The price of a hat is reduced by a% from its regular price, and then is raised by a% from the reduced price. If the resulting price is an 81% discount from the regular price, what is a?
To find the value of "a," we need to set up an equation based on the given information and solve for "a."
Let's say the regular price of the hat is denoted as "P."
The price is first reduced by "a%," so the new price is (P - (a/100)P).
Then, this reduced price is raised by "a%," so the resulting price is (P - (a/100)P) + ((a/100)(P - (a/100)P)).
According to the problem, this resulting price is an 81% discount from the regular price.
So, we can set up the equation:
(P - (a/100)P) + ((a/100)(P - (a/100)P)) = (1 - 81/100)P
Now, let's simplify and solve for "a":
(P - (a/100)P) + (a/100)(P - (a/100)P) = (1 - 0.81)P
After simplifying and removing like terms:
P - (a/100)P + (a/100)P - (a^2/10000)P = 0.19P
The "P" terms cancel out:
(1 - a/100 - a^2/10000)P = 0.19P
Dividing both sides by "P":
1 - a/100 - a^2/10000 = 0.19
Now, let's solve this quadratic equation:
1 - a/100 - a^2/10000 - 0.19 = 0
Rearranging and simplifying:
-a^2/10000 - a/100 + 0.81 = 0
Multiply all terms by 10000 to get rid of the fractions:
-a^2 - 100a + 8100 = 0
Solving this quadratic equation will give us two possible values for "a."