If we want to make two changes and end up back where we started, the two numbers we need to multiply need to be one another's inverses. For instance, 100 * 3/2 * 2/3 = 100.
This is a little harder to see with percents, because we might write the above like this: 100 increased by 50% then decreased by 33 1/3% equals 100. That's why it can be easier to figure out what number or fraction we're actually multiplying by (e.g. 50% increase = multiply by 1.5; 33 1/3% decrease = multiply by .6666). We can use these formulas:
Increase by x%: Multiply by (100+x)/100
Decrease by x%: Multiply by (100-x)/100
After that, we can often convert to fractions or decimals and find an inverse from there.
Here's an example:
x increased by 60% then decreased by y% equals x. What is y?
x increased by 60% = x * 160/100 = x * 16/10 = x * 8/5
So what do we need to do to get back to x? Multiply by the inverse of 8/5 . . . 5/8.
x * 8/5 * 5/8 = x
So we know that reducing by y is the same as multiplying by 5/8.
5/8 = .625 = 62.5% If we multiplied by 62.5%, y must be the remaining 37.5%.
We can translate the original sentence and plug this in, just to be see how it works:
x increased by 60% then decreased by y% equals x.
x * 160/100 * (100-y)/100 = x
x * 160/100 * (100-37.5)/100 = x
x * 160/100 * 62.5/100 = x
x * 8/5 * 5/8 = x
x = x
It works!
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Dmitry Farber | Manhattan GMAT Instructor | New York
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