## Smart Abstraction Technique and Advanced Profit and Loss Concepts Help Easy Solution

A man sells two watches at same price profit and loss problem: Learn how to solve a profit loss question easily with smart problem solving techniques.

We will explain how to solve a profit loss question easily using smart problem solving techniques: the abstraction method and advanced profit and loss concepts.

### How to Solve a Profit Loss Question Easily

A man sells two wristwatches, one at a profit of 30% and the other at a loss of 30%, but each at the same selling price of Rs. 400. The net profit or loss is:

- 6% profit
- 0% (no profit, no loss)
- 9% loss
- 1% profit

### Conventional Solution

#### Basic concepts of Profit and loss

**Profit** of 30% in selling the first watch at Rs.400 **or the loss** of 30% in selling the second watch at Rs.400 are calculated **on the basis of their cost prices**.

$\text{Profit} = \text{Selling price} - \text{Cost price}$, and,

$\text{Loss} = \text{Cost price} - \text{Selling price}$.

Thus, the selling price of the first watch is 30% (of its cost price) more than its cost price, and the selling price of the second watch is 30% (of its cost price) less than its cost price.

#### Initial analysis

Let Cost prices of the two watches be CP1 and CP2 and the common selling price SP.

So total cost price TCP = CP1 + CP2, and

Total sale price is TSP = 2SP = Rs.800.

If TCP is greater than TSP there is a loss otherwise a gain.

#### Find the two cost prices in terms of the given sale price

For the **first watch sale** at a profit of 30%,

$SP = CP1 + .3\times{CP1} = 1.3\times{CP1}$

Or, $1.3\times{CP1} = SP = Rs.400$.

So, $CP1 = \displaystyle\frac{Rs.400}{1.3}$.

For the** second watch sale **at a loss of 30%,

$SP = CP2 - .3\times{CP2} = 0.7\times{CP2}$

Or, $0.7\times{CP2} = SP = Rs.400$

Or, $CP2 = \displaystyle\frac{Rs.400}{0.7}$.

#### Find the loss or gain at the Last Stage (postpone calculation)

So, **Total cost of the two watches** is,

$TCP = \displaystyle\frac{Rs.400}{1.3} + \displaystyle\frac{Rs.400}{0.7} = \displaystyle\frac{Rs.800}{0.91}$,

whereas **total sale price was** $TSP = Rs.400 + Rs.400 = Rs.800$. This is less than the total cost price TCP.

Thus, there has been a **net loss in the whole transaction,**

\begin{align} Loss &= \displaystyle\frac{Rs.800}{0.91} - Rs.800 \\ & = Rs.800\times{\displaystyle\frac{.09}{0.91}} \\ & =\frac{Rs.800\times{.09}}{0.91} \end{align}

This loss is on the total cost of $\displaystyle\frac{Rs.800}{0.91}$,

The **percentage loss is**,

$\frac{\displaystyle\frac{Rs.800\times{.09}}{0.91}}{\displaystyle\frac{Rs.800}{0.91}}= .09=9{\%}$.

**Answer:** c: 9%.

### How to solve profit and loss problem in a few steps by Abstraction technique

**Even in the conventional solution, you have not calculated the actual values of the two fractional cost prices**, keeping those in fraction form only. Finally, there was no need to calculate any awkward fraction because, in percentage ratio calculation the prices were canceled out leaving only the ratio of proportions.

This characteristic enables us to observe in general,

In profit and loss problems, wherever the selling prices are same in two sale situations, and final requirement is a percentage (in terms of whatever, sale or cost price),

we may ignore the same selling price altogether.The price values will finally cancel out.

This is an example of **direct application of abstraction technique**, which says,

When in two entities or situations

there is a common set of properties, you canfocus only on the special properties different in each, leaving out the common properties.

In simple terms, **abstraction means generalization**, *focusing only on the core of the problem that is important*, shedding unnecessary details and thereby **decreasing unwanted clutter of information.**

Sensing this property you have delayed fraction calculation till the last stage and achieved a clean solution, **even though the method followed was conventional.**

#### Reason for the abstraction to work in this case

The reason why abstraction works in this case is the **rich advanced concept** that,

Profit or loss percentage both are in terms of cost prices, and as the final requirement is also a percentage, which is a ratio, the common price will finally be canceled out. A corollary of this concept is:

whatever be the prices, if the percentages reamain same, the answer will always be the same.Think over.

#### Rich advanced concepts

In any topic area, problems can be solved using the very basic concepts. **You know the use of the basic concepts in profit and loss problems already.**

When you are comfortable with the basic concepts, you can form what we call * rich advanced concepts derived *from the basic concepts itself and use the rich concepts sometimes to

**solve problems in the specific area in a few steps only.**

**Rich concepts in Profit and Loss**

If CP = Cost price and $x{\%}$ = Profit, in terms of Sale price SP,

$CP = \displaystyle\frac{SP}{1 + 0.01\times{x}}$.

Similarly, if $x{\%}$ = Loss, in terms of Sale price SP,

$CP = \displaystyle\frac{SP}{1 - 0.01\times{x}}$.

These relationships are so simple and can so easily be derived from the basic concepts of Profit and Loss topic, it is useful to remember and use these rich or derived concepts when SP and profit are given but CP is unknown.

### Fastest solution using advanced profit and loss concepts based on abstraction of basic concepts

Being aware now of these simple but rich concepts, you will apply the concepts on the same problem. As you are also aware of the fact that only percentages are important, prices are not, you think and express in terms of the terms like CP1, CP2 and SP not considering their numerical values at all.

With these two powerful new concepts under the belt, form your problem expression directly moving to the final stage,

$CP1 + CP2 = SP\left(\displaystyle\frac{1}{1.3} + \displaystyle\frac{1}{0.7}\right) = 2SP\times{\displaystyle\frac{1}{0.91}}$.

This total cost is larger than total sale price of 2SP.

So the loss is,

$2SP\left(\displaystyle\frac{1}{0.91} - 1\right)=2SP\left(\displaystyle\frac{.09}{.91}\right)$.

Or, $Loss = \displaystyle\frac{2SP}{0.91}\times{.09}=9{\%}$ of total cost price.

2SP is part of the total cost and loss both, thus canceled out or **abstracted out** in our terminology.

This result can be reached in fraction of a minute comfortably if you follow this path with full awareness of the rich concepts of Profit and Loss and the Abstraction technique.

Read a brief version of the article here.

**Resources that should be useful for you**

* 7 steps for sure success in SSC CGL tier 1 and tier 2 competitive tests* or

*to access all the valuable student resources that we have created specifically for SSC CGL, but*

**section on SSC CGL****generally for any hard MCQ test.**

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