Commercial Arithmetic 1
Interactive lessons, worked examples, and problem-solving techniques to help Kenyan students master KJSEA, KPSEA, KCSE, and IGCSE mathematics.
First Principles
Objective: Master everyday commercial transactions by breaking down Profit, Loss, Discount, and Simple Interest to foundational percentage relationships.
The Anchor: Understanding the Base
Every commercial transaction compares a changing amount to a base value (representing \(100\%\)). The secret to never getting confused is pinpointing the base:
• Profit or Loss is always calculated on the Cost Price (CP).
• Discount is always calculated on the Marked Price (MP).
• Simple Interest is always calculated on the Principal (P).
(a) Concrete Scenario: The Open-Air Market
Consider Mama Mwangi stocking 50 kg bags of rice in Gikomba market. She purchases a bag for \(\text{KSh } 4{,}000\) (Cost Price). To make room for bargaining, she writes \(\text{KSh } 5{,}200\) on the price tag (Marked Price). When a loyal customer bargains, she agrees to a \(10\%\) discount off the tagged price. Does she still make a profit?
(b) Visualizing Financial Flows
Think of money as a proportional bar:
- Marked Price: The full tag value (\(100\%\) of MP).
- Discount: A slice cut away from the Marked Price: \(\text{Discount} = 10\% \times 5{,}200 = \text{KSh } 520\).
- Selling Price: The money received: \(5{,}200 - 520 = \text{KSh } 4{,}680\).
- Profit: The surplus above the Cost Price: \(4{,}680 - 4{,}000 = \text{KSh } 680\).
🪙 Market Stall Ledger
Adjust the values and watch the profit, discount, and interest update live.
Key Formulas
1. Profit and Loss
\[\text{Profit} = \text{Selling Price (SP)} - \text{Cost Price (CP)}\]\[\text{Loss} = \text{Cost Price (CP)} - \text{Selling Price (SP)}\]\[\text{Profit \%} = \left(\frac{\text{Profit}}{\text{Cost Price}}\right) \times 100\%\]\[\text{Loss \%} = \left(\frac{\text{Loss}}{\text{Cost Price}}\right) \times 100\%\]2. Marked Price & Discount
\[\text{Discount} = \frac{\text{Discount \%}}{100} \times \text{Marked Price (MP)}\]\[\text{Selling Price (SP)} = \text{Marked Price (MP)} - \text{Discount}\]\[\text{Discount \%} = \left(\frac{\text{Discount}}{\text{Marked Price}}\right) \times 100\%\]3. Simple Interest (S.I.) & Total Amount
\[I = \frac{P \times R \times T}{100}\]\[A = P + I\]Where:• \(P\) = Principal (initial sum borrowed or invested in KSh)
• \(R\) = Rate of interest per annum (\(\%\))
• \(T\) = Time duration in years
• \(A\) = Total accumulated amount to repay
Worked Examples
Example 1 (Easy): Discount on Goods
A trader in Eldoret marks a solar lamp at \(\text{KSh } 3{,}200\). During a promotional week, he offers a \(15\%\) discount. Calculate the final price a customer pays.
Step-by-Step Solution:- Identify the base: Marked Price \(MP = \text{KSh } 3{,}200\).
- Find the discount amount:\[\text{Discount} = \frac{15}{100} \times 3{,}200 = 15 \times 32 = \text{KSh } 480\]
- Compute the selling price:\[\text{Selling Price} = 3{,}200 - 480 = \text{KSh } 2{,}720\]
Example 2 (Medium): Profit Percentage from Cost
A dairy farmer buys a milking machine for \(\text{KSh } 45{,}000\). She spends \(\text{KSh } 3{,}000\) on transport and installation. She later sells it for \(\text{KSh } 57{,}600\). Determine her percentage profit.
Step-by-Step Solution:- Find the total cost price (including overheads):\[\text{Total Cost Price} = 45{,}000 + 3{,}000 = \text{KSh } 48{,}000\]
- Calculate the profit earned:\[\text{Profit} = 57{,}600 - 48{,}000 = \text{KSh } 9{,}600\]
- Calculate profit percentage relative to total cost price:\[\text{Profit \%} = \left(\frac{9{,}600}{48{,}000}\right) \times 100\% = \left(\frac{1}{5}\right) \times 100\% = 20\%\]
Example 3 (Hard): Loan Repayment with Simple Interest
Otieno borrows \(\text{KSh } 80{,}000\) from a microfinance institution charging a simple interest rate of \(12\%\) per annum. If he settles the loan fully after \(2\frac{1}{2}\) years, calculate the total amount he pays back.
Step-by-Step Solution:- Extract values: Principal \(P = 80{,}000\), Rate \(R = 12\%\), Time \(T = 2.5\text{ years}\).
- Calculate simple interest accrued:\[I = \frac{P \times R \times T}{100} = \frac{80{,}000 \times 12 \times 2.5}{100} = 800 \times 30 = \text{KSh } 24{,}000\]
- Calculate total repayment amount:\[A = P + I = 80{,}000 + 24{,}000 = \text{KSh } 104{,}000\]
Common Mistakes
Misconception 1: Calculating Profit Percentage on Selling Price
Mistake: If an article bought for \(\text{KSh } 800\) is sold for \(\text{KSh } 1{,}000\), a student writes: \(\text{Profit \%} = \frac{200}{1000} \times 100 = 20\%\).
Correction: Profit measures return on capital invested (Cost Price). Thus: \(\text{Profit \%} = \frac{200}{800} \times 100 = 25\%\).
Why it feels right: The selling price is the number seen on the cash register receipt, so students mistakenly adopt it as the whole.
Misconception 2: Applying Discount to the Cost Price
Mistake: Calculating discount using the cost price rather than the marked price.
Correction: Discount is a customer-facing reduction. A shopkeeper discounts the price tag (Marked Price), not the wholesale purchase price (Cost Price).
Misconception 3: Forgetting Time Units in Simple Interest
Mistake: Using months directly as \(T\) in \(I = \frac{PRT}{100}\) (e.g., using \(T = 6\) for 6 months).
Correction: The annual rate \(R\) requires time \(T\) to be in years. Always convert months to years: \(6\text{ months} = \frac{6}{12} = 0.5\text{ years}\).
Real World
Agribusiness & Microfinance in Kenya
Every commercial venture in East Africa—from tea farming in Kericho to hardware retail in Nakuru—relies on commercial arithmetic to maintain cash flow and evaluate profitability.
🌾 SACCO Agricultural Loans
Farmers borrow from SACCOs to purchase certified seeds and fertilizer before rains. Calculating simple interest ensures they project whether harvest revenues will comfortably clear the debt.
📱 Mobile Money Agency Float
M-Pesa and banking agents earn percentage commissions on deposits and withdrawals. Accurately tracking percentage margins determines if an agency kiosk is sustainable.
Scenario: A youth cooperative borrows \(\text{KSh } 200{,}000\) at \(8\%\) simple interest per annum for 2 years to start a poultry project. The project yields \(\text{KSh } 270{,}000\) in net sales. After repaying the loan principal and interest (\(\text{KSh } 200{,}000 + \text{KSh } 32{,}000 = \text{KSh } 232{,}000\)), the cooperative retains \(\text{KSh } 38{,}000\) in pure profit!
Practice