How do you find percent increase and decrease in reverse?

Answer 1

The Short Answer:If a number (x) is increased by a percentage (p) to receive a number (y), the formula to find x is:

x=y/(1+(p/100))

If a number (x) is decreased by a percentage (p) to receive a number (y), the formula to find xis:

x=y/(1-(p/100))

Examples:A Decrease in Percentage:Let's say that a you purchased a shirt for $13.50 and you know it was on sale for 10% off that week. You want to know what the regular price is before returning to the store.

Using the formula above, the problem looks like this:

x=13.50/(1-(10/100)

We start in the inner most parenthesis where and find that 10/100=0.1:

x=13.50/(1-0.1)

1-0.1 is 0.9:

x=13.50/0.9

And finally 13.50/0.9=15:

x=15

Now we know that the original price of the shirt was $15.00!

Let's try reversing an increase in percentage now.

An Increase in Percentage:You borrowed some money from a friend and he says you now owe him $26.25 after 5% interest. You have decided that on top of not playing Xbox with you this weekend, he is only going to get the amount you borrowed... If only you could remember how much that was.

Using the formula above, the problem looks like this:

x=26.25/(1+(5/100)

We start in the inner most parenthesis where and find that 5/100=0.05:

x=26.25/(1+0.05)

1-0.05 is 0.95:

x=26.25/1.05And finally 26.25/1.05=25:

x=25

You pay him the $25.00 you owe him and remove him from your friends list.

Why Does This Work?Assuming that you know a number ( x) has been raised to some percentage (p) and resulted in a new value (y), the first thing you'll need to do is convert the percentage to the corresponding ratio/proportion (r) with the below formula:

r=p/100

This means for 10% r=.1 and for 33% r=.33. This allows us to look at what was actually done to the number xto get the result y:

y=x+xr

We can factor the shared x out of the right side as below:

y=x(1+r)

If we then divide both sides by (1+r) the result is:

y/(1+r)=x

The only difference between an increase in a percentage and a decrease in percentage is the sign of r as shown below:

y=x-xr

We can factor the shared x out of the right side as before:

y=x(1-r)

If we then divide both sides by (1-r) the result is:

y/(1-r)=x