Profit, Loss & Discount

The setup

Two articles are sold for the same price: one at a profit of x%, the other at a loss of x%. Most people guess "no profit no loss". The answer is always a net loss.

The net loss formula

$\text{Net loss\%} = \frac{x^2}{100}\% \quad \text{of the total cost price}$ 20% and 20% → 4% loss. 25% → $6.25\%$. 10% → 1%. $16\frac{2}{3}\% = \frac{1}{6}$ → $\left(\frac{1}{6}\right)^2 = \frac{1}{36} = 2\frac{7}{9}\%$.

Why there must be a loss

Loss% and gain% act on two different cost prices. The article sold at a loss has the bigger CP, so its percentage bites harder. With common SP S: $CP_{\text{profit}} = \frac{S}{1 + x/100}, \qquad CP_{\text{loss}} = \frac{S}{1 - x/100}$ and the excess of $CP_{\text{loss}}$ over S is always greater than the excess of S over $CP_{\text{profit}}$, because dividing by a number below 1 moves further than multiplying by a number above 1.

Full reconstruction (when rupees are asked)

Two watches sold at ₹960 each, at 20% profit and 20% loss:

• $CP_1 = 960 \div 1.2 = ₹800$; $CP_2 = 960 \div 0.8 = ₹1{,}200$.
• Total CP $= ₹2{,}000$, total SP $= ₹1{,}920$ → loss $= ₹80 = 4\%$ of ₹2,000. The formula checks out.

Useful special cases

• SP ₹1,980 each at ±10%: CPs 1,800 and 2,200 → loss ₹40.
• SP ₹600 each at ±20%: CPs 500 and 750 → loss ₹50.
• The loss per cent depends only on x, never on the SP.
• Unequal percentages (say +20% and −17.5%) break the $x^2/100$ shortcut — rebuild both CPs from the SP and add.

Where the loss hides — a fuller look

Take SP ₹100 on each article at ±20%. The profitable article cost $100 \div 1.2 = ₹83.33$, so the gain is ₹16.67. The loss-making article cost $100 \div 0.8 = ₹125$, so the loss is ₹25. Same ₹100 selling price, but the loss (₹25) beats the gain (₹16.67) — the gap $25 - 16.67 = ₹8.33$ is exactly 4% of the total CP ₹208.33. The bigger the x, the wider the gap: at ±50% the loss is $\frac{2500}{100} = 25\%$ of CP.

A three-step routine for every version

1. Name the common SP (call it S, or use the rupee value given).
2. Recover each CP with its own chip: divide by $1 + \frac{x}{100}$ for the gain item and by $1 - \frac{x}{100}$ for the loss item.
3. Add the CPs, add the SPs, and read off the profit or loss.

How the options are built

• "No profit, no loss" — the instinct answer; always wrong here.
• $\frac{x^2}{200}\%$ — halves the correct loss.
• $\frac{2x^2}{100}\%$ — doubles it.
• The correct $\frac{x^2}{100}\%$. For rupee versions, one option usually applies the loss per cent to the SP instead of the CP.

Two quick checks

• The loss per cent must depend only on x: change the SP and the percentage does not move.
• The loss item's CP is always the bigger one — if your reconstruction says otherwise, a chip was inverted.

Quick revision

• Equal ±x% on the same SP → loss $\frac{x^2}{100}\%$.
• CPs: divide the common SP by $1 \pm \frac{x}{100}$.
• Loss in rupees = total CP − total SP; check it equals $\frac{x^2}{100}\%$ of total CP.
• The result is never zero — "no profit no loss" is the trap.