## Use basic and rich concepts and avoid calculations

### Problem

A man sells two wrist watches one at a profit of 30% and another at a loss of 30%, but each at a same selling price of Rs.400. The net profit or loss is,

- 6%
- 0%
- 9%
- 1%

### Most basic profit and loss concept

Profit or loss percentage is on Cost Price or CP.

#### Given

Here, in two transactions, CPs are different and not known, but selling price, Rs.400 is same.

Also given, in one case, loss is 30% and in the other gain or profit is 30%.

#### Required

- Net profit or loss percentage considering one person made both the transactions.
- The effect of two purchases and two sales together.

In other words, Total Cost Price and Total Sale Price will give us Net profit or Net loss. As Total Sale price is Rs.800, we need to have an idea of Total Cost Price.

Notice that I have avoided the use of “we need to calculate the total cost price”. This is the turning point, the point of choosing the right and speedy path to solution. We won’t calculate Prices. Let’s see how.

### Procedural approach

You may proceed to find the actual sale prices from the basic formula of profit and loss.

In the case of profit,

$\text{Profit}= \text{30% of CP1}$, we assume CP1 as the first cost price

$= SP – CP1$, we assume SP as the sale price which is Rs.400, but we won’t bring in the figure of Rs.400 right now,

Or, $0.3CP1=400 – CP1$

Or. $1.3CP1 = 400$

Or, $CP1=\displaystyle\frac{400}{1.3}$, if you are fond of doing calculations, go ahead do it.

Similarly, you can easily visualize, in case of loss, the cost price CP2 as,

$CP2=\displaystyle\frac{400}{0.7}$, as $SP=(1-0.3)CP2=0.7CP2$.

This is easy isn’t it?

So, total cost price in terms of single known sale price SP is,

$CP1+ CP2= 400\left(\displaystyle\frac{1}{1.3}+\displaystyle\frac{1}{0.7}\right)=\displaystyle\frac{800}{0.91}$

As total sale price is 800, you can see that total cost price is more than total sale price, resulting in a net loss. How much? That is also easy.

$\text{Net loss percentage}=\displaystyle\frac{(CP1+CP2) -800}{CP1+CP2}$; by definition loss percentage is on the total cost price, so it will come as the denominator; actual loss is total cost price minus total sale price.

$=1- 0.91=0.09=9$%

Simple isn’t it?

Two important points here.

First, notice that we need not have used the sale price figure of Rs.400 at all. It finally canceled out. It could have been any figure, say Rs.243189. Answer of 9% would have remained same.

Reason behind this surprising observation lies in the fact that, given profit and loss are percentages or ratios, sale prices on two transactions are same, and the required net loss or profit is also a percentage, a ratio. In any problem if this situation happens, you can completely avoid any calculation of actual values. Calculation will only be a simple fraction sum. The fixed sale price will cancel out.

The second important point is the way we arrived at the solution.

This is called procedural method of math problem solving. This is the way you are used to solve your school math problems. And if you adopt this approach you would lose extremely valuable time in exam hall.

This is not a school exam you are preparing for; this is a hard MCQ based competitive test where you must solve each such problem under a minute.

### Efficient way to solve without any writing

**The deduction will be in mind, as far as possible.**

**Basic profit and loss concept** we know as,

$\text{Profit percentage}=\displaystyle\frac{\text{Sale Price} – \text{Cost Price}}{\text{Cost Price}}$.

Similarly for loss when Cost price will be more than the Sale price.

Based on this basic formula we will always use the **Rich concept of Profit and Loss in such problems as**,

**$SP=CP1+Profit=CP1+0.3CP1=1.3CP1$**. We won’t deduce, we would know and use this relation.

Similarly for lossy transaction,

**$SP=CP2-Loss=CP2-0.3CP2=0.7CP2$.**

Knowing this, total cost price immediately comes out as,

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

You may write down this step if you need, but with just a look at this relation, knowing the loss relation, it is just a few seconds to arrive at the answer as loss of 9%.

Important takeaways:

**Avoid calculations**as far as possible if given values and required values all are in ratios or percentages.**Use Rich concept:**30% profit means, SP=1.3CP, and 30% loss means, SP=0.7CP.

Simple but valuable time saving use of concepts.