A small monthly payment does not tell you the total price. To compare two classroom offers, first turn each offer into one complete amount. Then calculate the difference and, only if asked, express that difference as a percentage of a clearly named amount. This order separates the arithmetic from the way an offer is advertised.
This lesson uses invented equipment prices in Kenyan shillings. Every example states that all charges are included. The calculations compare nominal totals only; they are not current offers, lending advice or recommendations about anyone’s budget. The cash and instalment cases below do not model an interest rate, tax band or change in the value of money over time. The later interest section introduces separate, explicitly fictional simple and annual-compound models.
Identify what each number means
The marked price is the displayed starting price. A cash discount reduces that price when the question says it applies. The cash price is the amount payable under that cash offer. A deposit is an initial payment under the instalment offer. Instalments are the remaining scheduled payments. The deposit contributes to the total; it is not a discount unless the question explicitly says so.
Let M be the marked price, d the discount percentage, D the deposit, n the number of equal instalments and I the amount of each instalment. The cash total is C = M(1 − d/100). The instalment total is H = D + nI. The extra amount paid under the instalment offer is H − C. Write these three lines before making a percentage comparison.
If a deposit is given as a percentage, identify its base first. “A deposit of 20% of the marked price” means 0.20M. It does not mean 20% of the discounted cash price. If the wording leaves the base unspecified, the data are incomplete; a calculation cannot repair an ambiguous condition.
Worked case 1: cash discount against twelve payments
A fictional equipment supplier gives two complete offers. The marked price is KSh 85,000. The cash discount is 8% of the marked price. Alternatively, the buyer pays a KSh 18,000 deposit now and twelve monthly instalments of KSh 6,500, with the first instalment one month later. All charges are included. Find both totals and the extra instalment cost as a percentage of the cash price.
The discount is 8/100 × 85,000 = KSh 6,800. Therefore the cash price is 85,000 − 6,800 = KSh 78,200. A useful check is that 92% of 85,000 also gives 78,200: paying after an 8% discount means paying the remaining 92%.
The twelve instalments total 12 × 6,500 = KSh 78,000. Add the deposit once: H = 18,000 + 78,000 = KSh 96,000. The extra amount is 96,000 − 78,200 = KSh 17,800. As a percentage of the cash price, it is 17,800/78,200 × 100 = 22.7621…%, or 22.76% to two decimal places.
The timeline has a KSh 18,000 deposit at month 0 and a KSh 6,500 payment at each month from 1 through 12. There are twelve instalments, not thirteen. Month 0 identifies the deposit separately; it is already included by the D term in D + nI.
| Step | Cash offer | Instalment offer |
|---|---|---|
| Starting information | Marked price KSh 85,000 | Deposit KSh 18,000 |
| Adjustment or remaining payments | Subtract KSh 6,800 discount | Add 12 × KSh 6,500 = KSh 78,000 |
| Complete amount | KSh 78,200 | KSh 96,000 |
| Comparison | Instalments cost KSh 17,800 more in the supplied model | |
Why the percentage base changes the answer
The same KSh 17,800 difference is 18.5416…% of KSh 96,000. Thus “instalments are 22.76% more than cash” and “cash is 18.54% less than instalments” describe the same two totals using different denominators. Reversing the comparison does not preserve the percentage.
Another possible calculation is 17,800/85,000 × 100 = 20.9411…%. That is the difference as a percentage of the marked price. It is a valid answer only when that is the requested base. A percentage without its reference amount leaves the comparison unfinished. Do not average these percentages: they answer different questions.
Worked case 2: find a missing monthly instalment
A second fictional item has a marked price of KSh 64,000 and a cash discount of 5%. Its instalment total is stated to be 15% more than its cash price. The deposit is KSh 13,920, followed by ten equal monthly instalments. All charges are included. Find each instalment.
Cash price: C = 64,000 × 0.95 = KSh 60,800. The extra 15% is based on this cash price, so the instalment total is H = 60,800 × 1.15 = KSh 69,920. Subtract the deposit before sharing the remaining amount across ten payments: I = (69,920 − 13,920)/10 = KSh 5,600.
Check forwards: 13,920 + 10 × 5,600 = 69,920, and 69,920 − 60,800 = 9,120 = 15% of 60,800. Dividing the whole KSh 69,920 by ten would incorrectly make the deposit payable again through the instalments.
Worked case 3: recover the marked price
For a third fictional offer, the discounted cash price is KSh 73,920 after a 12% discount on the marked price. The alternative is a KSh 20,000 deposit and eight instalments of KSh 7,400. All charges are included. Find the marked price and compare the complete offers.
After a 12% reduction, the cash price represents 88% of the marked price. Therefore 0.88M = 73,920 and M = 73,920/0.88 = KSh 84,000. Check: 12% of 84,000 is 10,080; subtracting gives 73,920.
The instalment total is 20,000 + 8 × 7,400 = KSh 79,200. It exceeds the cash price by KSh 5,280, which is 5,280/73,920 × 100 = 7.1428…%, or 7.14%. Adding 12% to the cash price would give KSh 82,790.40, not the marked price, because that would use the smaller cash amount as the base.
Three practice tasks with full solutions
Try each calculation before reading its solution. All figures below are fictional and include every charge. State the percentage base in words as well as in the fraction.
Practice 1: compare two complete prices
An item is marked KSh 48,000 with a 7.5% cash discount. Its alternative is a KSh 9,000 deposit plus nine monthly instalments of KSh 4,600. Find the extra instalment cost as a percentage of the cash price.
The discount is KSh 3,600, so C = KSh 44,400. The instalment total is 9,000 + 9 × 4,600 = KSh 50,400. The difference is KSh 6,000. Using cash as the base gives 6,000/44,400 × 100 = 13.5135…%, approximately 13.51%.
Practice 2: work backwards to an instalment
The cash price is KSh 52,500. The instalment total is 20% above cash, with a KSh 15,000 deposit and twelve equal monthly instalments. Calculate each instalment.
H = 52,500 × 1.20 = KSh 63,000. The outstanding payments total 63,000 − 15,000 = KSh 48,000, so each instalment is KSh 4,000. Check: 15,000 + 12 × 4,000 = 63,000.
Practice 3: reverse a discount, then compare
The cash price after a 16% discount is KSh 63,000. The other offer is KSh 12,000 plus ten instalments of KSh 5,700. Find the marked price and the percentage extra over cash.
M = 63,000/0.84 = KSh 75,000. The instalment total is 12,000 + 57,000 = KSh 69,000. The KSh 6,000 difference is 6,000/63,000 × 100 = 9.5238…%, approximately 9.52% of cash.
Profit, loss and fictional interest comparisons
The cash and instalment cases above compare complete nominal prices. Extend that comparison with the following invented trading and interest models. Rates and charges here are mathematical assumptions, not available products, financial advice or predictions. No tax, fees, withdrawals or changing rates apply unless stated.
Worked trading case: discount can leave profit or loss
A trader buys one item for KSh 2,400. Transport allocated to this item costs KSh 100, making the stated total cost KSh 2,500. The marked selling price is KSh 3,200. With a 15% discount, selling price = 3,200 × 0.85 = KSh 2,720. Profit = 2,720 − 2,500 = KSh 220, or 220/2,500 × 100 = 8.8% of total cost.
Compare a 25% discount on the same marked price: the customer pays KSh 2,400, leaving a loss of KSh 100, or 4% of total cost. This is why a discount percentage is not a profit percentage. Their bases differ. The first offer’s profit margin on selling price would instead be 220/2,720 × 100 ≈ 8.09%; that is a different requested measure. Reversing the check, 2,500 × 1.088 = 2,720 and 2,500 × 0.96 = 2,400.
Worked interest case: same principal, rate and duration
Compare two fictional balances starting at principal P = KSh 20,000 for two complete years. Model S uses simple interest at 5% per year on the original principal. Model C adds interest annually at 5% per year, so the next year also earns interest on the first year’s interest. Nothing is deposited or withdrawn.
Simple interest I = Prt = 20,000 × 0.05 × 2 = KSh 2,000; final amount A = P + I = KSh 22,000. Annual compound amount A = P(1 + r)t = 20,000 × 1.05² = KSh 22,050, of which KSh 2,050 is interest. Year-by-year: 20,000 becomes 21,000, then 21,000 + 1,050 = 22,050. The extra KSh 50 is 5% of the first year’s KSh 1,000 interest.
This comparison is valid because principal, annual rate, duration and cash-flow assumptions match. For nine months of the simple-interest model, t = 9/12 years and I = KSh 750. Do not put t = 9 alongside an annual rate. The annual-compounding model above specifies two complete years; it does not state a convention for part-years. Nor is the extra instalment cost in the earlier examples an annual interest rate: a percentage price difference alone does not define a reducing-balance loan schedule.
Independent comparisons
- An item has total cost KSh 1,800 and marked price KSh 2,500. Compare a 20% discount with a 30% discount. Find the profit or loss and its percentage of total cost in each case.
- Using a fictional principal of KSh 15,000 at 8% per year for two complete years, compare simple interest with annually compounded interest. State both final amounts and their difference. There are no other cash flows or charges.
Explained answers
- At 20% off, selling price = KSh 2,000, profit = KSh 200, and profit percentage = 200/1,800 × 100 = 11⅑%. At 30% off, selling price = KSh 1,750, loss = KSh 50, and loss percentage = 50/1,800 × 100 = 2 7/9%. The discount base is KSh 2,500 in both routes; the profit/loss base is KSh 1,800.
- Simple interest = 15,000 × 0.08 × 2 = KSh 2,400, so final amount = KSh 17,400. Compound amount = 15,000 × 1.08² = KSh 17,496, including KSh 2,496 interest. Difference = KSh 96, equal to 8% of the first year’s KSh 1,200 interest.
Review by labelling every percentage base, keeping the interest rate’s period consistent with time, and distinguishing interest earned from the final balance. A correct number without those assumptions may answer a different question.
Frequently asked questions
Is the marked price always the cash price?
No. A stated cash discount changes it. Calculate the discounted cash price before comparing complete offers.
Do I add the deposit to every instalment?
No. Add the deposit once, then add the total of all scheduled instalments.
What if there is a separate delivery charge?
Include it in the applicable total if the question supplies it. These examples explicitly include all charges.
Which denominator gives the percentage extra?
Use the amount named after “of” or “compared with.” Extra over cash uses the cash price as denominator.
Can I undo a discount by adding the same percentage?
No. Divide the discounted amount by its remaining proportion, such as 0.88 after a 12% discount.
When should I round?
Keep exact money totals and full intermediate percentages. Round the final answer to the precision requested.
Does the smaller total settle a real purchase decision?
It settles only this supplied nominal-total comparison. Timing, affordability and actual contractual terms are outside the classroom model.
Check your next comparison
Write down the percentage base, payment count and full total for each route. The Mathematics topical revision collection provides further resource navigation. Some contents may require membership; no particular worksheet or download is promised here.
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