3.1 Buying and selling; discounts, profit and loss, margins and mark-ups

3.1.1 Calculating Discounts

Discounts are reductions in the selling price offered by sellers to buyers as incentives or promotions. They are commonly used in retail shops, banks, and cooperatives to encourage purchases or reward loyal customers. The discount amount and the price after discount are key figures to compute in business transactions.

The formula to calculate the amount of discount is:

$$\text{Discount} = \frac{\text{Discount Rate}}{100} \times \text{Original Price}$$

The selling price after discount is:

$$\text{Selling Price} = \text{Original Price} - \text{Discount}$$

Worked Examples

Example 1: A retail store in Nairobi offers a 10% discount on a laptop originally priced at Ksh 50,000. Find the discount amount and the selling price.

Given: Original Price = Ksh 50,000, Discount Rate = 10%

$$\text{Discount} = \frac{10}{100} \times 50,000$$

$$= 0.10 \times 50,000$$

$$= 5,000$$

$$\text{Selling Price} = 50,000 - 5,000$$

$$= 45,000$$

Answer: Discount = Ksh 5,000, Selling Price = Ksh 45,000

Example 2: A cooperative sells maize bags at Ksh 3,200 each but offers a 7.5% discount during harvest season. Calculate the discounted price.

Given: Original Price = Ksh 3,200, Discount Rate = 7.5%

$$\text{Discount} = \frac{7.5}{100} \times 3,200$$

$$= 0.075 \times 3,200$$

$$= 240$$

$$\text{Selling Price} = 3,200 - 240$$

$$= 2,960$$

Answer: Discounted Price = Ksh 2,960

Example 3: At a county hospital pharmacy, a medicine pack costs Ksh 1,500. During a health campaign, a 12% discount is given. Find the discount and new price.

Given: Original Price = Ksh 1,500, Discount Rate = 12%

$$\text{Discount} = \frac{12}{100} \times 1,500$$

$$= 0.12 \times 1,500$$

$$= 180$$

$$\text{Selling Price} = 1,500 - 180$$

$$= 1,320$$

Answer: Discount = Ksh 180, Selling Price = Ksh 1,320

Example 4: A hotel offers a 15% discount on a room originally priced at Ksh 8,000 per night during off-season. Calculate the discount and final price.

Given: Original Price = Ksh 8,000, Discount Rate = 15%

$$\text{Discount} = \frac{15}{100} \times 8,000$$

$$= 0.15 \times 8,000$$

$$= 1,200$$

$$\text{Selling Price} = 8,000 - 1,200$$

$$= 6,800$$

Answer: Discount = Ksh 1,200, Selling Price = Ksh 6,800

Example 5: A university bookstore sells textbooks priced at Ksh 2,400 each. During a clearance sale, a 20% discount is offered. Calculate the discount and the price payable.

Given: Original Price = Ksh 2,400, Discount Rate = 20%

$$\text{Discount} = \frac{20}{100} \times 2,400$$

$$= 0.20 \times 2,400$$

$$= 480$$

$$\text{Selling Price} = 2,400 - 480$$

$$= 1,920$$

Answer: Discount = Ksh 480, Selling Price = Ksh 1,920

Additional Worked Examples

Example 6: 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?

Given: Selling Price = Ksh 12,000, Discount Rate = 16%

Let Marked Price = \( M \).

Discount = \( \frac{16}{100} \times M \)

Selling Price = Marked Price - Discount

So,$$ 12,000 = M - (\frac{16}{100} \times M) $$$$ 12,000 = M (1 - 0.16) $$$$ 12,000 = M \times 0.84 $$$$ M = \frac{12,000}{0.84} $$$$ M = 14,285.71 $$ Answer: Marked Price = Ksh 14,285.71

Example 7: A farmer was allowed a cash discount of Ksh 175 on farm implements worth Ksh 3,500. What was the percentage discount?

Given: Discount = Ksh 175, Original Price = Ksh 3,500

Percentage Discount:$$ \text{Discount \%} = \frac{\text{Discount}}{\text{Original Price}} \times 100 $$$$ = \frac{175}{3,500} \times 100 $$$$ = 0.05 \times 100 $$$$ = 5\% $$ Answer: Percentage Discount = 5%

Example 8: A school bought textbooks worth Ksh 27,027 from a bookseller. If the bookseller allowed a discount of 10%, what was the cost of the books without the discount?

Given: Selling Price (after discount) = Ksh 27,027, Discount Rate = 10%

Let Marked Price = \( M \).

Selling Price = Marked Price - Discount

Discount = \( \frac{10}{100} \times M \)

So,$$ 27,027 = M - (\frac{10}{100} \times M) $$$$ 27,027 = M \times 0.90 $$$$ M = \frac{27,027}{0.90} $$$$ M = 30,030 $$ Answer: Cost without discount = Ksh 30,030

Example 9: 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 as follows:

Calculate:
(a) The total cash discount on all the items bought.
(b) The total amount paid for the items.
(c) The amount of money he would have paid if no discount was allowed.
(d) The total percentage discount.

Solution:

(a) Total cash discount:

• Pangas: \(3 \times 180 = 540\), Discount: \(0.08 \times 540 = 43.20\)
• Jembes: \(4 \times 350 = 1,400\), Discount: \(0.05 \times 1,400 = 70\)
• Bow-saw: \(1 \times 250 = 250\), Discount: \(0.05 \times 250 = 12.50\)
• Tool box: \(1 \times 2,500 = 2,500\), Discount: \(0.07 \times 2,500 = 175\)
• Fork jembes: \(3 \times 400 = 1,200\), Discount: \(0.05 \times 1,200 = 60\)

Total cash discount:$$ 43.20 + 70 + 12.50 + 175 + 60 = 360.70 $$ Answer: Total cash discount = Ksh 360.70

(b) Total amount paid for the items:

Total price before discount:$$ 540 + 1,400 + 250 + 2,500 + 1,200 = 5,890 $$ Total amount paid = Total price before discount - Total cash discount$$ 5,890 - 360.70 = 5,529.30 $$ Answer: Total amount paid = Ksh 5,529.30

(c) Amount if no discount was allowed:$$ 5,890 $$ Answer: Ksh 5,890

(d) Total percentage discount:$$ \frac{360.70}{5,890} \times 100 = 6.12\% $$ Answer: Total percentage discount = 6.12%

3.1.2 Computing Profit and Loss

Profit and loss calculations are crucial for determining the financial outcome of buying and selling activities. Profit occurs when selling price exceeds cost price, while loss happens when cost price is higher than selling price. These calculations help businesses like banks, retail outlets, and farms monitor performance and make pricing decisions.

The formulas for profit and loss are:

$$\text{Profit} = \text{Selling Price} - \text{Cost Price} \quad \text{(if Selling Price > Cost Price)}$$

$$\text{Loss} = \text{Cost Price} - \text{Selling Price} \quad \text{(if Cost Price > Selling Price)}$$

Profit or loss percentage is calculated as:

$$\text{Profit \%} = \frac{\text{Profit}}{\text{Cost Price}} \times 100$$

$$\text{Loss \%} = \frac{\text{Loss}}{\text{Cost Price}} \times 100$$

Worked Examples

Example 1: A retail shop buys a smartphone for Ksh 25,000 and sells it at Ksh 30,000. Find the profit and profit percentage.

Given: Cost Price = Ksh 25,000, Selling Price = Ksh 30,000

$$\text{Profit} = 30,000 - 25,000$$

$$= 5,000$$

$$\text{Profit \%} = \frac{5,000}{25,000} \times 100$$

$$= 0.20 \times 100$$

$$= 20\%$$

Answer: Profit = Ksh 5,000, Profit Percentage = 20%

Example 2: A SACCO buys office supplies for Ksh 15,000 but sells them at Ksh 13,500. Calculate the loss and loss percentage.

Given: Cost Price = Ksh 15,000, Selling Price = Ksh 13,500

$$\text{Loss} = 15,000 - 13,500$$

$$= 1,500$$

$$\text{Loss \%} = \frac{1,500}{15,000} \times 100$$

$$= 0.10 \times 100$$

$$= 10\%$$

Answer: Loss = Ksh 1,500, Loss Percentage = 10%

Example 3: A farm sells a tractor for Ksh 1,200,000 which was bought at Ksh 1,350,000. Calculate the loss and loss percentage.

Given: Cost Price = Ksh 1,350,000, Selling Price = Ksh 1,200,000

$$\text{Loss} = 1,350,000 - 1,200,000$$

$$= 150,000$$

$$\text{Loss \%} = \frac{150,000}{1,350,000} \times 100$$

$$= 0.1111 \times 100$$

$$= 11.11\%$$

Answer: Loss = Ksh 150,000, Loss Percentage = 11.11%

Example 4: A county government office buys 100 chairs at Ksh 4,500 each. They sell all at Ksh 5,000 each. Calculate total profit and profit percentage.

Given: Cost Price per chair = Ksh 4,500, Selling Price per chair = Ksh 5,000, Quantity = 100

$$\text{Profit per chair} = 5,000 - 4,500 = 500$$

$$\text{Total Profit} = 500 \times 100 = 50,000$$

$$\text{Profit \%} = \frac{500}{4,500} \times 100 = 11.11\%$$

Answer: Total Profit = Ksh 50,000, Profit Percentage = 11.11%

Example 5: A hotel buys 50 bedsheets at Ksh 2,000 each and sells them at Ksh 1,800 each. Find total loss and loss percentage.

Given: Cost Price per bedsheet = Ksh 2,000, Selling Price per bedsheet = Ksh 1,800, Quantity = 50

$$\text{Loss per bedsheet} = 2,000 - 1,800 = 200$$

$$\text{Total Loss} = 200 \times 50 = 10,000$$

$$\text{Loss \%} = \frac{200}{2,000} \times 100 = 10\%$$

Answer: Total Loss = Ksh 10,000, Loss Percentage = 10%

3.1.3 Understanding Margins

Margin is the difference between selling price and cost price expressed as a percentage of the selling price. It indicates the proportion of the selling price that is profit. Businesses use margin to set prices that cover costs and yield desired profits.

The margin formula is:

$$\text{Margin \%} = \frac{\text{Selling Price} - \text{Cost Price}}{\text{Selling Price}} \times 100$$

Worked Examples

Example 1: A retail store sells a jacket for Ksh 4,000. The cost price was Ksh 3,000. Calculate the margin percentage.

Given: Selling Price = Ksh 4,000, Cost Price = Ksh 3,000

$$\text{Margin \%} = \frac{4,000 - 3,000}{4,000} \times 100$$

$$= \frac{1,000}{4,000} \times 100$$

$$= 0.25 \times 100$$

$$= 25\%$$

Answer: Margin = 25%

Example 2: A cooperative sells sugar at Ksh 120 per kg. The cost price is Ksh 90 per kg. Find the margin percentage.

Given: Selling Price = Ksh 120, Cost Price = Ksh 90

$$\text{Margin \%} = \frac{120 - 90}{120} \times 100$$

$$= \frac{30}{120} \times 100$$

$$= 0.25 \times 100$$

$$= 25\%$$

Answer: Margin = 25%

Example 3: A university bookstore sells a calculator for Ksh 2,500, which cost Ksh 2,000. Calculate the margin.

Given: Selling Price = Ksh 2,500, Cost Price = Ksh 2,000

$$\text{Margin \%} = \frac{2,500 - 2,000}{2,500} \times 100$$

$$= \frac{500}{2,500} \times 100$$

$$= 0.20 \times 100$$

$$= 20\%$$

Answer: Margin = 20%

Example 4: A county government office sells office desks at Ksh 15,000 each. The cost price is Ksh 12,000. Find the margin.

Given: Selling Price = Ksh 15,000, Cost Price = Ksh 12,000

$$\text{Margin \%} = \frac{15,000 - 12,000}{15,000} \times 100$$

$$= \frac{3,000}{15,000} \times 100$$

$$= 0.20 \times 100$$

$$= 20\%$$

Answer: Margin = 20%

Example 5: A retail outlet sells a blender for Ksh 7,500. The cost price is Ksh 6,500. Calculate the margin.

Given: Selling Price = Ksh 7,500, Cost Price = Ksh 6,500

$$\text{Margin \%} = \frac{7,500 - 6,500}{7,500} \times 100$$

$$= \frac{1,000}{7,500} \times 100$$

$$= 0.1333 \times 100$$

$$= 13.33\%$$

Answer: Margin = 13.33%

3.1.4 Calculating Mark-Ups

Mark-up is the amount added to the cost price to determine the selling price. It is expressed as a percentage of the cost price. Mark-ups ensure businesses cover costs and achieve target profits.

The formula for mark-up percentage is:

$$\text{Mark-up \%} = \frac{\text{Selling Price} - \text{Cost Price}}{\text{Cost Price}} \times 100$$

Alternatively, to find selling price using mark-up:

$$\text{Selling Price} = \text{Cost Price} + (\frac{\text{Mark-up \%}}{100} \times \text{Cost Price})$$

Worked Examples

Example 1: A retail shop buys a TV for Ksh 20,000 and wants a 25% mark-up. Calculate the selling price.

Given: Cost Price = Ksh 20,000, Mark-up = 25%

$$\text{Selling Price} = 20,000 + (\frac{25}{100} \times 20,000)$$

$$= 20,000 + 5,000$$

$$= 25,000$$

Answer: Selling Price = Ksh 25,000

Example 2: A bank buys office chairs at Ksh 8,000 each and applies a 15% mark-up. Find the selling price.

Given: Cost Price = Ksh 8,000, Mark-up = 15%

$$\text{Selling Price} = 8,000 + (\frac{15}{100} \times 8,000)$$

$$= 8,000 + 1,200$$

$$= 9,200$$

Answer: Selling Price = Ksh 9,200

Example 3: A hotel buys bed linens at Ksh 3,500 per set and marks them up by 20%. Calculate the selling price.

Given: Cost Price = Ksh 3,500, Mark-up = 20%

$$\text{Selling Price} = 3,500 + (\frac{20}{100} \times 3,500)$$

$$= 3,500 + 700$$

$$= 4,200$$

Answer: Selling Price = Ksh 4,200

Example 4: A cooperative buys maize at Ksh 2,800 per bag and wants to sell it with a 12.5% mark-up. Find the selling price.

Given: Cost Price = Ksh 2,800, Mark-up = 12.5%

$$\text{Selling Price} = 2,800 + (\frac{12.5}{100} \times 2,800)$$

$$= 2,800 + 350$$

$$= 3,150$$

Answer: Selling Price = Ksh 3,150

Example 5: A county government office buys computers at Ksh 45,000 each and applies an 18% mark-up. Calculate the selling price.

Given: Cost Price = Ksh 45,000, Mark-up = 18%

$$\text{Selling Price} = 45,000 + (\frac{18}{100} \times 45,000)$$

$$= 45,000 + 8,100$$

$$= 53,100$$

Answer: Selling Price = Ksh 53,100

Chapter Summary

This chapter covered essential commercial mathematics topics applied in business contexts. It began with buying and selling concepts, focusing on calculating discounts, profit and loss, margins, and mark-ups to determine accurate pricing and profitability. Next, the chapter addressed commissions and salaries, explaining how to compute piece rates, hourly wages, gross and net pay, including deductions such as Pay As You Earn (PAYE). The section on bills calculations demonstrated how to work out charges for utilities like water and electricity using given rates and consumption data. Interest calculations were explored through simple and compound interest formulas, highlighting their differences and applications in finance. The chapter also examined depreciation and appreciation of assets, teaching methods to evaluate changes in asset values over time. Hire purchase arrangements were explained, showing how to calculate instalments and total costs under credit sales. Finally, foreign currency exchange transactions were discussed, enabling learners to perform conversions and understand exchange rate impacts on international trade.