = c

• x is our variable – the unknown value we want to find.
• a is the coefficient – the number multiplying our variable.
• b and c are constants – the fixed numbers in the problem.

Our goal is always to figure out the value of x.

The Golden Rule: Keep it Balanced!

Think of an equation as a perfectly balanced scale. If you add a weight to one side, you must add the exact same weight to the other side to keep it balanced. The same is true in algebra. Whatever you do to one side of the equals sign (=), you must do the exact same thing to the other side. This is the most important rule for solving equations.

Step-by-Step: How to Solve for the Unknown

Our mission is to get the variable (x) all by itself on one side of the equation. We do this in two simple steps:

1. Isolate the Variable Term: Move any constants that are on the same side as the variable to the other side. You do this by performing the opposite operation. If a number is added, you subtract it from both sides. If it's subtracted, you add it to both sides.
2. Solve for the Variable: Once the variable term is alone, you can solve for the variable itself. If the variable is multiplied by a number (its coefficient), you divide both sides by that number.

Let's see it in action: 2x + 4 = 10

1. Isolate the term '2x': The constant on the same side is + 4. The opposite is - 4. So, we subtract 4 from both sides.
   2x + 4 - 4 = 10 - 4
   2x = 6
2. Solve for 'x': x is being multiplied by 2. The opposite is to divide by 2. So, we divide both sides by 2.
   2x / 2 = 6 / 2
   x = 3

We have successfully found our unknown value!

Worked Example: "Putting it to Work in Zimbabwe"

Scenario: Rudo runs a small poultry business in Mutare, where she raises and sells broiler chickens. She wants to calculate her break-even point for the month.

Her Costs:

• Fixed Costs: These are costs that don't change no matter how many chickens she raises. This includes rent for the chicken run and electricity. Her fixed costs are $60 USD per month.
• Variable Costs: These costs depend on the number of chickens. This includes day-old chicks, feed, and vaccines. This costs her $2 USD per chicken.

Her Revenue:

• Selling Price: She sells each fully grown chicken for $5 USD.

The Question: How many chickens (x) does Rudo need to sell in a month to cover all her costs?

1. Step 1: Set up the equation.
   The break-even point is where Total Revenue equals Total Costs.
   Total Revenue = Selling Price × Number of Chickens = 5x
   Total Costs = (Variable Cost × Number of Chickens) + Fixed Costs = 2x + 60
   So, the equation is: 5x = 2x + 60
2. Step 2: Isolate the variable term.
   We need to get all the x terms on one side. We can subtract 2x from both sides to keep the equation balanced.
   5x - 2x = 2x - 2x + 60
   3x = 60
3. Step 3: Solve for the variable, x.
   x is being multiplied by 3. To get x alone, we divide both sides by 3.
   3x / 3 = 60 / 3
   x = 20

Answer: Rudo must raise and sell 20 chickens in a month to break even. If she sells more than 20, she will make a profit. If she sells fewer than 20, she will make a loss.

Common Mistakes to Avoid

• Forgetting to balance: Only performing an operation (like subtracting a number) on one side of the equation. This is the most common error.
• Sign Errors: When you move a positive term across the equals sign, it becomes negative. Forgetting this can lead to the wrong answer.
• Confusing Costs: Mixing up fixed and variable costs when setting up the equation from a word problem.

Using Ratios, Rates, and Proportions for Business Comparison

What's the Big Idea?

Have you ever heard a farmer in Mashonaland West proudly say they harvested 8 tonnes of maize per hectare? Or seen a sign at a bureau de change in Harare showing the exchange rate between the USD and the new ZiG? Maybe you've followed a recipe for maputi that says to use one cup of salt for every 20 cups of maize. These are all real-life examples of ratios, rates, and proportions. They are the mathematical tools we use to compare things.

In business, comparison is everything. We need to know if we are doing well compared to last month, if our prices are fair, or if our resources are being used efficiently. Ratios and rates help us do just that. A ratio compares quantities of the same kind (e.g., profit to investment), while a rate compares different kinds of quantities (e.g., cost in dollars per kilogram).

Key Vocabulary

• Ratio: A comparison of two or more quantities that are measured in the same unit.
• Rate: A special ratio that compares two quantities with different units (e.g., kilometres per hour).
• Percentage: A special kind of ratio where a quantity is compared to 100.
• Proportion: A statement that two ratios or rates are equal to each other.
• Direct Proportion: A relationship where two quantities increase or decrease together at the same rate.
• Inverse Proportion: A relationship where one quantity increases while the other quantity decreases.

Ratios: The Art of Comparison

A ratio compares two quantities. If a bag of grain contains 10kg of maize and 5kg of sorghum, the ratio of maize to sorghum is 10 to 5. We can write this as 10:5 or 10/5. Just like fractions, we should always simplify ratios. The simplest form of the ratio is 2:1.

Fractions, Decimals, and Percentages

These are different ways to express parts of a whole.

• Fraction to Percentage: (Numerator ÷ Denominator) × 100. So, (1 ÷ 2) × 100 = 50%.
• Percentage to Decimal: Divide by 100. So, 50% ÷ 100 = 0.5.

Direct vs. Inverse Proportion

1. Direct Proportion: More of one means more of the other. (e.g., the more money you pay for fuel, the more litres you get).
2. Inverse Proportion: More of one means less of the other. (e.g., the more builders working on a wall, the less time it will take).

Worked Example:

Scenario: Two friends, Tatenda and Farai, start a car washing business. Tatenda invests 100 USD. Farai contributes 50 USD. They agree to share any profit according to the ratio of their investment. They make a profit of $90 USD.

The Question: How should they divide the $90 profit fairly?

1. Step 1: Write down the investment ratio.
   Ratio = 100 : 50
2. Step 2: Simplify the ratio.
   100 ÷ 50 = 2; 50 ÷ 50 = 1. The simplified ratio is 2:1.
3. Step 3: Find the total number of parts.
   Total Parts = 2 + 1 = 3 parts.
4. Step 4: Calculate the value of one part.
   Value of one part = $90 ÷ 3 = $30 USD.
5. Step 5: Distribute the profit.
   Tatenda's Share: 2 parts × $30 = $60 USD.
   Farai's Share: 1 part × $30 = $30 USD.

Answer: Tatenda should receive $60 USD and Farai should receive $30 USD.

Discounts and Commissions: The Language of Sales

What's the Big Idea?

Have you ever walked into an Edgars store and seen a sign that says, "Massive Clearance! 30% OFF!"? That reduction in price is called a discount. Now, think about the person who sells you a data plan for Econet. They often earn a commission, which is a bonus they get based on the value of the sales they make. Discounts and commissions are two of the most common applications of percentages in the business world.

Key Vocabulary

• Discount: A reduction, usually a percentage, taken off the original price.
• Commission: A payment given to a salesperson, calculated as a percentage of the sales they have generated.
• Marked Price: The initial price before any discount.
• Sale Price: The final price a customer pays after the discount.

Calculating Discounts and Commissions

1. To find the Discount Amount:
   Discount Amount = Marked Price × Discount Rate (%)
   Sale Price = Marked Price - Discount Amount
2. To find the Commission Amount:
   Commission Amount = Total Sales × Commission Rate (%)
   Total Earnings = Basic Salary + Commission Amount

Worked Example: "Putting it to Work in Zimbabwe"

Scenario: Fungai works at a real estate agency. He earns a basic monthly salary of $300 USD. He also earns a 2% commission on the value of any properties he sells. This month, he sold a house for $45,000 USD.

The Question: What were Fungai's total earnings for the month?

1. Step 1: Calculate the commission amount.
   Commission Amount = $45,000 × 2% = $45,000 × 0.02 = $900 USD.
2. Step 2: Calculate his total earnings.
   Total Earnings = Basic Salary + Commission Amount = $300 + $900 = $1,200 USD.

Answer: Fungai's total earnings for the month were $1,200 USD.

Hire Purchase and Currency Conversion

What's the Big Idea?

Imagine you need a new generator for your home, but the cash price is $500 USD. The shop offers a deal: pay a small amount today, take the generator home, and pay the rest in monthly amounts. This is Hire Purchase. Now, what if your uncle from South Africa sends you R2,000? You can't use Rand, so you need to convert it to US Dollars. This is currency conversion.

Key Vocabulary

• Hire Purchase (HP): A method of buying goods through an initial deposit, followed by regular payments (instalments).
• Deposit: The upfront payment made to begin a hire purchase agreement.
• Instalment: One of a series of regular payments.
• Exchange Rate: The value of one currency for the purpose of conversion to another.

Procedure of Solving Problems Involving Hire Purchase

1. Step 1: Calculate the Deposit. (Percentage of the cash price).
2. Step 2: Calculate the Total Amount Paid in Instalments. (Monthly instalment × number of payments).
3. Step 3: Calculate the Total Hire Purchase Price. (Deposit + Total of the instalments).
4. Step 4: Find the Extra Cost. (Total HP price - Cash price).

Currency Conversion Methods

There are only two operations to remember: multiply or divide.

• Method 1: To find out how much FOREIGN currency you will get, you MULTIPLY.
  Example: You have $100 USD and want ZAR (Rate: 1 USD = 18 ZAR). You get 100 × 18 = 1,800 ZAR.
• Method 2: To find out how much LOCAL currency (USD) you will get, you DIVIDE.
  Example: You have 1,800 ZAR and want USD. You get 1,800 ÷ 18 = 100 USD.

Connecting the Dots: You have now mastered the essential calculations for everyday commerce. We are now ready to move from the mathematics of buying and selling to the mathematics of borrowing and investing, which leads to the next topic: Simple Interest.