sequential percent changes
- a percent increase followed by a percent decrease of the same amount does not return to the original value.
- common errors include assuming that sequential percent changes can be directly added or subtracted.
- the correct approach involves calculating the product of multipliers for each percent change.
- examples illustrate that sequential changes require careful calculation to avoid predictable mistakes.
- understanding and applying the concept of multipliers is key to solving these types of gmat problems accurately.
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.
