## Sequential Percent Changes

- A same percent increase followed by a decrease does not return to the original value.
- Never add or subtract percent changes directly; this approach is always incorrect.
- Use multipliers to accurately calculate the effect of sequential percent changes.
- Common mistakes include assuming that a percent up followed by the same percent down cancels out, and adding or subtracting percentages to find the net change.
- Correctly using multipliers reveals the true impact of sequential percent changes, avoiding predictable mistakes.

**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.**