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Sequential Percent Changes


Mike McGarry
Lesson by Mike McGarry
Magoosh Expert

Summary
Understanding sequential percent changes is crucial for avoiding common mistakes on the GRE, particularly in problems involving multiple percent increases and decreases.
  • Sequential percent changes do not cancel each other out; an increase followed by a decrease of the same percentage does not return to the original value.
  • The correct approach to solving these problems involves using multipliers for each percent change and multiplying them together.
  • Common mistakes include assuming that sequential percent changes can be added or subtracted to find the overall effect.
  • Using multipliers reveals the true impact of sequential percent changes, often leading to counterintuitive results.
  • Recognizing and avoiding these common pitfalls can significantly improve performance on the GRE.
Chapters
00:00
The Trap of Sequential Percent Changes
01:09
The Correct Approach: Multipliers
04:38
Applying the Multiplier Method

Q: Why is 1.3 the multiplier for a 30% increase? How do we find the multiplier?

Let's review what a multiplier is. Say I have some number "x."

A "multiplier" is a number I multiply by x in order to take a certain percentage of x or increase or decrease x by a certain percentage. 

100% of x would be just 1 * x = x 

70% of x would be 70/100 * x = 0.7x

225% of x would be 225/100 * x = 2.25x

3% of x would be 3/100 * x = 0.03x 

0.17% of x would be 0.17/100 * x = 0.0017x

Now, that's just taking a percentage "of" x. 

If we want to *increase* x by a percentage or express a percentage more than x, we just add the percentage increase to 1. 

Examples:

70% increase in x = 100% of x + 70% of x = 1x+ 0.7x = (1 + 0.7)x = 1.7x 

So 1.7x represents a 70% increase in x. 

43% increase in x would be: x + 0.43x = 1.43x 

200% increase in x would be: x + 2x = 3x

If we want to decrease x by a percentage or express a percentage less than x, we just subtract that percentage from 1: 

70% decrease in x = 100% of x - 70% of x = 1x - 0.7x = (1 - 0.7)x = 0.3x So 0.3x represents a 70% decrease in x. 

Notice that this multiplier .3 is the same as (30% of x). 

43% decrease in x would be: x - 0.43x = 0.57x a 98% decrease in x would be: x - 0.98x = 0.02x

So, the multiplier for a 30% increase in x is:

x + 30/100 * x = x + 0.3x = (1 + 0.3)x = 1.3x

This makes sense, because 1.3x is greater than x, and when we increase x by 30% we should have more than x.

0.3x would be 30% OF x

30/100 * x = 0.3x

So if we had 100, 30% of 100 would be 0.3 * 100 = 30. 

But increasing 100 by 30% would be 100 + (0.3 * 100) = 1.3 * 100.


Q: How do we know that .78 represents a 22% decrease and .84 represents a 16% decrease? How do we know whether we have an increase or a decrease? 

When we have sequential percent changes, we can express each percent increase, decrease, or "of" of a number as a multiplier. The sequential product of all the multipliers together will either be a number less than one  or a number greater than one.

If the product is a decimal less than one, we have a decrease. 

And percent decrease is:

(1 - product)

So when we get .78, we know that's a decrease, and the amount of decrease is: (1 - .78) = .22, which is 22%. So we have a 22% decrease.

When we get .84, we know that's a decrease, and the amount of decrease is:

(1 - .84) = .16, which is 16%. So we have a 16% decrease.

If the product is decimal greater than one, we have an increase

And the percent increase is:

(product - 1)

So say we had a a product of 1.68. That's an increase, and the amount of increase is:

(1.68 - 1) = .68, which is 68%, so we have a 68% increase.

If our product is one exactly, then that's just 100% of our original, so we had no change.

More examples:

Resulting product of .77: decrease of (1 - .77) = .23 = 23%
Resulting product of .01: decrease of (1 - .01) = .99 = 99%

Resulting product of 1.9: increase of (1.9 - 1) = .9 = 90%
Resulting product of 8.4: increase of (8.4 - 1) = 7.4 = 740%
Resulting product of 2: increase of (2 - 1) = 1 = 100% <---(that's an increase of 100%...i.e., an exact doubling of our original)

Example: What is the multiplier for: 60% of X increased by 60% then decreased by 60%?

.6X = 60% of X

...increased by 60% = 1.6(.6X)

...decreased by 60% = .4(1.6(.6X)) 

= .384X 

This final number represents 38.4% of our original X. Or we could say that it is a (1 - .384) × 100%  = 61.6% decrease in X.