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.