By the end of this chapter, you will be able to:
Mastering these skills will help you confidently handle everyday commercial math tasks that are essential in the business world.
Buying and selling transactions form the foundation of commercial activities across all sectors in Kenya, from retail businesses to county government procurement. Understanding discounts, profit and loss, margins, and mark-ups is essential for professionals to make informed decisions, price goods accurately, and maintain financial viability. This chapter develops the mathematical skills needed to calculate these measures precisely, ensuring effective commercial management in diverse professional contexts.
Buying and selling involve exchanging goods or services for money, where pricing strategies significantly impact profitability. Discounts incentivize purchases and affect the final selling price, while profit, loss, margins, and mark-ups determine financial outcomes of transactions. Mastery of these concepts enables professionals in sectors such as banking, healthcare, agriculture, and hospitality to optimize pricing and revenue.
Discounts reduce the original price to encourage sales or reward customers. The discount amount is calculated based on the percentage discount offered, and the net price is the amount payable after applying the discount.
The formula for calculating the discount amount is:
$$ \text{Discount} = \frac{\text{Discount Rate}}{100} \times \text{Original Price} $$
The net price is:
$$ \text{Net Price} = \text{Original Price} - \text{Discount} $$
Example 1: A county referral hospital orders medical supplies worth Ksh 50,000 with a 10% discount. Calculate the discount and net price.
Given: Original Price = Ksh 50,000, Discount Rate = 10%
$$ \text{Discount} = \frac{10}{100} \times 50,000 $$
$$ = 0.10 \times 50,000 = 5,000 $$
$$ \text{Net Price} = 50,000 - 5,000 = 45,000 $$
Answer: Discount = Ksh 5,000; Net Price = Ksh 45,000
Example 2: A retail store offers a 15% discount on a laptop priced at Ksh 80,000. Find the discount and net price.
Given: Original Price = Ksh 80,000, Discount Rate = 15%
$$ \text{Discount} = \frac{15}{100} \times 80,000 $$
$$ = 0.15 \times 80,000 = 12,000 $$
$$ \text{Net Price} = 80,000 - 12,000 = 68,000 $$
Answer: Discount = Ksh 12,000; Net Price = Ksh 68,000
Example 3: A SACCO sells shares valued at Ksh 120,000 with a 5% discount for early payment. Calculate the discount and net price.
Given: Original Price = Ksh 120,000, Discount Rate = 5%
$$ \text{Discount} = \frac{5}{100} \times 120,000 $$
$$ = 0.05 \times 120,000 = 6,000 $$
$$ \text{Net Price} = 120,000 - 6,000 = 114,000 $$
Answer: Discount = Ksh 6,000; Net Price = Ksh 114,000
Example 4: A hotel offers a 12.5% discount on a conference hall booking fee of Ksh 40,000. Determine the discount and net price.
Given: Original Price = Ksh 40,000, Discount Rate = 12.5%
$$ \text{Discount} = \frac{12.5}{100} \times 40,000 $$
$$ = 0.125 \times 40,000 = 5,000 $$
$$ \text{Net Price} = 40,000 - 5,000 = 35,000 $$
Answer: Discount = Ksh 5,000; Net Price = Ksh 35,000
Example 5: A university bookstore sells textbooks worth Ksh 25,000 with a 7% discount. Calculate the discount and net price.
Given: Original Price = Ksh 25,000, Discount Rate = 7%
$$ \text{Discount} = \frac{7}{100} \times 25,000 $$
$$ = 0.07 \times 25,000 = 1,750 $$
$$ \text{Net Price} = 25,000 - 1,750 = 23,250 $$
Answer: Discount = Ksh 1,750; Net Price = Ksh 23,250
Example 6: A farmer bought three pangas, four jembes, one bow-saw, three fork jembes and one tool box from a hardware shop. The prices and discounts are:
| Item | Price per unit (Ksh) | Discount (%) | Quantity |
|---|---|---|---|
| Panga | 180 | 8 | 3 |
| Jembe | 350 | 5 | 4 |
| Bow-saw | 250 | 5 | 1 |
| Tool box | 2,500 | 7 | 1 |
| Fork jembe | 400 | 5 | 3 |
(a) Calculate the total cash discount on all items bought.
For each item:
Panga: $$\text{Discount} = \frac{8}{100} \times 180 \times 3 = 0.08 \times 180 \times 3 = 43.2 \text{ Ksh}$$
Jembe: $$\text{Discount} = \frac{5}{100} \times 350 \times 4 = 0.05 \times 350 \times 4 = 70 \text{ Ksh}$$
Bow-saw: $$\text{Discount} = \frac{5}{100} \times 250 \times 1 = 0.05 \times 250 = 12.5 \text{ Ksh}$$
Tool box: $$\text{Discount} = \frac{7}{100} \times 2,500 \times 1 = 0.07 \times 2,500 = 175 \text{ Ksh}$$
Fork jembe: $$\text{Discount} = \frac{5}{100} \times 400 \times 3 = 0.05 \times 400 \times 3 = 60 \text{ Ksh}$$
Total cash discount:$$43.2 + 70 + 12.5 + 175 + 60 = 360.7 \text{ Ksh}$$
Answer: Total cash discount = Ksh 360.7
(b) Calculate the total amount paid for the items.
Total price before discount:$$\text{Total} = (180 \times 3) + (350 \times 4) + (250 \times 1) + (2,500 \times 1) + (400 \times 3)$$$$= 540 + 1,400 + 250 + 2,500 + 1,200 = 5,890 \text{ Ksh}$$
Total amount paid:$$5,890 - 360.7 = 5,529.3 \text{ Ksh}$$
Answer: Total amount paid = Ksh 5,529.3
(c) Calculate the amount that would have been paid if no discount was allowed.
Answer: Ksh 5,890
(d) Calculate the total percentage discount.
$$\text{Total percentage discount} = \frac{360.7}{5,890} \times 100 = 6.12\%$$
Answer: Total percentage discount = 6.12%
Example 7: Jane paid Ksh 12,000 for a TV set after she was allowed a discount of 16%. What was the marked price of the TV?
Let marked price = \(M\).
Net price = Ksh 12,000
Discount rate = 16%
Discount = \(\frac{16}{100} \times M\)
Net price = \(M - \frac{16}{100} M = M \times (1 - 0.16) = M \times 0.84\)
So,$$0.84 M = 12,000$$$$M = \frac{12,000}{0.84} = 14,285.71$$
Answer: Marked price = Ksh 14,285.71
Example 8: A farmer was allowed a cash discount of Ksh 175 on farm implements worth Ksh 3,500. What was the percentage discount?
$$\text{Percentage discount} = \frac{175}{3,500} \times 100 = 5\%$$
Answer: Percentage discount = 5%
Example 9: An umbrella and a pen are sold at a discount of 8% and 3% respectively. The cost of the umbrella is four times that of the pen. Calculate the overall discount offered on the two commodities.
Let cost of pen = \(x\), umbrella = \(4x\).
Discount on pen: \(0.03x\)
Discount on umbrella: \(0.08 \times 4x = 0.32x\)
Total cost = \(x + 4x = 5x\)
Total discount = \(0.03x + 0.32x = 0.35x\)
Overall discount percentage:$$\frac{0.35x}{5x} \times 100 = 7\%$$
Answer: Overall discount = 7%
Profit and loss measure the financial result of buying and selling goods or services. Profit occurs when the selling price exceeds the cost price, while loss occurs when the selling price is less than the cost price.
The formulas are:
$$ \text{Profit} = \text{Selling Price} - \text{Cost Price} \quad \text{(if Selling Price > Cost Price)} $$
$$ \text{Loss} = \text{Cost Price} - \text{Selling Price} \quad \text{(if Selling Price < Cost Price)} $$
Profit or loss percentage is calculated as:
$$ \text{Profit \%} = \frac{\text{Profit}}{\text{Cost Price}} \times 100 $$
$$ \text{Loss \%} = \frac{\text{Loss}}{\text{Cost Price}} \times 100 $$
Example 1: A retail shop buys goods for Ksh 30,000 and sells them for Ksh 35,000. Calculate the profit and profit percentage.
Given: Cost Price = Ksh 30,000, Selling Price = Ksh 35,000
$$ \text{Profit} = 35,000 - 30,000 = 5,000 $$
$$ \text{Profit \%} = \frac{5,000}{30,000} \times 100 = 16.67\% $$
Answer: Profit = Ksh 5,000; Profit Percentage = 16.67%
Example 2: A cooperative buys farm produce for Ksh 45,000 and sells at Ksh 42,000. Calculate the loss and loss percentage.
Given: Cost Price = Ksh 45,000, Selling Price = Ksh 42,000
$$ \text{Loss} = 45,000 - 42,000 = 3,000 $$
$$ \text{Loss \%} = \frac{3,000}{45,000} \times 100 = 6.67\% $$
Answer: Loss = Ksh 3,000; Loss Percentage = 6.67%
Example 3: A county government office buys office equipment for Ksh 150,000 and sells it for Ksh 165,000. Find the profit and profit percentage.
Given: Cost Price = Ksh 150,000, Selling Price = Ksh 165,000
$$ \text{Profit} = 165,000 - 150,000 = 15,000 $$
$$ \text{Profit \%} = \frac{15,000}{150,000} \times 100 = 10\% $$
Answer: Profit = Ksh 15,000; Profit Percentage = 10%
Example 4: A hotel purchases furniture for Ksh 200,000 but sells it for Ksh 180,000. Calculate the loss and loss percentage.
Given: Cost Price = Ksh 200,000, Selling Price = Ksh 180,000
$$ \text{Loss} = 200,000 - 180,000 = 20,000 $$
$$ \text{Loss \%} = \frac{20,000}{200,000} \times 100 = 10\% $$
Answer: Loss = Ksh 20,000; Loss Percentage = 10%
Example 5: A retail business buys stock for Ksh 75,000 and sells it at Ksh 90,000. Calculate the profit and profit percentage.
Given: Cost Price = Ksh 75,000, Selling Price = Ksh 90,000
$$ \text{Profit} = 90,000 - 75,000 = 15,000 $$
$$ \text{Profit \%} = \frac{15,000}{75,000} \times 100 = 20\% $$
Answer: Profit = Ksh 15,000; Profit Percentage = 20%
Margin refers to the percentage of the selling price that is profit. It is a key indicator in pricing strategies and financial analysis.
The formula for margin percentage is:
$$ \text{Margin \%} = \frac{\text{Profit}}{\text{Selling Price}} \times 100 $$
Example 1: A retail shop sells a product at Ksh 50,000 and makes a profit of Ksh 10,000. Calculate the margin percentage.
Given: Selling Price = Ksh 50,000, Profit = Ksh 10,000
$$ \text{Margin \%} = \frac{10,000}{50,000} \times 100 = 20\% $$
Answer: Margin Percentage = 20%
Example 2: A farm cooperative sells produce at Ksh 120,000 and earns a profit of Ksh 24,000. Find the margin percentage.
Given: Selling Price = Ksh 120,000, Profit = Ksh 24,000
$$ \text{Margin \%} = \frac{24,000}{120,000} \times 100 = 20\% $$
Answer: Margin Percentage = 20%
Example 3: A county government office sells furniture for Ksh 180,000 and makes a profit of Ksh 30,000. Calculate margin percentage.
Given: Selling Price = Ksh 180,000, Profit = Ksh 30,000
$$ \text{Margin \%} = \frac{30,000}{180,000} \times 100 = 16.67\% $$
Answer: Margin Percentage = 16.67%
Example 4: A hotel sells a conference package for Ksh 100,000 with a profit of Ksh 15,000. Find the margin percentage.
Given: Selling Price = Ksh 100,000, Profit = Ksh 15,000
$$ \text{Margin \%} = \frac{15,000}{100,000} \times 100 = 15\% $$
Answer: Margin Percentage = 15%
Example 5: A retail business sells goods at Ksh 80,000 and makes a profit of Ksh 20,000. Calculate the margin percentage.
Given: Selling Price = Ksh 80,000, Profit = Ksh 20,000
$$ \text{Margin \%} = \frac{20,000}{80,000} \times 100 = 25\% $$
Answer: Margin Percentage = 25%
Mark-up is the percentage added to the cost price to arrive at the selling price. It is crucial for setting prices that cover costs and generate profit.
The formula for mark-up percentage is:
$$ \text{Mark-up \%} = \frac{\text{Profit}}{\text{Cost Price}} \times 100 $$
The selling price can also be calculated as:
$$ \text{Selling Price} = \text{Cost Price} + \text{Profit} $$
Or using mark-up percentage:
$$ \text{Selling Price} = \text{Cost Price} \times (1 + \frac{\text{Mark-up \%}}{100}) $$
Example 1: A retail store buys goods at Ksh 40,000 and wants a 25% mark-up. Calculate the selling price.
Given: Cost Price = Ksh 40,000, Mark-up = 25%
$$ \text{Selling Price} = 40,000 \times (1 + \frac{25}{100}) $$
$$ = 40,000 \times 1.25 = 50,000 $$
Answer: Selling Price = Ksh 50,000
Example 2: A cooperative buys farm produce at Ksh 90,000 and uses a 20% mark-up. Find the selling price.
Given: Cost Price = Ksh 90,000, Mark-up = 20%
$$ \text{Selling Price} = 90,000 \times (1 + \frac{20}{100}) $$
$$ = 90,000 \times 1.20 = 108,000 $$
Answer: Selling Price = Ksh 108,000
Example 3: A hotel purchases furniture for Ksh 150,000 and applies a 30% mark-up. Calculate the selling price.
Given: Cost Price = Ksh 150,000, Mark-up = 30%
$$ \text{Selling Price} = 150,000 \times (1 + \frac{30}{100}) $$
$$ = 150,000 \times 1.30 = 195,000 $$
Answer: Selling Price = Ksh 195,000
Example 4: A retail business buys stock for Ksh 75,000 and applies a 40% mark-up. Find the selling price.
Given: Cost Price = Ksh 75,000, Mark-up = 40%
$$ \text{Selling Price} = 75,000 \times (1 + \frac{40}{100}) $$
$$ = 75,000 \times 1.40 = 105,000 $$
Answer: Selling Price = Ksh 105,000
Example 5: A county government office buys office equipment at Ksh 200,000 and adds a 15% mark-up. Calculate the selling price.
Given: Cost Price = Ksh 200,000, Mark-up = 15%
$$ \text{Selling Price} = 200,000 \times (1 + \frac{15}{100}) $$
$$ = 200,000 \times 1.15 = 230,000 $$
Answer: Selling Price = Ksh 230,000
A retail store sells a smartphone with an original price of Ksh 60,000 at a 12% discount. Calculate the discount amount and the net price. (5 marks)
A county hospital buys medical equipment for Ksh 250,000 and sells it at Ksh 275,000. Calculate the profit and profit percentage. (5 marks)
A SACCO sells shares at Ksh 500,000 with a profit margin of 18%. Calculate the profit made. (5 marks)
A hotel buys furniture for Ksh 180,000 and applies a 22% mark-up. Calculate the selling price. (5 marks)
A cooperative purchases farm produce for Ksh 120,000 but sells it at a 7% loss. Calculate the selling price and loss amount. (5 marks)
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Create a free accountThis chapter covers essential commercial mathematics concepts starting with buying and selling, where calculations of discounts, profit and loss, margins, and mark-ups are explained. It then explores commissions and salaries, detailing how to compute piece rates, hourly wages, gross and net pay, including deductions such as PAYE. The chapter proceeds to bills calculations, focusing on water and electricity charges and how to accurately determine consumption costs. Simple and compound interest calculations are presented next, showing how money grows over time through different interest methods. The concepts of depreciation and appreciation of assets highlight how asset values change, affecting financial decisions. Hire purchase transactions are examined, illustrating how installment payments are structured and calculated. Finally, the chapter addresses foreign currency exchange transactions, explaining how to convert currencies and manage exchange rates in business contexts.
A retail shop in Nairobi buys a batch of 50 laptops at Ksh 40,000 each. If the shop wants to sell each laptop at a 20% profit margin on cost price, what is the selling price per laptop? (2 marks)
A SACCO agent earns a commission of 3% on all loans disbursed. If the agent disbursed loans totaling Ksh 2,500,000 in a month, calculate the total commission earned. (2 marks)
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