Commercial arithmetic problems

A shopkeeper buys a bicycle, spends money fixing it up, and sells it on — did he gain or lose, and by how much? A shirt is marked ₹1,200 and offered "15% off" — what does the customer actually pay? Commercial arithmetic is the set of tools that answers exactly these questions: cost price (CP), selling price (SP), marked price (MP), overhead expenses, profit, loss and discount. Every transaction has two separate comparisons hiding inside it — profit or loss is always measured against what the seller paid (CP), while discount is always measured against what the tag says (MP). Keeping those two bases straight, and knowing which formula to reach for when a problem gives you some quantities and hides others, is the whole game. This article compresses the ideas into the formulas you need, one full derivation of the discount-to-marked-price rearrangement, and a worked example that chains overheads, profit and discount together.

The core insight

Profit and loss are always calculated as a percentage of the cost price, while discount is always calculated as a percentage of the marked price — mixing up these two bases is the single most common commercial-arithmetic error.

The core idea

Profit and loss compare what a seller paid (CP) with what the article actually sold for (SP):

\text{Profit} = SP - CP \quad (\text{when } SP > CP), \qquad \text{Loss} = CP - SP \quad (\text{when } CP > SP)

where: $SP$ is the selling price and $CP$ is the cost price, both in the same currency. Profit or loss is then always converted to a percentage of the cost price, never of the selling price:

\text{Profit\%} = \frac{\text{Profit}}{CP} \times 100\%, \qquad \text{Loss\%} = \frac{\text{Loss}}{CP} \times 100\%

If a shopkeeper spends extra money on repairs, transport or packing after buying an article, that overhead expense is folded into the cost price before any profit or loss is worked out — it is money spent to bring the article to a sellable state, so it belongs on the "cost" side of the ledger:

CP_{\text{effective}} = \text{Purchase price} + \text{Overheads}

Discount is a different comparison altogether: it measures how far the selling price falls below the price printed on the tag, the marked price (MP):

\text{Discount} = MP - SP, \qquad \text{Discount\%} = \frac{\text{Discount}}{MP} \times 100\%

where: $MP$ is the marked (listed) price and the discount percentage is always taken of the marked price. The figure below keeps the two journeys — the seller's CP-to-SP journey and the shopper's MP-to-SP journey — visually apart, which is exactly the separation that word problems try to blur.

Figure

Two parallel number lines showing cost price rising to selling price as profit on top and marked price falling to selling price as discount on the bottom

Rearranging for SP or MP from a discount

The most direct formula gives discount from MP and SP, but problems often flip the question and ask for SP or MP directly. This derivation, more than 150 characters long as befits its scope, sets out to rearrange the basic discount relation so that the selling price can be recovered from a marked price and a discount percentage, and — reversed a second time — so that the marked price can be recovered from a target selling price and a discount percentage, which is the step students find hardest to trust.

Derivation AFinding SP or MP from a discount percentage

The goal is to start from the two defining relations $\text{Discount} = MP - SP$ and $\text{Discount\%} = \dfrac{\text{Discount}}{MP} \times 100\%$ , and rearrange them into two ready-to-use formulas — one that gives SP directly from MP and a discount percentage, and one that gives MP directly from a target SP and a discount percentage — without ever computing the discount amount as a separate step.

Step 1. Express the discount amount in terms of MP and the discount percentage. From $\text{Discount\%} = \dfrac{\text{Discount}}{MP} \times 100\%$ , multiply both sides by $MP$ and divide by $100\%$ to get $\text{Discount} = MP \times \dfrac{\text{Discount\%}}{100}$ .

Step 2. Substitute this into $SP = MP - \text{Discount}$ . This gives $SP = MP - MP \times \dfrac{\text{Discount\%}}{100} = MP\left(1 - \dfrac{\text{Discount\%}}{100}\right)$ . This is the forward formula: given a marked price and a discount percentage, the selling price falls out in one line.

Step 3. Reverse the formula to solve for MP instead. Starting from $SP = MP\left(1 - \dfrac{\text{Discount\%}}{100}\right)$ , divide both sides by the bracket $\left(1 - \dfrac{\text{Discount\%}}{100}\right)$ , which can also be written as $\dfrac{100 - \text{Discount\%}}{100}$ . Dividing by a fraction means multiplying by its reciprocal, so $MP = SP \times \dfrac{100}{100 - \text{Discount\%}} = \dfrac{SP \times 100}{100 - \text{Discount\%}}$ .

Step 4. Check the two formulas are consistent by plugging one into the other with a round number: take $MP = 1000$ and $\text{Discount\%} = 20$ . Then $SP = 1000\left(1 - \dfrac{20}{100}\right) = 800$ . Feeding this SP back into the MP formula, $MP = \dfrac{800 \times 100}{100 - 20} = \dfrac{80000}{80} = 1000$ , exactly recovers the starting marked price, confirming the two rearrangements are true inverses of each other.

SP = MP\left(1 - \frac{\text{Discount\%}}{100}\right), \qquad MP = \frac{SP \times 100}{100 - \text{Discount\%}}

where: $SP$ is the selling price, $MP$ is the marked price, and $\text{Discount\%}$ is the discount rate applied to the marked price. Analogous rearrangements hold for profit and loss on the cost-price side: $SP = CP\left(1 + \dfrac{\text{Profit\%}}{100}\right)$ or $SP = CP\left(1 - \dfrac{\text{Loss\%}}{100}\right)$ .

Worked example

A trader buys a bicycle for ₹2,400 and spends ₹150 on repairs, then sells it for ₹2,805. Separately, a shirt is marked at ₹1,200 and discounted 15% in a sale. Find the trader's profit percentage, the shirt's sale price, and the marked price the shopkeeper would need to set to sell the same shirt at ₹1,200 after a 20% discount.

Step 1. Find the trader's effective cost price, folding in the overhead. $CP_{\text{effective}} = 2400 + 150 = ₹2550$ .

Step 2. Compare with the selling price. Since $SP = 2805 > CP = 2550$ , this is a profit: $\text{Profit} = 2805 - 2550 = ₹255$ , so $\text{Profit\%} = \dfrac{255}{2550} \times 100\% = 10\%$ .

Step 3. Find the shirt's discount and sale price using $\text{Discount} = 15\%$ of $1200 = \dfrac{15}{100} \times 1200 = ₹180$ , so $SP = 1200 - 180 = ₹1020$ .

Step 4. Now work backwards for the second shirt scenario: the desired SP is ₹1,200 and the discount rate is 20%, so use the rearranged formula from the derivation above. $MP = \dfrac{SP \times 100}{100 - \text{Discount\%}} = \dfrac{1200 \times 100}{100 - 20} = \dfrac{120000}{80} = ₹1500$ . Check: 20% of ₹1,500 is ₹300, and $1500 - 300 = ₹1200$ , which matches.

The trader made a profit of 10%, the sale shirt sells for ₹1,020, and the shopkeeper must mark the second shirt at ₹1,500.

Common traps

Using the wrong base. Profit% and loss% are always on CP; discount% is always on MP — never swap them, and never use SP as a base for either.
Forgetting overheads. Repair, transport or packing costs must be added to the purchase price to get the effective CP before profit or loss is calculated.
Assuming SP always sits between CP and MP. A large discount can push SP below CP even though the article was marked up — always calculate, never assume.
Stacking successive discounts wrongly. "20% off, then 10% off" is not a flat 30% off — the second discount applies to the already-reduced price, not the original marked price.

Ask the tutor: if a word problem mixes overheads, profit and discount in one scenario and you are not sure which base to use at each step, ask the AI tutor to walk through that specific problem's numbers with you.

Why is discount percentage calculated on the marked price instead of the cost price?

Because discount describes what happens between the price on the tag and what the customer finally pays — it has nothing to do with what the shopkeeper originally paid. The marked price is the reference point a customer actually sees, so "15% off" naturally means 15% of that printed number. Profit and loss, by contrast, describe the seller's own transaction, comparing what was paid (CP) with what was received (SP), so their percentage naturally uses CP as the base. Keeping the two questions separate — "what did the shopkeeper gain" versus "how much cheaper is this than the tag" — is what keeps the bases straight.

If a shopkeeper gives a discount, does that mean the sale is always a loss?

No. Discount only tells you how SP compares with MP, not how SP compares with CP. A shopkeeper typically marks an article well above the cost price precisely so that even after a generous discount, SP still lands above CP and a profit remains. In the worked example, the bicycle trader made a 10% profit with no discount involved at all, while the shirt example shows a discounted SP that could still sit comfortably above its own cost price — the two comparisons are simply independent unless a problem links them explicitly.

How do I know whether to use the forward SP formula or the reversed MP formula?

Look at which two quantities the problem already gives you. If you know the marked price and the discount percentage and need the selling price, use $SP = MP\left(1 - \text{Discount\%}/100\right)$ directly. If instead you know the selling price you want to end up with and the discount percentage being offered, and need to find what marked price achieves that, use the reversed formula $MP = SP \times 100/(100 - \text{Discount\%})$ . The safest habit is to write down what you are given and what you are asked for before picking a formula, rather than memorising which one to use.

Commercial arithmetic problems

The core idea

Rearranging for SP or MP from a discount

Worked example

Common traps

Before this

Where this leads