If the diameter of circle C is 3 times the diameter of circle D, then

If the diameter of circle $C$ is $3$ times the diameter of circle $D$, then the area of circle $C$ is how many times the area of circle $D$ ?

So, you were trying to be a good test taker and practice for the GRE with PowerPrep online. Buuuut then you had some questions about the quant section—specifically question 18 of Section 4 of Practice Test 1. Those questions testing our knowledge of Circles can be kind of tricky, but never fear, PrepScholar has got your back!

Survey the Question

Let’s search the problem for clues as to what it will be testing, as this will help shift our minds to think about what type of math knowledge we’ll use to solve this question. Pay attention to any words that sound math-specific and anything special about what the numbers look like, and mark them on your paper.

We can see that we’re asked about circles, diameters, and areas, so this question likely tests what we know about Circles. Let’s keep this in mind as we proceed.

What Do We Know?

Let’s carefully read through the question and make a list of the things that we know.

 

  1. We want to calculate the ratio of the area of one circle to another circle
  2. We know the ratio of the diameters between the two circles

 

Develop a Plan

Let’s start with a top-down approach to this question. We want to know how many times larger the area of circle $C$ is compared to the area of circle $D$. Whenever we see the phrase “how many times larger,” we know that we are looking for the ratio of the two values in question. So let’s go ahead and write the answer to this question as this ratio:

$${\Area \of \Circle \C}/{\Area \of \Circle \D}$$

Next, we know that the area of a circle is $π(\Radius)^2$, so let’s go ahead and plug that equation in for circle $C$ and circle $D$.

$${\Area \of \Circle \C}/{\Area \of \Circle \D} = {π (\Radius_C\;)^2}/{π(\Radius_D\;)^2}$$

We know that the radius is just the diameter of a circle divided by $2$. So if the diameter of circle $C$ is three times larger than the diameter of circle $D$, then the radius of circle $C$ is three times larger than the radius of circle $D$.

$$\Radius_C = 3·\Radius_D$$

So seems as if we have a clear path forward. We’ll just start substituting in equations, simplifying, and see what comes out at the end.

Solve the Question

Let’s start with our ratio and plug in $3·\Radius_D$ for $\Radius_C$:

$${\Area \of \Circle \C}/{\Area \of \Circle \D} = {π (3·\Radius_D)^2}/{π(\Radius_D)^2}$$

Next, we can further simplify this fraction by canceling out the $π$ and $\Radius_D$ from both the top and bottom of the right fraction.

$$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\Area \of \Circle \C}/{\Area \of \Circle \D} = {π (3·{\Radius_X})^2}/{π({\Radius_X})^2}$$
$$\;\,{\Area \of \Circle \C}/{\Area \of \Circle \D} = 3^2$$
$${\Area \of \Circle \C}/{\Area \of \Circle \D} = 9$$

Well that wasn’t so bad! We now have our final answer.

The correct answer is $9$.

What Did We Learn

When solving problems of the “how many times larger is Y than Z” variety, we should start by writing a ratio for what the problem wants us to solve. In this problem we started by writing the ratio of the area of one circle to the area of another circle:

$${\Area \of \Circle \C}/{\Area \of \Circle \D}$$

 

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