Profit Loss and Discount

 

Profit, Loss and Discount

 

In our daily life, we come across so many financial transactions. We purchase and sale of some items. Shopkeepers and businessmen has different situations of profit and loss. This is very much applied aspect of mathematics in our daily life. Following are some important terms and formulae based on this chapter.

 

COST PRICE (CP)

The price at which an item is purchased, is known as Cost Price (CP) of the item.

            \[CP=\frac{SP\times 100}{(100+P%)}\]   (In case of profit)

\[CP=\frac{SP\times 100}{(100-L%)}\]    (In case of loss)

 

SELLING PRICE (SP)

 The price of an item at which the item is sold, is known as Selling Price (SP) of the item.

\[SP=\frac{CP\times (100+P%)}{100}\]   (In case of profit)

\[SP=\frac{CP\times (100-L%)}{100}\]    (In case of loss)

 

PROFIT

If the selling price of an item is more than cost price of that item, then there is a profit or gain on selling the item.

Profit or gain \[=SP-CP;\] Profit percentage \[\text{=}\frac{\text{Profit}}{\text{CP}}\text{ }\!\!\times\!\!\text{ 100}\]

 

LOSS

If the selling price of an item is less than cost price of that item, then there is a loss on selling the item.

\[Loss=CP-SP\]

Loss percentage \[=\frac{Loss}{CP}\times 100\]

 

Overhead Charges

Those charges which are extra expenditures on purchased items apart from actual cost price, are known as overhead charges. Such charges include freight charges, rent, salary of employee etc. If these charges are given in the question, then these are considered with cost price.

 

Marked Price (MP) or MRP

To avoid the loss due to bargaining by the customer and get the profit over the cost price, trader increases the cost price by a certain value, this increase in value over cost price is known as markup and the increased price (i.e., CP + Markup) is called the marked price or printed price or list price of the goods.

 

Marked Price = CP + Markup

Marked Price = CP + (Per cent markup of CP)

Generally, goods are sold at marked price, if there is no further discount offered, then in this case selling price equals to marked price.

 

DISCOUNT

When we go to market to buy some items from a shop, the shopkeeper attracts his customers by reducing marked price of the item. Due to this, items seems cheaper to the customers. The reduction in the marked price of item is known as discount. There is a fixed rate of discount on any item. By this rate, discount is calculated on marked price of the item.

\[\text{Discount=}\frac{\text{Marked price }\!\!\times\!\!\text{ Rate of discount}}{\text{100}}\]

If there is a discount of r% on the marked price of an item, then selling price of an item

\[=\text{Marked price}-\text{Discount}\]

\[\text{=Marked}\,\,\text{Price }\!\!\times\!\!\text{ }\left( \text{1-}\frac{\text{r}}{\text{100}} \right)\]

           

Remember

\[CP<SP\le MP\](When Profit Earned)

\[SP<CP\le MP\](When Loss Incurred)

\[CP=SP\le MP\](No Profit/No Loss)

Quicker One

Ø   If marked Price of an item is Rs. x and the successive discounts rates are \[{{\text{r}}_{1}}%,\] \[{{\text{r}}_{2}}%,\] \[{{\text{r}}_{3}}%,\] and so on, then selling price of the item

\[=x\times \left( 1-\frac{{{r}_{1}}}{100} \right)\left( 1-\frac{{{r}_{2}}}{100} \right)\left( 1-\frac{{{r}_{3}}}{100} \right)...\]

Ø   If a shopkeeper wants a profit of R% after allowing a discount of r%, then Marked Price (MP) of the item

\[=CP\times \left( \frac{100+R}{100-r} \right)\]  Cost Price (CP) of the item \[=MP\left( \frac{100-r}{100+R} \right)\]

Ø   Single discount equivalent to two successive discount  \[{{x}_{1}}%\] and\[\because \]

Ø   Single discount equivalent to three successive discounts \[{{x}_{1}}%,\] \[{{y}_{2}}%\] and \[{{z}_{3}}%\]

\[=\left[ 1-\left( 1-\frac{{{x}_{1}}}{100} \right)\,\left( 1-\frac{{{y}_{2}}}{100} \right)\,\left( 1-\frac{{{z}_{3}}}{100} \right) \right]\times 100%\]

Ø   A merchant fixes the marked price of an article in such a way that after allowing a discount of r%, he earns a profit of R%, then marked price of the article is \[\left( \frac{r+R}{100-r}\times 100 \right)%\] more than its cost price.

Ø   If a shopkeeper allows a discount of r% on an item and marked price on the article is r% more than the cost price, then profit or loss per cent in this transaction \[=\frac{r\times (100-{{r}_{1}})}{100}-{{r}_{1}}\] Positive value shows a profit while negative value shows loss.

Ø   If cost price of x articles is equal to selling price of y articles and \[x>y,\] then there is always a profit.

Profit percentage \[=\frac{x-y}{y}\times 100%\] But if \[x<y,\] then there is always a loss.

Loss percentage \[=\frac{y-x}{y}\times 100%\]

Ø   If n the part of some items is sold at r% profit (loss), then required loss (gain) per cent in selling rest of the items in order that there is neither profit nor loss in whole transaction, is \[\frac{nr}{1-n}%.\]

Ø   If selling price of x articles is equal to the profit of selling price o f y, then

Profit percentage \[=\frac{y}{x-y}\times 100\]

Loss percentage \[=\frac{y}{x+y}\times 100\]

Ø   If on selling x articles for Rs. 1, there is a profit (loss) of r%. then number of articles purchased for Rs. 1 will be \[\frac{x\,\,(100\pm r)}{100}.\]

Ø   If on purchasing x articles for Rs. 1, there is a profit (loss) of r%, then number of articles sold for Rs. 1 will be \[x\left( \frac{1\times 100}{100\pm r} \right).\]

Ø   If a person purchased y articles for Rs. x and sold them at a rate of x articles for Rs. y, then

(i) If \[x>y,\] then loss percentage \[=\frac{(x+y)(x-y)\times 100}{{{x}^{2}}}%\]

(ii) If \[x<y,\] then gain percentage \[=\frac{(y+x)(y-x)\times 100}{{{x}^{2}}}%\]

Ø   If on selling an article for Rs. x, the profit percentage is equal to the cost price of the article, then the cost price  of the article \[=-\,50+10\sqrt{25+x}.\]

Ø   If a man purchases a items for Rs. x and sells b items for Rs. y, then profit or loss percentage

\[=\frac{ay-bx}{bx}\times 100%\]

Note         \['+'\] ve for profit and \['-'\] ve for loss.