Four concentric circles have radii 10, 20, 30, and 40. What is the ratio of the sum of the areas of the shaded annuli (rings between circles) to the area of the largest circle?

Rationale

The ratio of the sum of the areas of the shaded annuli to the area of the largest circle is 3:4.

The area of each circle is calculated using the formula (A = pi r^2). The areas of the shaded annuli can be found by subtracting the area of the smaller circle from the area of the larger circle for each pair of concentric circles. The area of the largest circle is then used to find the ratio.

A (1:4) YOUR ANSWER

This option suggests a ratio of 1:4, which does not accurately reflect the calculated areas of the annuli. The areas of the annuli must collectively represent a larger proportion of the largest circle's area than the ratio indicates.

B (3:4) CORRECT

This choice correctly represents the ratio of the total area of the shaded annuli to the area of the largest circle. The total area of the annuli is calculated as \(A(10) + A(20) + A(30) - A(40)\) yielding a ratio of 3 parts to 4 parts of the area of the largest circle, which matches the correct calculations.

C (1:2) YOUR ANSWER

This option suggests a ratio of 1:2, implying that the area of the shaded annuli is only half of the area of the largest circle. This is an incorrect interpretation of the area calculations, as the sum of the annuli's areas is greater than half of the largest circle's area.

D (2:3) YOUR ANSWER

This choice implies a ratio of 2:3, which undervalues the area of the shaded annuli compared to the largest circle. The calculations indicate a larger ratio, confirming that the area of the annuli significantly exceeds this option.

Conclusion

The correct ratio of the sum of the areas of the shaded annuli to the area of the largest circle is 3:4. This conclusion arises from accurate area calculations of both the annuli and the largest circle, demonstrating that the shaded regions collectively occupy a significant portion of the total area of the largest circle, confirming option B as the correct answer.

Four concentric circles have radii 10, 20, 30, and 40. What is the ratio of the sum of the areas of the shaded annuli (rings between circles) to the area of the largest circle?

The ratio of the sum of the areas of the shaded annuli to the area of the largest circle is 3:4.

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