Successive Percentage Changes

1. The Core Idea of Successive Change

Imagine a product's price increases by 20%, and then, due to a new tax, it increases again by 30%. Is the total increase 50%? Not quite. The second percentage change is calculated on the new, already increased price, not the original one. This is the essence of successive percentage change.

This simple diagram illustrates the concept using multipliers, which we'll explore shortly.

To build a strong intuition for this, let's watch a segment from a video by Gagan Pratap Sir, a well-known educator for SSC exams. He explains the concept using a very simple "100 method," which makes the logic crystal clear.

As you saw, a 20% increase followed by a 30% increase results in a 56% net increase. The term Net Percentage Change (or single equivalent change) refers to the single percentage that would produce the same final result as the series of successive changes. Our main goal is to learn how to calculate this net change efficiently.

2. Methods for Calculating Successive Change

For your exam, you need methods that are both conceptually sound and fast. We will cover two primary methods: the Ratio Method and the Formula Method.

Method 1: The Ratio Method (The Conceptual Powerhouse)

The video you just watched beautifully transitions from the "100 method" to the idea of "multiplying factors." This is the key.

• An increase of x% is the same as multiplying the original value by $(1+\frac{x}{100})$.
• A decrease of y% is the same as multiplying by $(1-\frac{y}{100})$.

Successive changes are just successive multiplications. The Ratio Method is a way of organizing this multiplication using fractions.

Let's continue with the same video to see how this powerful method is derived and used.

To solidify this, let's review the formal structure from a text resource.

Let's walk through an example using the Ratio Method:

The price of an item is first increased by 25%, then decreased by 20%. Find the net percentage change.

1. Convert percentages to fractions:

   • Increase of 25% = $+\frac{1}{4}$. A value of 4 becomes $4+1=5$. The ratio of Initial:Final is 4:5.
   • Decrease of 20% = $-\frac{1}{5}$. A value of 5 becomes $5-1=4$. The ratio of Initial:Final is 5:4.

2. Set up the successive ratios and multiply:

3. Calculate the net change:
   • The initial value was 20, and the final value is also 20.
   • The change is $20-20=0$.
   • Therefore, the net percentage change is 0%.

The Ratio Method is extremely versatile, especially when dealing with three or more changes or with percentages that correspond to simple fractions (like 16.66%, 14.28%, etc.).

Method 2: The Formula Method (The Shortcut)

For two successive changes, there is a very direct and fast formula. This is a must-know for SSC CGL.

If a value is changed first by $x\%$ and then by $y\%$, the net percentage change is:

Remember to use a positive sign for an increase and a negative sign for a decrease.

Let's see this formula in action with Gagan Pratap Sir.

Key Shortcut: When a value is first increased by $x\%$ and then decreased by $x\%$ (or vice-versa), the net result is always a decrease of $\frac{x^2}{100}\%$.

Let's re-solve our previous example using the formula: Price increased by 25% ($x=+25$), then decreased by 20% ($y=-20$).

The result is the same. The formula is often faster for two changes involving integers.

3. Strategic Application: Choosing Your Method

For the SSC CGL exam, speed is everything. Knowing which method to use for a given problem is as important as knowing the methods themselves.

• Use the Ratio Method when percentages convert to simple fractions (10%, 25%, 33.33%, etc.) or when you have three or more successive changes.
• Use the Formula Method when you have two successive changes and the percentages are integers that don't have simple fractions (e.g., 12%, 23%).

Rakesh Yadav Sir, another expert in this field, provides excellent advice on tackling highly calculative problems by choosing the right approach.

4. Practice Problems

Let's apply these methods to a few typical exam-style questions. You can find these examples in the "Problems on Successive Percentage Change" section of the GeeksforGeeks article.

Problem 1: Shopkeeper's Discount
A shopkeeper gives two successive discounts of 20% and 30%. What is the final discount? (This is Problem 6 in the article)

• Method 1: Ratio

  • -20% = $-\frac{1}{5}$ ⇒ Ratio is 5 : 4
  • -30% = $-\frac{3}{10}$ ⇒ Ratio is 10 : 7
  • Overall: (5 × 10) : (4 × 7) = 50 : 28. To make it easier, let's simplify to 25 : 14.
  • Discount = $25-14=11$.
  • Percentage Discount = $\frac{11}{25}\times 100\%=44\%$.

• Method 2: Formula

  • $x=-20$, $y=-30$
  • Net Change = $-20+(-30)+\frac{(-20)(-30)}{100}$
  • = $-50+\frac{600}{100}=-50+6=-44\%$.
  • The negative sign indicates a decrease, so the final discount is 44%.

Problem 2: Area of a Rectangle
The length of a rectangle is increased by 10% and its breadth is decreased by 7%. What is the net change in its area? (This is Problem 5 in the article)

• Since Area = Length × Breadth, this is a successive change problem. The formula is quick here.
• $x=+10$, $y=-7$
• Net Change = $10-7+\frac{(10)(-7)}{100}=3-\frac{70}{100}=3-0.7=+2.3\%$.
• The area increases by 2.3%.

Conclusion

In this lesson, we have laid a crucial foundation for your quantitative aptitude journey. You've learned not just how to calculate successive percentage changes, but also how to think strategically about them.

Here are the key takeaways:

• Successive percentage change means applying a percentage change to the new value, not the original. The net effect is not a simple sum of the percentages.
• You have mastered two primary methods:
  • The Ratio Method: Conceptually strong, best for simple fractions and for more than two changes.
  • The Formula Method ($x+y+\frac{xy}{100}$): A fast shortcut for two successive changes, especially with integers.
• You learned the high-yield shortcut for a +x% change followed by a -x% change, which always results in an $\frac{x^2}{100}\%$ decrease.

The Ratio Method you learned today is a direct application of ratios. In our next lesson, we will dive deeper into the world of Ratio and Proportion, exploring how to use these concepts to solve a wide variety of comparison and distribution problems you'll encounter in the SSC CGL.