## Marked price, Sale price, Cost price, Discount, Profit and Free item offer concepts simplified

### Basic concepts on Profit and loss

$\text{Profit amount}=\text{Sale price}-\text{Cost price}$.

When the $\text{Profit amount}$ is expressed as a percentage, the * Percentage reference value* is the $\text{Cost price}$, that is,

$\text{Profit percentage}=\displaystyle\frac{\text{Profit amount}\times{100}}{\text{Cost price}}$.

When $\text{Sale price}$ is less than the $\text{Cost price}$, * a loss* is incurred with the loss as,

$\text{Loss amount}=\text{Cost price}-\text{Sale price}$.

When the $\text{Loss amount}$ is expressed as a percentage, the * Percentage reference value* again is the $\text{Cost price}$, that is,

$\text{Loss percentage}=\displaystyle\frac{\text{Loss amount}\times{100}}{\text{Cost price}}$.

The $\text{Discount percentage}$, if any, is * applied as a percentage on the Percentage reference* value of $\text{Marked price}$, to obtain the $\text{Discount amount}$.

The $\text{Marked price}$ is then reduced by the $\text{Discount amount}$ to obtain the $\text{Sale price}$.

These are briefly the basic concepts in the topic area of Competitive test Profit an loss. *The concepts are apparently simple.*

Time and again though we encounter problems on Profit and loss that pose some difficulties to the problem solver.

In this session we have chosen such a problem to show how even a confusing and time consuming Profit and loss problem can be solved easily in a few simple steps if we use Problem analysis, Problem solving strategies and the basic concepts on Profit and Loss and other related topics such as Percentage.

### Problem

With cost price per item as Rs.1002, a shopkeeper offers 4% discount on the marked price and 1 item free with every 15 items purchased, achieving a profit of 35%. The marked price is then increased above the cost price by,

- 39%
- 40%
- 20%
- 50%

### Problem analysis first stage - deducing rich concept of Free item offer from basic concepts

The crucial problem faced first is how to deal with what actually is meant by, "1 item free with every 15 items purchased". By deductive reasoning we decide that unless * this free item offer is converted to its equivalent form in terms of cost price and sale price* we cannot proceed with the solution.

#### Rich profit and loss concept of free item offer

By taking recourse to Context awareness or understanding what actually happens, we transform the statement, "**1 item free with every 15 items purchased"**, to,

16 items sold at the Sale price of 15 items.

In other words, adding more meaning we can say,

A batch of 16 items Costing the Purchase Price of 16 items is sold at the Sale price of 15 items.

This basically is the core of the * Free item offer concept*, and is a rich Profit and loss concept that is deduced using context awareness.

With the new form of free offer statement transformed to known quantities of cost price and sale price, our * final objective* will be,

To convert the marked price in terms of the cost price,

so that we can find the **desired difference between the two.**

### Problem analysis second stage, use of abstraction technique to avoid calculation of actual price values

After crossing the first barrier of free offer problem, * the conventiional approach* would be to

*, and*

**calculate the actual sale price from the cost price***But that*

**then the actual marked price from the sale price.**

**involves time concuming costly calculations.**Instead, we use * a rich percentage concept* applicable especially to a large number of profit and loss problems, and avoid time consuming calculations using

**abstraction technique.**#### A rich percentage concept

In any problem where all quantity values including the final target quantity are in terms of percentages of a given value and the relations between the quantities are linear, actual value of any quantity can be done away with, replacing all quantity values by representative substitution variable symbols.

This concept is embodied in **Actual value independence concepts.**

#### Application of abstraction technique and substitution technique based on the rich percentage concept of all values in terms of percentages of a single value

Recognizing that our problem satisfies the condition embodied in the rich percentage concept of all values in terms of percentages, we decide not to go by the conventional path of calculating the actual sale price and then the marked price.

Instead, using abstraction and substitution together we name the cost price of 1 item by $CP$, sale price of 1 item by $SP$ and marked price of 1 item by $MP$, with the confidence that in the final step the variable representing cost price will be eliminated leaving only the desired percentage increase from cost price to obtain the marked price.

* Note here* that the prices are all in terms of 1 item. This is necessary to deal with the free offer problem.

### Problem analysis second stage, visualization of the path of reaching the desired objective in two steps

As the target is to find the marked price, we need to obtain it by using the discount along with the sale price, and again we must get the sale price using the free offer and the profit percentage, along with the cost price. It is a clearly defined *two step process* -* first the sale price then the marked price*,

*both in terms of the cost price.*

### Problem solving execution first stage, finding Sale price in terms of Cost price

If $CP$ is cost of one item, $SP$ is sale price of 1 item, and $P$ profit on 1 item, we have from the * profit statement* and

*,*

**Free item offer**$\text{Profit on 16 items} = \text{Sale price of 15 items} - \text{Cost price of 16 items}$, as 1 item is given free which anyway had its purchase cost, but no sale earning.

Or, $16P=15SP-16CP$

$=0.35\times{16CP}$, as the profit is 35% on the actually sold 16 items,

Or, $15SP=1.35\times{16CP}$,

Or, $SP=0.09\times{16CP}=1.44CP$.

### Problem solving execution second stage, finding Marked price in terms of Cost price

We cannot find the marked price directly from the cost price as the discounted marked price equal to the sale price stands in between. So first we must express the sale price in terms of the marked price, and then using the earlier obtained expression of sale price in terms of cost price we can reach the marked price in terms of cost price.

As the discount 4% is on the marked price reducing it to the sale price, using the rich percentage concept we can say,

$SP=0.96MP$, not only have we converted the 4% discount to its equivalent decimal, but also reduced the marked price by the discount on it to get $0.96MP$ equating it to $SP$.

Or, $MP=\displaystyle\frac{SP}{0.96}$

$=\displaystyle\frac{1.44CP}{0.96}$

$=\displaystyle\frac{3CP}{2}$.

So the difference beween $MP$ and $CP$ is $\frac{1}{2}CP=50\text{% of }CP$.

**Answer.** Option d: 50%.

### Solution analysis

The * first barrier* faced in this problem was to understand the

*and deal with it. Understanding what actually happened in terms of selling and profit accrued thereby, we could deduce first the equivalent form of free offer in terms of cost price and the sale price.*

**meaning of the free offer**At the * second stage of analysis* we recognize that the problem satisfies

*, and use*

**the rich percentage concept of all quantities in terms of percentages***to represent all prices for 1 item by corresponding variable symbols.*

**abstraction and substitution**In the * third stage of analysis*, we identify the strategy of reaching the target marked price in two steps - first sale price from cost price and profit, and second - marked price from sale price and discount. In the second step we need to apply another rich percentage concept based on concept of discount.

Accordingly in corresponding implementation of the strategies, the * sale price in terms of cost price is deduced using cost price, profit and free offer relation*.

*Our target all along is to evaluate the marked price in terms of the cost price.*

In the * final implementation of strategic steps*, the

*, this path being conceptually simpler to understand.*

**sale price is expressed as a percentage of marked price reduced by discount percentage first**Then the **sale price is replaced by its cost price equivalent to obtain the marked price in terms of the cost price.**

Difference between marked price and cost price as a percentage of cost price is now only a simple step away. * Finally our desired answer was in terms of a percentage*, verifying the effectiveness of the use of rich percentage concept in

**avoiding calculation of all actual price values.*** This analytical minimum calculation process can be applied to a large number of profit and loss problems* to reach to solution in only a few simple steps.

**Note:** Problem understanding, problem solving strategy formulation and use of the basic and rich concepts enabled reaching the solution in a few simple steps avoiding all time consuming calculations. **The calculations were only on the percentages and not on the actual values of the prices at all.**