The Quest for 700: Weekly GMAT Challenge (Answer)

Yesterday, Manhattan GMAT posted a 700 level GMAT question on our blog. Today, they have followed up with the answer:

First, we figure out the area of the smallest circle. A₁ =r² = 1² = .

Now, we find the area of the second smallest circle (n = 2). A₂ = A₁ + (2(2) – 1) = + 3 = 4. This means that the radius of the second smallest circle is 2 (since the area is r²).

The third smallest circle has area A₃ = A₂ + (2(3) – 1) = 4 + 5 = 9. This means that the radius of this circle is 3.

Finally, the fourth smallest circle (that is, the largest circle) has area A₄ = A₃ + (2(4) – 1) = 9 + 7 = 16. This means that the radius of this circle is 4.

The sum of all the areas is + 4 + 9 + 16 = 30.

The sum of all the circumferences is 2 times the sum of all the radii. The sum of all the radii is 1 + 2 + 3 + 4 = 10, so the circumferences sum up to 20.

Thus, the sum of all the areas, divided by the sum of all the circumferences, is 30/(20) = 1½.