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Compiled by: Samuel Maina
Commercial mathematics is essential in cooperative management for accurate financial transactions and decision-making. Understanding discounts enables cooperative managers to calculate correct pricing, improve cash flow, and negotiate better deals with suppliers and customers. This chapter focuses on the three main types of discounts encountered in cooperative business transactions: cash discounts, trade discounts, and quantity discounts.
Discounts reduce the price payable by a buyer, either to encourage prompt payment, bulk purchases, or to facilitate trade. Each discount type has unique calculation methods and applications in cooperatives. Mastery of these calculations ensures precise invoicing and financial control.
Cash discounts are reductions offered to buyers who pay their invoices promptly, usually within a specified period. This encourages quick payment and improves the cooperative’s liquidity. The cash discount is typically expressed as a percentage of the invoice amount and applies after trade discounts if any.
The formula for calculating the amount payable after a cash discount is:
$$ \text{Amount Payable} = \text{Invoice Amount} \times (1 - \frac{\text{Cash Discount \%}}{100}) $$
Example 1: A cooperative sells goods worth Ksh 50,000 with a 5% cash discount if payment is made within 10 days. Calculate the amount payable if the buyer pays within the discount period.
Given: Invoice Amount = Ksh 50,000, Cash Discount = 5%
$$ \text{Amount Payable} = 50,000 \times (1 - \frac{5}{100}) $$
$$ = 50,000 \times 0.95 $$
$$ = 47,500 \text{ Ksh} $$
Answer: Ksh 47,500
Example 2: A SACCO issues an invoice of Ksh 120,000 with a 3% cash discount for payment within 7 days. If the buyer pays after 7 days, what amount is payable? What if payment is made within the discount period?
Given: Invoice Amount = Ksh 120,000, Cash Discount = 3%
Payment after 7 days (no discount):
$$ \text{Amount Payable} = 120,000 \text{ Ksh} $$
Payment within 7 days:
$$ \text{Amount Payable} = 120,000 \times (1 - \frac{3}{100}) $$
$$ = 120,000 \times 0.97 $$
$$ = 116,400 \text{ Ksh} $$
Answer: After 7 days: Ksh 120,000; Within 7 days: Ksh 116,400
Example 3: A cooperative sells farming equipment worth Ksh 250,000 with a 4% cash discount if paid within 15 days. Calculate the discount amount and the amount payable if the buyer pays on time.
Given: Invoice Amount = Ksh 250,000, Cash Discount = 4%
Discount amount:
$$ \text{Discount} = 250,000 \times \frac{4}{100} $$
$$ = 10,000 \text{ Ksh} $$
Amount payable:
$$ = 250,000 - 10,000 $$
$$ = 240,000 \text{ Ksh} $$
Answer: Discount = Ksh 10,000; Amount Payable = Ksh 240,000
Example 4: A retailer buys goods worth Ksh 80,000 from a cooperative with a 2.5% cash discount. Calculate the amount payable when the buyer pays within the discount period.
Given: Invoice Amount = Ksh 80,000, Cash Discount = 2.5%
$$ \text{Amount Payable} = 80,000 \times (1 - \frac{2.5}{100}) $$
$$ = 80,000 \times 0.975 $$
$$ = 78,000 \text{ Ksh} $$
Answer: Ksh 78,000
Example 5: A cooperative issues an invoice of Ksh 150,000 with a 6% cash discount if payment is made within 20 days. Calculate the amount payable if the buyer pays within the discount period and the discount amount.
Given: Invoice Amount = Ksh 150,000, Cash Discount = 6%
Discount amount:
$$ = 150,000 \times \frac{6}{100} $$
$$ = 9,000 \text{ Ksh} $$
Amount payable:
$$ = 150,000 - 9,000 $$
$$ = 141,000 \text{ Ksh} $$
Answer: Discount = Ksh 9,000; Amount Payable = Ksh 141,000
Trade discount is a reduction offered by sellers to buyers, usually retailers or wholesalers, to encourage bulk buying or to reward loyalty. Unlike cash discounts, trade discounts are deducted from the list price before invoicing and are not conditional on prompt payment.
The formula for the invoice amount after trade discount is:
$$ \text{Invoice Amount} = \text{List Price} \times (1 - \frac{\text{Trade Discount \%}}{100}) $$
If multiple trade discounts apply, they are applied successively, not added.
Example 1: A cooperative sells goods with a list price of Ksh 100,000 and offers a 10% trade discount. Calculate the invoice amount.
Given: List Price = Ksh 100,000, Trade Discount = 10%
$$ \text{Invoice Amount} = 100,000 \times (1 - \frac{10}{100}) $$
$$ = 100,000 \times 0.90 $$
$$ = 90,000 \text{ Ksh} $$
Answer: Ksh 90,000
Example 2: A retailer purchases goods with a list price of Ksh 250,000 from a cooperative offering successive trade discounts of 10% and 5%. Calculate the invoice amount.
Given: List Price = Ksh 250,000, Trade Discounts = 10%, 5%
First discount:
$$ = 250,000 \times (1 - \frac{10}{100}) = 250,000 \times 0.90 = 225,000 $$
Second discount applied on Ksh 225,000:
$$ = 225,000 \times (1 - \frac{5}{100}) = 225,000 \times 0.95 = 213,750 $$
Answer: Ksh 213,750
Example 3: A cooperative offers trade discounts of 15% and 7% on goods with a list price of Ksh 500,000. Calculate the final invoice amount.
Given: List Price = Ksh 500,000, Trade Discounts = 15%, 7%
First discount:
$$ = 500,000 \times (1 - \frac{15}{100}) = 500,000 \times 0.85 = 425,000 $$
Second discount:
$$ = 425,000 \times (1 - \frac{7}{100}) = 425,000 \times 0.93 = 395,250 $$
Answer: Ksh 395,250
Example 4: A cooperative offers a single trade discount of 12% on goods priced at Ksh 80,000. Calculate the invoice amount.
Given: List Price = Ksh 80,000, Trade Discount = 12%
$$ = 80,000 \times (1 - \frac{12}{100}) = 80,000 \times 0.88 = 70,400 $$
Answer: Ksh 70,400
Example 5: A retailer receives goods with a list price of Ksh 300,000 and is offered successive trade discounts of 8%, 4%, and 3%. Calculate the invoice amount.
Given: List Price = Ksh 300,000, Trade Discounts = 8%, 4%, 3%
First discount:
$$ = 300,000 \times (1 - \frac{8}{100}) = 300,000 \times 0.92 = 276,000 $$
Second discount:
$$ = 276,000 \times (1 - \frac{4}{100}) = 276,000 \times 0.96 = 264,960 $$
Third discount:
$$ = 264,960 \times (1 - \frac{3}{100}) = 264,960 \times 0.97 = 256,810.20 $$
Answer: Ksh 256,810.20
Example 6: A farmer buys three pangas at Ksh 180 each (8% discount), four jembes at Ksh 350 each (5% discount), one bow-saw at Ksh 250 (5% discount), three fork jembes at Ksh 400 each (5% discount), and one tool box at Ksh 2,500 (7% discount) from a hardware shop. Calculate:
(a) The total cash discount on all 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.
(a) Total cash discount:
Total cash discount:$$ 43.20 + 70 + 12.50 + 60 + 175 = 360.70 \text{ Ksh} $$
Answer: Total cash discount = Ksh 360.70
(b) Total amount paid after discount:
Total paid:$$ 496.80 + 1,330 + 237.50 + 1,140 + 2,325 = 5,529.30 \text{ Ksh} $$
Answer: Total amount paid = Ksh 5,529.30
(c) Amount if no discount:$$ 540 + 1,400 + 250 + 1,200 + 2,500 = 5,890 \text{ Ksh} $$
Answer: Amount without discount = Ksh 5,890
(d) Total percentage discount:$$ \text{Total discount} = 360.70 \text{ Ksh}\newline \text{Percentage discount} = \frac{360.70}{5,890} \times 100 = 6.12\% $$
Answer: Total percentage discount = 6.12%
Example 7: Calculate the marked price on a bag of cement selling at Ksh 570 after a discount of 5% is offered.
Let marked price be \(M\).
Discounted price:$$ 570 = M \times (1 - \frac{5}{100}) = M \times 0.95 $$
So,$$ M = \frac{570}{0.95} = 600 \text{ Ksh} $$
Answer: Marked price = Ksh 600
Example 8: An umbrella and a pen are sold at a discount of 8% and 3% respectively. If the cost of the umbrella is four times that of the pen, calculate the overall discount offered on the two commodities.
Let pen price = \(x\), umbrella price = \(4x\)
Discount on pen: \(x \times 0.03 = 0.03x\)
Discount on umbrella: \(4x \times 0.08 = 0.32x\)
Total price before discount: \(x + 4x = 5x\)
Total discount: \(0.03x + 0.32x = 0.35x\)
Overall discount percentage:$$ \frac{0.35x}{5x} \times 100 = 7\% $$
Answer: Overall discount = 7%
Quantity discounts are price reductions based on the volume or quantity of goods purchased. They encourage larger purchases and reduce handling costs. Quantity discounts are often structured in tiers, with higher discounts for larger quantities.
The amount payable after quantity discount is:
$$ \text{Amount Payable} = \text{Unit Price} \times \text{Quantity} \times (1 - \frac{\text{Quantity Discount \%}}{100}) $$
Example 1: A cooperative sells maize at Ksh 120 per 90-kg bag. A quantity discount of 5% applies for purchases of 50 bags or more. Calculate the amount payable for 60 bags.
Given: Unit Price = Ksh 120, Quantity = 60, Quantity Discount = 5%
Total before discount:
$$ = 120 \times 60 = 7,200 \text{ Ksh} $$
Discount amount:
$$ = 7,200 \times \frac{5}{100} = 360 \text{ Ksh} $$
Amount payable:
$$ = 7,200 - 360 = 6,840 \text{ Ksh} $$
Answer: Ksh 6,840
Example 2: A cooperative sells sugar at Ksh 100 per 50-kg bag with a 10% discount for orders of 100 bags or more. Calculate the amount payable for 120 bags.
Given: Unit Price = Ksh 100, Quantity = 120, Quantity Discount = 10%
Total before discount:
$$ = 100 \times 120 = 12,000 \text{ Ksh} $$
Discount amount:
$$ = 12,000 \times \frac{10}{100} = 1,200 \text{ Ksh} $$
Amount payable:
$$ = 12,000 - 1,200 = 10,800 \text{ Ksh} $$
Answer: Ksh 10,800
Example 3: A cooperative offers coffee at Ksh 250 per kilogram. For purchases of 500 kg and above, a 7% quantity discount applies. Calculate the amount payable for 600 kg.
Given: Unit Price = Ksh 250, Quantity = 600, Quantity Discount = 7%
Total before discount:
$$ = 250 \times 600 = 150,000 \text{ Ksh} $$
Discount amount:
$$ = 150,000 \times \frac{7}{100} = 10,500 \text{ Ksh} $$
Amount payable:
$$ = 150,000 - 10,500 = 139,500 \text{ Ksh} $$
Answer: Ksh 139,500
Example 4: A cooperative sells fertilizer at Ksh 1,200 per bag with a 12% discount for bulk purchases above 200 bags. Calculate the amount payable for 250 bags.
Given: Unit Price = Ksh 1,200, Quantity = 250, Quantity Discount = 12%
Total before discount:
$$ = 1,200 \times 250 = 300,000 \text{ Ksh} $$
Discount amount:
$$ = 300,000 \times \frac{12}{100} = 36,000 \text{ Ksh} $$
Amount payable:
$$ = 300,000 - 36,000 = 264,000 \text{ Ksh} $$
Answer: Ksh 264,000
Example 5: A cooperative sells milk at Ksh 80 per litre with a 4% quantity discount for orders of 1,000 litres or more. Calculate the amount payable for 1,200 litres.
Given: Unit Price = Ksh 80, Quantity = 1,200, Quantity Discount = 4%
Total before discount:
$$ = 80 \times 1,200 = 96,000 \text{ Ksh} $$
Discount amount:
$$ = 96,000 \times \frac{4}{100} = 3,840 \text{ Ksh} $$
Amount payable:
$$ = 96,000 - 3,840 = 92,160 \text{ Ksh} $$
Answer: Ksh 92,160
A cooperative sells goods worth Ksh 75,000 and offers a 6% cash discount for payment within 14 days. Calculate the amount payable if the buyer pays within the discount period. (5 marks)
A retailer receives an invoice with a list price of Ksh 180,000 and is offered successive trade discounts of 12% and 8%. Calculate the invoice amount. (6 marks)
A cooperative sells sugar at Ksh 90 per 50-kg bag with a 7% quantity discount for purchases of 80 bags or more. Calculate the amount payable for 100 bags. (6 marks)
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Create a free accountThis chapter covered various types of discounts including cash, trade, and quantity discounts, explaining how each affects the final price in commercial transactions. It detailed the calculation of commissions as a percentage of sales or profits earned by agents or salespersons. Different methods of calculating wages were discussed, focusing on piece rate and hourly rate systems, followed by practical approaches to computing total wages and salaries. The chapter then explored the principles of simple and compound interest, illustrating their application in financial calculations. Profit margin and mark-up concepts were examined to distinguish between cost-based and selling price-based profit calculations. Methods to calculate gross pay and net pay were presented, incorporating deductions and allowances. The depreciation and appreciation of assets were addressed to understand changes in asset values over time. Finally, the chapter concluded with procedures for determining hire purchase prices and handling foreign exchange transactions in business contexts.
A cooperative society offers a cash discount of 5% on an invoice amount of Ksh 120,000 if payment is made within 10 days. Calculate the amount payable after the cash discount. (3 marks)
A supplier gives a trade discount of 12% followed by a quantity discount of 8% on goods worth Ksh 250,000. Calculate the net price the cooperative will pay after both discounts. (4 marks)
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