Question

The ratio of the radii of two circles is \(4 : 9 .\) What is the ratio of their circumferences? F. \(2 : 3\) G. \(4 : 9\) H. \(16 : 81\) J. \(4 : 8 \pi\) K. \(9 : 18 \pi\)

Short answer

Answer: The ratio of the circumferences of the two circles is 4:9.

Step-by-step solution

Write down the ratio of the radii

We are given that the ratio of the radii of the two circles is \(4:9\).

Write the formula for the circumference of a circle

The formula for the circumference of a circle is \(C = 2\pi r\).

Write the circumferences with their radii ratio

Let the radius of the first circle be \(4x\) and the radius of the second circle be \(9x\). Then, using the formula for the circumference, the circumference of the first circle is \(C_1 = 2\pi (4x) = 8\pi x\) and the circumference of the second circle is \(C_2 = 2\pi (9x) = 18\pi x\).

Find the ratio of the circumferences

To find the desired ratio of their circumferences, divide the circumference of the first circle by the circumference of the second circle: \(\frac{C_1}{C_2} = \frac{8\pi x}{18\pi x}\).

Simplify the ratio

We can cancel out \(\pi x\) from the numerator and denominator, giving us: \(\frac{8}{18}\). Simplify the fraction to get the ratio of the circumferences: \(\frac{8}{18} = \frac{4}{9}\). The ratio of the circumferences of the two circles is \(4:9\), which corresponds to the answer choice \(\boxed{\text{(G)}}\).

Key concepts

Circle Circumference Formula

The circle is a fundamental shape in geometry, and its properties are governed by specific formulas. One important property is the circumference, which is the distance around the circle. The formula to calculate the circumference is quite straightforward:

\[\begin{equation} C = 2\text{\textpi} r \text{,} \end{equation}\]where \(C\) is the circumference, \(\text{\textpi}\) (pi) is a constant approximately equal to 3.14159, and \(r\) is the radius of the circle. The radius is the distance from the center of the circle to any point on its edge.

When dealing with problems involving the circumference, it's critical to understand that all circumferences are directly proportional to their respective radii. This means if you have two circles with different radii, you can find the ratio of their circumferences by applying this formula to each circle, as the constant \(\text{\textpi}\) will remain the same for both.

Ratio and Proportion

The concept of ratio and proportion is integral to understanding relationships between different quantities. A ratio is a way to compare two quantities by division, expressing how much of one thing there is compared to another. For instance, if the ratio of the radii of two circles is \(4:9\), it tells us that for every unit of length the first radius has, the second radius has \(\frac{9}{4}\) times that length.

Proportion, on the other hand, is an equation that states two ratios are equal. It is used to solve for an unknown quantity when you have a set of equivalent ratios. If we know the radii are in proportion and have a common ratio, we can conclude that the circumferences will also be in proportion because circumference is calculated directly using the radius. For circles, this means that if the radii have a certain ratio, the circumferences will share that same ratio, as the circle circumference formula shows us.

Simplifying Fractions

Simplifying fractions is an essential skill in mathematics that allows us to express fractions in their simplest form. To simplify a fraction, we look for the greatest common divisor (GCD) of the numerator and the denominator—this is the largest number that divides both without leaving a remainder.

Once the GCD is found, both the numerator and the denominator are divided by this number to reduce the fraction to simpler terms. For example, in the solution to our original problem, we simplified the fraction \(\frac{8}{18}\) by dividing both the numerator and the denominator by 2, the GCD in this case. This process resulted in the simplified ratio \(\frac{4}{9}\). Simplifying fractions makes it easier to work with and understand them, especially when comparing or adding fractions together.