Percentages, Ratio, Profit & Loss

A percentage tells you how many parts out of 100. A ratio compares quantities. Profit is the amount earned above the buying cost.

Convert each percentage change to a multiplier. Choose a base such as 100 when no actual amount is needed.

A notebook costs ₹80. The shop gives a 25% discount. What do you pay? 1. 25% means one quarter: 25 ÷ 100 = 1 ÷ 4. 2. One quarter of ₹80 is ₹20. 3. Subtract the discount: ₹80 − ₹20 = ₹60.

The whole bar represents the marked price of ₹80. One quarter of its length is the ₹20 discount, and the remaining three quarters represent ₹60 paid. The base of this percentage is the marked price. For profit percentage, first identify the cost price instead of assuming the same base.

An item costs 800. It is marked 50% above cost and discounted twice, by 10% and 20%. Marked price=1200. Selling price=1200 x 0.9 x 0.8=864. Profit=64, so profit percentage=64/800 x 100=8%. Combined discount=1-0.9 x 0.8=28%, not 30%. Reverse problems divide by the multipliers in the opposite order.

Always ask: percentage of which amount? A discount uses the marked price; profit percentage usually uses the cost price. Profit percentage normally uses cost as its base; discount uses marked price. Write these bases before calculating.

Percent means out of 100. A ratio compares quantities using parts: 2:3 means two parts of one thing for every three of another. Profit is selling price minus cost price. A profit percentage is usually measured against cost price.

1. Decide what the whole or base is. 2. Convert a percentage to a fraction over 100. 3. For a ratio, add the parts and find the value of one part. 4. For profit, find the difference before finding its percentage. 5. For repeated percentage changes, apply the second change to the new value.

A notebook costs 80 rupees and sells for 100 rupees. Profit = 100 - 80 = 20 rupees. Profit percentage = 20 / 80 x 100 = 25%. A 20% increase followed by a 20% decrease takes 100 to 120 to 96, not back to 100.

The base matters. Dividing the profit by selling price would answer a different question.

Convert each percentage change to a multiplier. Choose a base such as 100 when no actual amount is needed.

An item costs 800. It is marked 50% above cost and discounted twice, by 10% and 20%. Marked price=1200. Selling price=1200 x 0.9 x 0.8=864. Profit=64, so profit percentage=64/800 x 100=8%. Combined discount=1-0.9 x 0.8=28%, not 30%. Reverse problems divide by the multipliers in the opposite order.

Profit percentage normally uses cost as its base; discount uses marked price. Write these bases before calculating.

p% means p/100 of an identified base. Percentage change = (new−old)/old × 100 for a positive nonzero base. A move from 20% to 25% is 5 percentage points, or a 25% relative increase.

Use multipliers: an increase of p% multiplies by 1+p/100; a decrease by 1−p/100. Successive signed changes a% and b% give a net change of a+b+ab/100 percent. If A is r% more than B, B is 100r/(100+r)% less than A. If A is r% less than B with 0≤r<100, B is 100r/(100−r)% more than A.

100 increased by 20% then reduced by 20% becomes 100 × 1.2 × 0.8 = 96. For fixed spending, a 25% price rise requires quantity to become 1/1.25 = 80% of the old amount, a 20% fall. Also, 36% of 1325 = (40%−4%) × 1325 = 530−53 = 477 exactly. Useful conversions 1/2=50%, 1/4=25%, 1/8=12.5%, 1/16=6.25%. The value of 1/3 is 33⅓%; 33.33% is only an approximation. Multiply recurring decimals only with appropriate rounding.

Scaling every length of a similar figure by k scales area by k² and volume by k³. Merely increasing the sides of an arbitrary quadrilateral does not establish that its area scales this way unless its shape is preserved.

Let CP be cost price, SP selling price and MP marked price. Profit = SP−CP; loss = CP−SP. Profit or loss percentage uses CP, while discount percentage uses MP. These bases must be positive.

A product costs 400 and is marked 600. A 20% discount gives SP=600×0.8=480. Profit is 80, so profit percentage is 80/400×100=20%. Two successive discounts of 10% and 20% leave 0.9×0.8=72% of MP: the effective discount is 28%.

If two articles sell for the same price, one at x% profit and one at x% loss with 0≤x<100, the combined loss percentage is x²/100. For equal selling prices of 120 and x=20, their costs are 100 and 150. Total cost 250 exceeds revenue 240 by 10, a 4% loss. If a seller charges for a stated weight W while delivering only w at the same cost per unit, gain percentage = (W−w)/w × 100. Charging for 1000 g while supplying 900 g gives 100/900×100≈11.11%, before any other costs.

The equal-selling-price loss shortcut does not apply to equal cost prices. Never divide a discount by cost price unless the question explicitly asks for that comparison.