Question

What percent increase is $4-$5?

Answer 1

Going from $4 to $5 is a 25% increase.

Explanation:

A percent increase is how much the amount gained is, relative to the amount we start with. It's the ratio of the amount gained to the initial amount.

If, for example, you start a chat group with `n` people, and in a week there are now `N` people, then the increase (in units of people) is

`N-n`

(This should seem easy.)

The relative increase is this difference of `N-n` (people) relative to the previous amount of `n` (people):

`(N-n)/n`

and so it has no units. It's like we're asking, "How many groups of size `n` could we make from this increase of `N-n`?"

Finally, the percent increase is just the relative increase, expressed as a percentage (which simply involves multiplying the relative increase by 100%):

`(N-n)/n xx 100%`

Using your question as an example, a dollar value has gone from $4 to $5. By the formula above, the percent increase is

`color(white)=(N-n)/nxx 100%`

`=($5-$4)/($4)xx100%`

`=($1)/($4)xx100%`

`= 0.25 xx 100%" "=" "25%`

So the percentage increase of going from $4 to $5 is 25%, because that $1 gain is 25% of the original $4.

Bonus:

Using some basic algebra, we can simplify the formula for relative increase to:

`(N-n)/n" "=" "N/n - n/n" "=" "N/n - 1`

In this way, the relative increase is the ratio of the new value to the old one, minus 1. Of course, the percent increase is still just this relative increase multiplied by 100%:

`(N/n -1 )xx 100%`

Both formulas give the right answer.

Answer 2

`25%`

Explanation:

If we start with `$4` and it increases to `$5`, we've increased the value by `$1, ($5-$4=$1)`. So what percent is this?

We can do this a couple of different ways. One way is to take the change and divide it by the original:

`color(blue)(ul(bar(abs(color(black)(($1)/($4)=.25=25%))))`

Another way we can do this is to set up an equation this way:

`"Starting number" xx "percentage increase" = "increase"`

`$4xx x = $1`

`color(blue)(ul(bar(abs(color(black)(x=($1)/($4)=.25=25%))))`

And yet another is to not calculate the difference first:

`"Starting number" xx "percentage"="new number"`

`$4 xx x =$5`

`x=($5)/($4)=1.25=125%`

`100%` is the original value, so the increase is the percentage less `100%`:

`color(blue)(ul(bar(abs(color(black)(125%-100%=25%))))`