Work-out commercial mathematics is essential for cooperative management professionals in Kenya to accurately calculate pricing, discounts, and payments in daily business transactions. Understanding different types of discounts helps cooperatives manage their cash flow, negotiate better deals with suppliers, and offer competitive prices to members. This chapter focuses on the three main types of discounts commonly encountered in cooperative business environments: cash discount, trade discount, and quantity discount, with practical worked examples for each.

1.1 Types of discounts

Discounts reduce the price payable by a buyer and are critical tools in managing cooperative finances. Each discount type serves a different purpose and is applied under different conditions. Precise calculation of these discounts ensures correct billing, accounting, and profitability analysis.

1.1.1 Cash discount

Cash discount is a price reduction offered by a seller to encourage the buyer to pay promptly, usually within a specified period. It motivates quick payment, improving the seller’s cash flow and reducing credit risk. The discount is calculated on the invoice amount before VAT or other taxes.

The formula to calculate the cash discount amount is

$$ \text{Cash Discount} = \text{Invoice Amount} \times \frac{\text{Discount Rate}}{100} $$

The amount payable after discount is

$$ \text{Amount Payable} = \text{Invoice Amount} - \text{Cash Discount} $$

Worked Examples

Example 1: A cooperative buys goods worth Ksh 50,000 with a 5% cash discount if paid within 10 days. Calculate the discount and amount payable.

Given: Invoice Amount = Ksh 50,000, Discount Rate = 5%

$$ \text{Cash Discount} = 50,000 \times \frac{5}{100} $$

$$ = 50,000 \times 0.05 $$

$$ = 2,500 \text{ Ksh} $$

$$ \text{Amount Payable} = 50,000 - 2,500 = 47,500 \text{ Ksh} $$

Answer: Ksh 2,500 discount; Ksh 47,500 payable

Example 2: A cooperative receives an invoice of Ksh 120,000 with a 3% cash discount. Calculate the amount to be paid if the discount is claimed.

Given: Invoice Amount = Ksh 120,000, Discount Rate = 3%

$$ \text{Cash Discount} = 120,000 \times \frac{3}{100} $$

$$ = 120,000 \times 0.03 $$

$$ = 3,600 \text{ Ksh} $$

$$ \text{Amount Payable} = 120,000 - 3,600 = 116,400 \text{ Ksh} $$

Answer: Ksh 3,600 discount; Ksh 116,400 payable

Example 3: A cooperative pays an invoice of Ksh 80,000 after 15 days. The invoice offers 4% cash discount if paid within 10 days. Calculate the amount payable.

Given: Invoice Amount = Ksh 80,000, Discount Rate = 4%, Payment after discount period

Since payment is after the discount period, no discount applies.

Answer: Ksh 80,000 payable

Example 4: A cooperative has an invoice of Ksh 45,000 with a 2.5% cash discount. Calculate the discount amount and final payment.

Given: Invoice Amount = Ksh 45,000, Discount Rate = 2.5%

$$ \text{Cash Discount} = 45,000 \times \frac{2.5}{100} = 45,000 \times 0.025 = 1,125 \text{ Ksh} $$

$$ \text{Amount Payable} = 45,000 - 1,125 = 43,875 \text{ Ksh} $$

Answer: Ksh 1,125 discount; Ksh 43,875 payable

Example 5: A cooperative receives a bill of Ksh 200,000 with 6% cash discount if paid within 7 days. Calculate the payable amount if paid within the discount period.

Given: Invoice Amount = Ksh 200,000, Discount Rate = 6%

$$ \text{Cash Discount} = 200,000 \times \frac{6}{100} = 200,000 \times 0.06 = 12,000 \text{ Ksh} $$

$$ \text{Amount Payable} = 200,000 - 12,000 = 188,000 \text{ Ksh} $$

Answer: Ksh 12,000 discount; Ksh 188,000 payable

Additional Worked Example: Calculating Percentage Discount

Example 6: A farmer was allowed a cash discount of Ksh 175 on farm implements worth Ksh 3,500. What was the percentage discount?

Given: Cash Discount = Ksh 175, Invoice Amount = Ksh 3,500

Formula:$$ \text{Percentage Discount} = \frac{\text{Cash Discount}}{\text{Invoice Amount}} \times 100 $$

Substitution:$$ \frac{175}{3,500} \times 100 = 5\% $$

Answer: 5% discount

Additional Worked Example: Calculating Marked Price Given Paid Price and Discount

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?

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

Let Marked Price = \( M \).

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

Answer: Marked price is Ksh 14,285.71

1.1.2 Trade discount

Trade discount is a reduction in the list price granted by the seller to the buyer, usually based on the buyer’s status or relationship, and is deducted before any invoice or tax is applied. It is common in bulk or regular cooperative purchases to encourage loyalty or reward volume buying.

The formula for trade discount is

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

The net price after trade discount is

$$ \text{Net Price} = \text{List Price} - \text{Trade Discount} $$

Worked Examples

Example 1: A cooperative buys goods listed at Ksh 75,000 and is given a 10% trade discount. Calculate the discount and net price.

Given: List Price = Ksh 75,000, Discount Rate = 10%

$$ \text{Trade Discount} = 75,000 \times \frac{10}{100} = 7,500 \text{ Ksh} $$

$$ \text{Net Price} = 75,000 - 7,500 = 67,500 \text{ Ksh} $$

Answer: Ksh 7,500 discount; Ksh 67,500 net price

Example 2: A cooperative receives a 15% trade discount on goods priced at Ksh 200,000. Find the net price.

Given: List Price = Ksh 200,000, Discount Rate = 15%

$$ \text{Trade Discount} = 200,000 \times \frac{15}{100} = 30,000 \text{ Ksh} $$

$$ \text{Net Price} = 200,000 - 30,000 = 170,000 \text{ Ksh} $$

Answer: Ksh 30,000 discount; Ksh 170,000 net price

Example 3: A cooperative purchases equipment priced at Ksh 150,000 with a 12% trade discount. Calculate the amount payable before taxes.

Given: List Price = Ksh 150,000, Discount Rate = 12%

$$ \text{Trade Discount} = 150,000 \times \frac{12}{100} = 18,000 \text{ Ksh} $$

$$ \text{Net Price} = 150,000 - 18,000 = 132,000 \text{ Ksh} $$

Answer: Ksh 18,000 discount; Ksh 132,000 net price

Example 4: A cooperative receives a 5% trade discount on a consignment valued at Ksh 90,000. Calculate the net amount.

Given: List Price = Ksh 90,000, Discount Rate = 5%

$$ \text{Trade Discount} = 90,000 \times \frac{5}{100} = 4,500 \text{ Ksh} $$

$$ \text{Net Price} = 90,000 - 4,500 = 85,500 \text{ Ksh} $$

Answer: Ksh 4,500 discount; Ksh 85,500 net price

Example 5: A cooperative orders goods with a list price of Ksh 250,000 and is given a 20% trade discount. Determine the net price.

Given: List Price = Ksh 250,000, Discount Rate = 20%

$$ \text{Trade Discount} = 250,000 \times \frac{20}{100} = 50,000 \text{ Ksh} $$

$$ \text{Net Price} = 250,000 - 50,000 = 200,000 \text{ Ksh} $$

Answer: Ksh 50,000 discount; Ksh 200,000 net price

Additional Worked Example: Calculating Marked Price Given Selling Price After Discount

Example 6: Calculate the marked price on a bag of cement selling at Ksh 570 after a discount of 5% is offered.

Let Marked Price = \( M \), Discount Rate = 5%, Selling Price after discount = Ksh 570.

Selling Price = Marked Price - Discount
Discount = \( M \times \frac{5}{100} \)
So:$$ 570 = M - 0.05M = 0.95M $$$$ M = \frac{570}{0.95} = 600 \text{ Ksh} $$

Answer: Marked price is Ksh 600

1.1.3 Quantity discount

Quantity discount is a reduction in price granted for purchasing large quantities of goods. It incentivizes bulk buying, which reduces handling and ordering costs. The discount is usually applied on the total price after trade discount.

The formula for quantity discount is

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

The price after quantity discount is

$$ \text{Price after Quantity Discount} = \text{Total Price} - \text{Quantity Discount} $$

Worked Examples

Example 1: A cooperative buys 1,000 units of a product at Ksh 50 per unit. The seller offers a 5% quantity discount on purchases above 500 units. Calculate the discount and total price.

Given: Unit Price = Ksh 50, Quantity = 1,000, Discount Rate = 5%

$$ \text{Total Price} = 50 \times 1,000 = 50,000 \text{ Ksh} $$

$$ \text{Quantity Discount} = 50,000 \times \frac{5}{100} = 2,500 \text{ Ksh} $$

$$ \text{Price after Quantity Discount} = 50,000 - 2,500 = 47,500 \text{ Ksh} $$

Answer: Ksh 2,500 discount; Ksh 47,500 payable

Example 2: A cooperative purchases 2,500 kg of fertilizer at Ksh 40 per kg with a 7% quantity discount for orders over 2,000 kg. Calculate the discount and final amount.

Given: Unit Price = Ksh 40, Quantity = 2,500, Discount Rate = 7%

$$ \text{Total Price} = 40 \times 2,500 = 100,000 \text{ Ksh} $$

$$ \text{Quantity Discount} = 100,000 \times \frac{7}{100} = 7,000 \text{ Ksh} $$

$$ \text{Price after Quantity Discount} = 100,000 - 7,000 = 93,000 \text{ Ksh} $$

Answer: Ksh 7,000 discount; Ksh 93,000 payable

Example 3: A cooperative buys 3,000 liters of milk at Ksh 60 per liter. The seller offers a 10% quantity discount for purchases above 2,500 liters. Calculate the amount payable.

Given: Unit Price = Ksh 60, Quantity = 3,000, Discount Rate = 10%

$$ \text{Total Price} = 60 \times 3,000 = 180,000 \text{ Ksh} $$

$$ \text{Quantity Discount} = 180,000 \times \frac{10}{100} = 18,000 \text{ Ksh} $$

$$ \text{Price after Quantity Discount} = 180,000 - 18,000 = 162,000 \text{ Ksh} $$

Answer: Ksh 18,000 discount; Ksh 162,000 payable

Example 4: A cooperative orders 800 bags of maize at Ksh 1,200 per bag with a 4% quantity discount on orders above 700 bags. Calculate the discount and total cost.

Given: Unit Price = Ksh 1,200, Quantity = 800, Discount Rate = 4%

$$ \text{Total Price} = 1,200 \times 800 = 960,000 \text{ Ksh} $$

$$ \text{Quantity Discount} = 960,000 \times \frac{4}{100} = 38,400 \text{ Ksh} $$

$$ \text{Price after Quantity Discount} = 960,000 - 38,400 = 921,600 \text{ Ksh} $$

Answer: Ksh 38,400 discount; Ksh 921,600 payable

Example 5: A cooperative purchases 5,000 units of sugar at Ksh 90 per unit with an 8% quantity discount on purchases exceeding 4,000 units. Calculate the amount payable.

Given: Unit Price = Ksh 90, Quantity = 5,000, Discount Rate = 8%

$$ \text{Total Price} = 90 \times 5,000 = 450,000 \text{ Ksh} $$

$$ \text{Quantity Discount} = 450,000 \times \frac{8}{100} = 36,000 \text{ Ksh} $$

$$ \text{Price after Quantity Discount} = 450,000 - 36,000 = 414,000 \text{ Ksh} $$

Answer: Ksh 36,000 discount; Ksh 414,000 payable

Chapter Summary

This chapter covered various types of discounts including cash, trade, and quantity discounts, explaining their application in commercial transactions. It detailed how commissions are calculated as a form of payment based on sales performance. Methods for calculating wages were explored, focusing on piece rate and hourly rate systems. The chapter also demonstrated how to compute wages and salaries accurately. It introduced simple and compound interest calculations essential for financial assessments. Concepts of profit margin and mark-ups were explained to aid in pricing decisions. Additionally, the calculation of gross pay and net pay was examined, highlighting deductions and allowances. The chapter concluded with discussions on depreciation and appreciation of assets, determining hire purchase prices, and handling foreign exchange transactions in business contexts.