Important Quantitative Aptitude Formulas on Profit and Loss

Important Formulas - Profit and Loss

 Cost price and selling price

Cost price (CP) is the price at which an article is purchased.

Selling price (SP) is the price at which an article is sold.

Profit and loss

If selling price is more than cost price, profit(gain) occurs.

If selling price is less than cost price, loss occurs.

In case of profit,

profit = selling price – cost price

selling price = cost price + profit

cost price = selling price - profit

In case of loss,

loss = cost price - selling price

selling price = cost price - loss

cost price = selling price + loss

Profit percentage and loss percentage

Profit percentage and loss percentage are always calculated on cost price unless otherwise stated.

In case of profit,

$\text{profit percentage}=\frac{\text{profit}\times 100}{\text{cost price}}$

$\text{selling price}=\text{cost price}+\frac{\text{cost price}\times \text{profit percentage}}{100}=\frac{\text{cost price}(100+\text{profit percentage})}{100}$

$\text{cost price}=\frac{100\times \text{selling price}}{100+\text{profit percentage}}$

Example: If an object is sold at a profit of $20\%,$
selling price $=120\%$ of cost price

In case of loss,

$\text{loss percentage}=\frac{\text{loss}\times 100}{\text{cost price}}$

$\text{selling price}=\text{cost price}-\frac{\text{cost price}\times \text{loss percentage}}{100}=\frac{\text{cost price}(100-\text{loss percentage})}{100}$

$\text{cost price}=\frac{100\times \text{selling price}}{100-\text{loss percentage}}$

Example: If an object is sold at a loss of $20\%$
selling price $=80\%$ of cost priceSelling at same price

Suppose a person sells two objects at the same price, one at a profit of 

$x_1\%$ and another at a profit of $x_2\%.$ Then,
net profit percentage $=\frac{100(x_1+x_2)+2x_1{x}_2}{200+x_1+x_2}$
Note:
(a) for loss, use -sign for $x_1$ and/or $x_2$ as applicable.
(b) If the formula evaluates to a -ve value, it means there is a net loss

 Suppose a trader sells two objects at the same price, one at a profit of $x\%$ and another at a loss of $x\%.$ Then he always incurs a net loss expressed as
net loss percentage $={\left(\frac{x}{10}\right)}^2$

Marked price and discount

The price on the label of an article is called its marked price(list price).

The deduction on the marked price is called discount.

selling price = marked price - discount

discount = marked price - selling price

marked price = selling price + discount

Note:
(a) If there is no discount, the marked price is equal to the selling price
(b) Discount is always calculated on marked price unless otherwise stated.

Discount percentageDiscount percentage is the discount expressed as a percent of marked price. 

discount percentage $=\frac{\text{discount}\times 100}{\text{marked price}}$

$\text{selling price}=\text{marked price}-\frac{\text{marked price}\times \text{discount percentage}}{100}=\frac{\text{marked price}(100-\text{discount percentage})}{100}$

$\text{marked price}=\frac{100\times \text{selling price}}{100-\text{discount percentage}}$

Successive discounts

If $d_1\%,d_2\%,d_3\%$ are successive discounts on marked price,
$\text{selling price}=\text{marked price}$ $\left(\frac{100-d_1}{100}\right)\left(\frac{100-d_2}{100}\right)\left(\frac{100-d_3}{100}\right)$

$\mathrm{Other\; important\; formulas1.If\; cost\; price\; of x\; articles\; is\; same}$as selling price of $y$ articles,
profit percentage $=\frac{(x-y)100}{y}$
Note: If the formula evaluates to a -ve value, it means there is a loss.

2. A trader purchases $m_1$ articles for $n_1$ and sells $m_2$ articles for $n_2$ Then,
profit percentage $=(\frac{m_1{n}_2}{m_2{n}_1}-1)100$
Note: If the formula evaluates to a -ve value, it means there is a loss.

3. A person buys two articles for $x.$ He sells one at a profit of $p\%$ and the other at a loss of $l\%$. If each item was sold at the same price, then

cost price of the article which was sold in profit
$=\frac{x(100-l)}{200+p-l}$

cost price of the article which was sold in loss
$=\frac{x(100+p)}{200+p-l}$

 4. A sells an object to B at a profit of $y_1\%,$ B sells it to C at a profit of $y_2\%$ and C sells it to D at a profit of $y_3\%$. Then,
$\text{cost price of D}$ $=\text{cost price of A}\left(\frac{100+y_1}{100}\right)$ $\left(\frac{100+y_2}{100}\right)\left(\frac{100+y_3}{100}\right)$
Note: for loss, use -sign for $y_1,y_2$ and/or $y_3$ as applicable.

5. A trader purchases a certain number of objects at $x$ objects for a rupee. He again purchases the same number of objects at $y$ objects for a rupee. He mixes them and sells each at $z$ objects for a rupee. Then,
profit percentage $=[\frac{2xy}{z(x+y)}-1]100$
Note: If the formula evaluates to a -ve value, it means there is a loss.

6. If an article is sold at a price of $s_1,$ a person loses a certain amount. If the article is sold at a price of $s_2$, he gains the same amount. Then,
$\text{cost price}=\frac{s_1+s_2}{2}$

 Using false weights

1. If a trader uses faulty measure of $f$ units instead of $t$ units and sells his goods at a profit of $x\%,$
net profit percentage $=\frac{100(t-f)+tx}{f}$
Note:
(a) for loss, use -sign for $x$
(b) If the formula evaluates to a -ve value, it means there is a net loss

2.If a trader uses faulty measure of $f$ units instead of $t$ units, but professes to sell his goods at cost price,
net profit percentage $=\frac{100(t-f)}{f}$
Note: If the formula evaluates to a -ve value, it means there is a net loss