maths > commercial-arithmetics

Formulas for profit-loss-discount

what you'll learn...

overview

Students need not memorize 20+ formulas anymore for the topics profit-loss, discount, and tax.

$\text{Profit%}=100×$ $\left(\text{SalePrice}-\text{costPrice}\right)$$\left(\textrm{S a \le P r i c e} - \textrm{C o s t P r i c e}\right)$$/\text{costPrice}$$/ \textrm{C o s t P r i c e}$

$\text{Discount%}=100×$ $\left(\text{MarkedPrice}-\text{SalePrice}\right)$$\left(\textrm{M a r k e \mathrm{dP} r i c e} - \textrm{S a \le P r i c e}\right)$$/\text{MarkedPrice}$$/ \textrm{M a r k e \mathrm{dP} r i c e}$

$\text{Tax%}=100×$ $\left(\text{BilledPrice}-\text{SalePrice}\right)$$\left(\textrm{B i l \le \mathrm{dP} r i c e} - \textrm{S a \le P r i c e}\right)$$/\text{SalePrice}$$/ \textrm{S a \le P r i c e}$

All these three formulas are very similar and has a simple explanation to the terms in numerator and denominator.

reference list

Students are stressed by many formulas. This is listed for reference, and explained later. No need to memorize any of these.

profit = sale price - cost price

loss = cost price - sale price

profit percentage = (sale price - cost price) * 100 / (cost price)

loss percentage = (cost price - sale price) * 100 / (cost price)

sale price = cost price * (100+profit%)/(100)

cost price = sale price * (100) / (100+profit%)

sale price = cost price * (100-loss%)/(100)

cost price = sale price * (100) / (100-loss%)

discount = marked price - sale price

discount percent = (marked price - sale price)*100 / marked price

sale price = markprice (100-discount%) / 100

marked price = sale price * 100 / (100- discount%)

tax = bill price - sale price

tax percent = (bill price - sale price)*100 / sale price

billprice = sale price (100+ tax%)/100

sale price = bill price * 100 /(100+tax%)

understanding tax inclusive

Summarizing all the terms learned so far.

•  Shopkeeper buys a pen for $40$$40$ coins. (cost price)

•  Shopkeeper spends $3$$3$ coins on transport or other shopkeeping expenses. (overhead expense)

•  Shopkeeper marks the price of the pen as $60$$60$ coins. (marked price)

•  Shopkeeper marks a $10$$10$ coins discount on the pen. (discount)

•  Customer buys the pen for $50$$50$ coins (bill price). The bill price is given as inclusive of taxes.

•  the shopkeeper pays $2$$2$ coins as tax (tax)

•  the sale-price is $50-2=48$$50 - 2 = 48$ coins.

The profit for the shopkeeper

= marked price - discount - cost price - overhead expenses - tax

$=60-10-40-3-2$$= 60 - 10 - 40 - 3 - 2$

$=5$$= 5$ coins (profit)

understanding tax extra

Summarizing all the terms learned with tax extra.

•  Shopkeeper buys a pen for $40$$40$ coins. (cost price)

•  Shopkeeper spends $3$$3$ coins on transport or other shopkeeping expenses. (overhead expense)

•  Shopkeeper marks the pen as $60$$60$ coins. (marked price)

•  Shopkeeper marks a $10$$10$ coins discount on the pen. (discount)

•  Customer buys the pen for $50$$50$ coins (sale price). The sale price is given as excluding taxes.

•  the customer pays $2$$2$ coins as tax (tax) to the shopkeeper. Effectively the customer pays $52$$52$ coins (bill price).

The profit for the shopkeeper

= marked price - discount - cost price - overhead expenses

$=60-10-40-3$$= 60 - 10 - 40 - 3$

$=7$$= 7$ coins (profit)

Note: The tax of $2$$2$ coins is collected from buyer and paid to government by the seller.

One need not memorize any formulas. Quickly follow through the story to recall formulas on the fly.

•  Loss is the negative profit.

•  Shopkeeper buys an article by cost price CP.

•  Shopkeeper sells the article at sale price SP.

•  Shopkeeper calculates the profit on the amount invested, which is cost price. So, profit percentage is given as percentage of cost price.
$\text{Profit%}=100×$ $\left(\text{SP}-\text{CP}\right)$$\left(\textrm{S P} - \textrm{C P}\right)$$/\text{CP}$$/ \textrm{C P}$

•  shopkeeper adds a tag to the article as marked price MP.

•  Discount is shown to the customer based on the price displayed to the customer, which is marked price. So, Discount percentage is given as percent of marked price.
$\text{Discount%}=100×$ $\left(\text{MP}-\text{SP}\right)$$\left(\textrm{M P} - \textrm{S P}\right)$$/\text{MP}$$/ \textrm{M P}$

•  For the sale, government collects tax. Shopkeeper collects the tax on behalf of the government and submits the tax amount.

•  Tax is added to the sale price in the bill and the total is called the billed price BP.

•  Tax is paid by the customer on the amount taken by the seller, which is the sale price. So, the tax percentage is given as a percentage of sale price.
$\text{Tax%}=100×$ $\left(\text{BP}-\text{SP}\right)$$\left(\textrm{B P} - \textrm{S P}\right)$$/\text{SP}$$/ \textrm{S P}$

all together

One need not memorize any formulas. Quickly follow through the story to recall formulas on the fly.

•  Loss is the negative profit.

•  Shopkeeper calculates the profit on the amount invested, which is cost price.
$\text{Profit%}=100×$ $\left(\text{SP}-\text{CP}\right)$$\left(\textrm{S P} - \textrm{C P}\right)$$/\text{CP}$$/ \textrm{C P}$

•  Discount is shown to the customer based on the price displayed to the customer, which is marked price.
$\text{Discount%}=100×$ $\left(\text{MP}-\text{SP}\right)$$\left(\textrm{M P} - \textrm{S P}\right)$$/\text{MP}$$/ \textrm{M P}$

•  Tax is paid by the customer on the amount taken by the seller, which is sale price.
$\text{Tax%}=100×$ $\left(\text{BP}-\text{SP}\right)$$\left(\textrm{B P} - \textrm{S P}\right)$$/\text{SP}$$/ \textrm{S P}$

The three formulas given above are easier to recall. Each has $3$$3$ variables and given any $2$$2$ quantities, use the equation as a linear equation (algebra) to solve for the third variable.

examples

Profit percentage is given as a percent of which of the following?
Remember the shopkeeper makes the profit on the investment of "cost-price".

Which of the following gives the formula for sale price?

The answer is "profit percent =$100×$$100 \times$ (SP-CP) $/$$/$ CP"

A book is sold for $22$$22$ coins with $10%$ profit included, what is the cost-price?

Solution:
sale price = $22$$22$
profit percentage = $10%$

profit percent =$100×$$100 \times$ (SP-CP) $/$$/$ CP

$10=100×\left(22-\text{CP}\right)/\text{CP}\right)$

This is a linear equation of one variable. It is easy to solve this for cost price.

A book is sold for $22$$22$ coins with $10%$ loss included, what is the cost-price?

Solution:
sale price = $22$$22$
profit percentage = $-10%$
Note: The loss is negative profit

profit percent =$100×$$100 \times$ (SP-CP) $/$$/$ CP

$-10=100×\left(22-\text{CP}\right)/\text{CP}\right)$

This is a linear equation of one variable. It is easy to solve this for cost price.

Discount percentage is given on which of the following?

The answer is "marked price". Remember discount is shown to a customer and customer sees the mark-price on the article.

Which of the following gives the formula for discount percentage?

The answer is "discount % $=100×$$= 100 \times$ (MP-SP) $/$$/$ MP".

A book is marked for $20$$20$ coins. The shop keeper sells for $16$$16$ coins. What is the discount percentage?

Solution:
marked price = $20$$20$
sale price = $16$$16$
discount % $=100×$$= 100 \times$ (MP-SP) $/$$/$ MP

discount % $=100×\left(20-16\right)/20$$= 100 \times \left(20 - 16\right) / 20$

This is a linear equation of one variable. It is easy to solve this for discount percent.

Tax percentage is calculate on which of the following?

The answer is "sale price". Remember tax is paid by the customer on the amount taken by the shopkeeper.

Which of the following gives the formula for tax percentage?

The answer is "tax % $=100×$$= 100 \times$ (BP-SP) $/$$/$ SP"

A book is sold for $20$$20$ coins inclusive of $10%$ tax. What is the sale price of the book?

Solution:
bill price = $20$$20$
tax percent = $10%$
tax % $=100×$$= 100 \times$ (BP-SP) $/$$/$ SP

$10=100×\left(20-\text{SP}\right)/\text{SP}$$10 = 100 \times \left(20 - \textrm{S P}\right) / \textrm{S P}$

This is a linear equation of one variable. It is easy to solve this for sale price.

summary

Students need not memorize 20+ formulas anymore for the topics profit-loss, discount, and tax.

$\text{Profit%}=100×$ $\left(\text{SalePrice}-\text{costPrice}\right)$$\left(\textrm{S a \le P r i c e} - \textrm{C o s t P r i c e}\right)$$/\text{costPrice}$$/ \textrm{C o s t P r i c e}$

$\text{Discount%}=100×$ $\left(\text{MarkedPrice}-\text{SalePrice}\right)$$\left(\textrm{M a r k e \mathrm{dP} r i c e} - \textrm{S a \le P r i c e}\right)$$/\text{MarkedPrice}$$/ \textrm{M a r k e \mathrm{dP} r i c e}$

$\text{Tax%}=100×$ $\left(\text{BilledPrice}-\text{SalePrice}\right)$$\left(\textrm{B i l \le \mathrm{dP} r i c e} - \textrm{S a \le P r i c e}\right)$$/\text{SalePrice}$$/ \textrm{S a \le P r i c e}$

All these three formulas are very similar and has a simple explanation to the terms in numerator and denominator.

Outline