Sequential Percent Changes
- A common misconception is that a percent increase followed by an equal percent decrease (or vice versa) returns the value to its original state, which is incorrect.
- The correct approach to solving problems with sequential percent changes involves using multipliers for each percent change and multiplying them together.
- Adding or subtracting percentage points in a sequence of percent changes is a predictable mistake that many test-takers make.
- Real-world examples illustrate how applying multipliers correctly reveals the actual outcome of sequential percent changes, often counterintuitive to common misconceptions.
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>#/p###
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 - 38.4 = 61.6% decrease in X.
