Successive Percentage Changes, Profit & Loss, and Discounts

1. The Compounding Effect of Successive Percentage Changes

A common trap in percentage problems is to simply add or subtract consecutive changes. For example, if a price increases by 20% and then by another 10%, the total increase is not 30%. Why? Because the second percentage change is calculated on the new, adjusted value.

Let's see this with an example:

• Initial Value: 100
• First increase (20%): $100+(20\%\text{of}100)=100+20=120$
• Second increase (10%): $120+(10\%\text{of}120)=120+12=132$

The final value is 132, which is a 32% increase from the original 100, not 30%. This is the essence of successive percentage change.

To understand this concept more deeply and learn a powerful shortcut, let's watch a segment of the following video.

The Net Percentage Change Formula

As explained in the video, a single formula can handle all scenarios of two successive percentage changes, $A\%$ and $B\%$:

$\text{Net Change \%}=A+B+\frac{AB}{100}$

A key point to remember:

• Use a positive sign for increases (e.g., profit, markup, price hike).
• Use a negative sign for decreases (e.g., discount, loss, price drop).

Let's revisit our initial example: a 20% increase followed by a 10% increase.

• $A=+20$, $B=+10$
• Net Change % = $20+10+\frac{20\times 10}{100}=30+\frac{200}{100}=30+2=32\%$. This matches our manual calculation and is much faster.

2. Successive Discounts

This concept is most frequently seen in problems involving discounts. A store might offer a 40% sale and then allow you to use a 20% off coupon on top of that.

This image provides a great visual breakdown of how successive discounts work.

Let's apply our formula to this scenario: a 40% discount followed by a 20% discount.

• $A=-40$, $B=-20$
• Net Change % = $-40+(-20)+\frac{(-40)\times (-20)}{100}$
• Net Change % = $-60+\frac{800}{100}=-60+8=-52\%$

The net result is a 52% discount, exactly as the image concludes.

3. The Language of Commerce: Profit, Loss, and Discounts

Now, let's formally define the terms used in almost all commercial math problems. Understanding these terms and the relationships between them is non-negotiable.

Let's summarize the flow of a typical transaction:

1. Cost Price (CP): The price at which a seller acquires an item. This is the seller's initial investment.
2. Marked Price (MP): The seller increases the CP by a certain percentage (called markup) to set a label price or list price.
3. Discount: The seller offers a reduction on the MP to entice buyers.
4. Selling Price (SP): The final price the customer pays after the discount. $SP=MP-\text{Discount}$.

The outcome for the seller is determined by comparing the final SP to the initial CP.

• If SP > CP, the seller makes a Profit.
• If SP < CP, the seller incurs a Loss.

The Golden Rules of Calculation Bases

This is the most common area for mistakes. Always remember:

• Profit % or Loss % is always calculated on the Cost Price (CP).
• Discount % is always calculated on the Marked Price (MP).

The Careers360 article you just looked at reinforces this in its "Common mistakes" table (section 5).

4. Connecting P&L to Successive Percentage Change

A typical profit and loss problem is often just a successive percentage change problem in disguise.

Example: A shopkeeper marks up an item by 40% and then offers a 25% discount. What is their net profit or loss percentage?

This is equivalent to a +40% change followed by a -25% change.

• $A=+40$, $B=-25$
• Net Change % = $40-25+\frac{40\times (-25)}{100}$
• Net Change % = $15-\frac{1000}{100}=15-10=5\%$

The shopkeeper makes a 5% profit.

This formula provides a much faster solution than the alternative method of assuming CP=100, calculating MP (140), then calculating SP (140 - 25% of 140 = 105), and finally finding the profit percentage $(105-100)/100$. While the CP=100 method is reliable and great for understanding, the formula is built for speed.

5. Expanding the Application

The power of the successive change concept extends beyond simple price changes. It applies to any relationship where quantities are multiplied together. Given your computer science background, you can think of it as analyzing the percentage change in the output of a function Z = X * Y when X and Y change by certain percentages.

Watch how this concept is applied to different scenarios in the same video.

The faulty weight problem is a classic CAT question type. The seller's profit comes from two sources:

1. Marking up the price (e.g., claiming to sell at CP but having a hidden markup).
2. Giving the customer less quantity than they paid for.

Each of these can be treated as a percentage gain, and the successive change formula can find the net profit.

Conclusion

Great job working through these concepts. You've now built a solid framework for a huge portion of the arithmetic questions on the CAT.

Key Takeaways:

• Successive percentage changes do not simply add up; they compound.
• The formula Net Change % = $A+B+AB/100$ is your fastest tool for solving problems with two successive changes (increases are +, decreases are -).
• Most problems involving markup and discount are successive percentage change problems.
• Always be clear about the base: Profit/Loss is calculated on Cost Price (CP), and Discount is calculated on Marked Price (MP).

In our next lesson, we will tackle Simple and Compound Interest. You'll see an immediate and powerful connection, as compound interest is nothing more than a series of successive percentage increases. The formula we learned today is, in fact, the basis for calculating compound interest over two years. Keep it fresh in your mind