Profit and Loss — Formulas, Shortcuts & Problems

Profit and loss compare cost price with selling price. If the selling price is higher, there is a profit. If it is lower, there is a loss. In standard school problems, the percentage is usually calculated using cost price as the base.

$$\text{Profit} = \text{SP} - \text{CP}, \qquad \text{Loss} = \text{CP} - \text{SP}$$

The percentage formulas are:

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

If $\text{SP} = \text{CP}$, there is no profit and no loss. If a question uses a different base, it should say so explicitly.

What Cost Price And Selling Price Mean

Cost price, or $\text{CP}$, is the amount paid to buy the item.

Selling price, or $\text{SP}$, is the amount received when the item is sold.

If $\text{SP} > \text{CP}$, the difference is profit. If $\text{SP} < \text{CP}$, the difference is loss. This is why profit and loss questions are really comparison questions first and percentage questions second.

Profit And Loss Formulas To Remember

These are the formulas used in most school and exam problems:

$$\text{Profit} = \text{SP} - \text{CP}$$ $$\text{Loss} = \text{CP} - \text{SP}$$ $$\text{Profit\%} = \frac{\text{Profit}}{\text{CP}} \times 100$$ $$\text{Loss\%} = \frac{\text{Loss}}{\text{CP}} \times 100$$

If the percentage is given and you need the selling price, these shortcut forms are faster:

$$\text{SP} = \text{CP}\left(1 + \frac{p}{100}\right)$$

for a profit of $p\%$, and

$$\text{SP} = \text{CP}\left(1 - \frac{l}{100}\right)$$

for a loss of $l\%$.

Worked Example: Find Profit And Profit Percent

Suppose a shopkeeper buys a bag for $500$ and sells it for $575$.

Step 1: compare selling price and cost price.

$$575 > 500$$

So the result is a profit.

Step 2: find the profit amount.

$$\text{Profit} = 575 - 500 = 75$$

Step 3: find the profit percent using cost price as the base.

$$\text{Profit\%} = \frac{75}{500} \times 100 = 15\%$$

So the shopkeeper made a profit of $75$ and a profit percent of $15\%$.

You can check the same result with the shortcut formula:

$$\text{SP} = \text{CP}\left(1 + \frac{15}{100}\right) = 500(1.15) = 575$$

That shortcut helps when the percentage is given first and you want the selling price quickly.

How To Solve Profit And Loss Questions Faster

If a question gives cost price and selling price, subtract first. Then decide whether the difference represents profit or loss.

If a question gives cost price and a profit or loss percent, go straight to the shortcut selling-price formula. That usually saves a step.

If a question asks for a percentage, pause and check the denominator. In standard profit and loss problems, it is cost price, not selling price.

Common Profit And Loss Mistakes

Using the wrong denominator

In standard textbook problems, profit percent and loss percent are calculated on cost price. Using selling price gives a different ratio.

Mixing up profit and loss

Do not use $\text{SP} - \text{CP}$ unless you already know the result should be profit. If the item sold for less, use $\text{Loss} = \text{CP} - \text{SP}$ directly.

Forgetting the no-profit-no-loss case

When $\text{SP} = \text{CP}$, the answer is neither profit nor loss. The percentage is $0\%$.

Applying a percent without checking the base

If the problem says "gain of $20\%$," that usually means $20\%$ of cost price. If the wording changes the base, the formula changes too.

Where Profit And Loss Is Used

You see profit and loss in school arithmetic, retail pricing, resale questions, and basic business math. The same idea appears whenever someone compares buying cost with selling value.

The arithmetic is simple, but careful reading matters. Most mistakes come from choosing the wrong base or mixing up cost price and selling price.

Try A Similar Profit And Loss Problem

Try your own version with cost price $800$ and selling price $720$. Decide first whether it is a profit or a loss, then find the amount and the percentage. After that, change only the selling price and see how the sign of the difference changes the whole problem.