Answer:
Explanation:
Let's call the original price of the item "x". We know that a tax of $36.25 was added to the price, so the total price of the item is:
x + $36.25
We want to find the original price "x", so we can set up an equation:
x + $36.25 = total price
We don't know what the total price is, but we do know that the tax rate is typically a percentage of the original price. Let's say the tax rate is "r" (where "r" is a decimal number between 0 and 1). Then the tax amount would be:
tax = r * x
We know that the tax amount is $36.25, so we can set up another equation:
r * x = $36.25
Now we have two equations:
x + $36.25 = total price
r * x = $36.25
We can solve for "x" by isolating it in the second equation:
x = $36.25 / r
Then we can substitute this expression for "x" into the first equation:
$36.25 / r + $36.25 = total price
We still don't know what "r" is, but we can use the fact that the total price is the original price plus the tax amount:
total price = x + tax
total price = x + r * x
total price = x * (1 + r)
Substituting this expression for "total price" into the first equation:
$36.25 / r + $36.25 = x * (1 + r)
Now we have an equation with only one variable, "r". We can solve for "r" using algebra:
$36.25 / r + $36.25 = x * (1 + r)
$36.25 + $36.25r = xr + xr^2
xr^2 + (x - $36.25)r - $36.25 = 0
This is a quadratic equation, which can be solved using the quadratic formula:
r = (-b ± sqrt(b^2 - 4ac)) / 2a
where a = x, b = x - $36.25, and c = -$36.25. Plugging these values in, we get:
r = (-x + $36.25 ± sqrt((x - $36.25)^2 + 4x($36.25))) / 2x
We can simplify this a bit by recognizing that x($36.25) is the same as $36.25x:
r = (-x + $36.25 ± sqrt((x - $36.25)^2 + $1441)) / 2x
We know that "r" is between 0 and 1, so we can ignore the negative root. We also know that the tax rate is typically less than 1 (i.e., less than 100%), so we can assume that "r" is a small number. We can use this assumption to simplify the expression under the square root:
(x - $36.25)^2 + $1441 ≈ x^2 - 2*$36.25x + $1441 + $1441
≈ x^2 - 2$36.25*x + $2882
Substituting this back into the formula for "r":
r ≈ (-x + $36.25 + sqrt(x^2 - 2*$36.25*x + $2882)) / 2x
Now we can use the fact that the tax amount is $36.25 to solve for "x":
r * x = $36
Substituting the expression for "r" that we derived above:
(-x + $36.25 + sqrt(x^2 - 2*$36.25*x + $2882)) / 2x * x = $36.25
Simplifying:
-x + $36.25 + sqrt(x^2 - 2*$36.25x + $2882) = 2$36.25
x^2 - 2*$36.25x + $2882 = (2$36.25 - $36.25)^2
x^2 - 2*$36.25*x + $2882 = $36.25^2
x^2 - 2*$36.25*x + $2882 - $36.25^2 = 0
Using the quadratic formula:
x = (-(-2*$36.25) ± sqrt((-2*$36.25)^2 - 41($2882 - $36.25^2))) / (2*1)
x = ($72.50 ± sqrt(5250.5625)) / 2
x ≈ $47.18 or $61.07
So the original price of the item was either $47.18 or $61.07. We can check our answer by adding the tax of $36.25 to each price:
$47.18 + $36.25 = $83.43
$61.07 + $36.25 = $97.32
The second calculation gives a more reasonable price for an item, so the original price was likely $61.07.