The area of a circle is increased by 800%

guerrero wrote:The area of a circle is increased by 800%. By what percent has the diameter of the circle increased?

(A) 100%
(B) 200%
(C) 300%
(D) 600%
(E) 800%

OAC

I think the correct answer is B

Let's work backwards from the answer choices.

Original circle:
Let the radius = 1
So, the area = (pi)(1^2) = pi
NOTE: The diameter = 2

Enlarged circle:
Let the radius = 3
So, the area = (pi)(3^2) = 9(pi)
NOTE: The diameter = 6

Notice that the area increases from 1(pi) to 9(pi)
This represents an increase of 800%

Also, the diameter increases from 2 to 6
This represents an increase of 200%
Answer = B

Cheers,
Brent

guerrero wrote: "The area of the circle is increased by 800%, thus the area is increased 9 times.

The area of a circle it proportional to the square of the diameter (area=pi d^2 / 4), therefore the diameter must increase 3 times (diameter increase 3 times = area increase 9 times), which is increase by 200%."

That's a legitimate/valid solution, but potentially hard to conceptualize for some students. So, I like the idea of playing it safe and doing some testing.

For this question, it's not a bad idea to start testing cases, beginning with the middle (C)

C) 300% increase
Let's say the original circle has radius 1 (i.e., diameter = 2)
The area of this circle is pi
If the diameter increases 300%, then the new diameter is 8, which means the radius = 4
If the radius of the enlarged circle is 4, the area is 16(pi)
An area increase from pi to 16(pi) represents a 1500% increase.
We want an 800% increase.
So C is not the correct answer. In fact, we're looking for a smaller increase. So, let's try . . .

B) 200% increase
Let's say the original circle has radius 1 (i.e., diameter = 2)
The area of this circle is pi
If the diameter increases 200%, then the new diameter is 6, which means the radius = 3
If the radius of the enlarged circle is 3, the area is 9(pi)
An area increase from pi to 9(pi) represents an 800% increase.
BINGO!

The answer is B

Cheers,
Brent