last week's MGRE Math Beast Challenge

I was indeed correct in answering (A) to last week's MGRE Math Beast Challenge. Here's MGRE's official explanation, in two parts: non-algebraic and algebraic. Neither explanation seems to rely on similar triangles, strangely enough.

NB: I have changed all subscript notation to capitals and lowercase: h₀ is now H; h₁ is now h; r₀ is now R; r₁ is now r.

The first explanation:

The best way to solve this problem may be one that involves little math. Notice that the cone is much wider at the top than it is at the bottom. Because the bottom of the cone is so narrow, filling up the cone to 40 percent of its height would fill less than 40 percent of its volume. The cone must be filled above 40 percent of h0 to reach 40 percent of the original volume, so h/H > 0.4.

The second explanation:

To solve this problem algebraically, we’ll use the volume formula provided.

Volume at H:

1000π = (1/3)π(R²)H

We’ll solve this for H, as it is one part of the expression in Quantity A:

H = 1000π • (3/πR²) = 3000/R²

Volume at h:

0.4(1000π) = 400π = (1/3)πr²h

We’ll solve this for h, as it is the other part of the expression in Quantity A:

h = 400π • 3/πr² = 1200/r²

Putting it together, Quantity A is

h/H = (1200/r²)/(3000/R²)

= (1200/3000) • (R²/r²)

= 0.4 • (R/r)²

To compare Quantity A to 0.4, the question is simply whether R/r is greater or less than 1. Similar to how H/h > 1, we can see that R/r > 1, so Quantity A is greater than 0.4.

To be honest, although I love the non-algebraic explanation, I don't like MGRE's algebraic explanation. They'd have done a lot better to go with similar triangles as their strategy. MGRE's algebraic explanation culminates in the ratios H/h and R/r, but the point is to compare h/H (the reciprocal of the aforementioned H/h) with 0.4. I think a few explanatory steps are missing in MGRE's writeup of this problem. "We can see that" is a non-explanation, and to say that "the question is... whether R/r is greater or less than 1" is really to say that no algebra was needed at all: we can compare visually, even if the drawing isn't done to scale. But having compared radii in this way, can the average GRE-taker leap deductively to a conclusion about h/H? I don't think so. Similar triangles give us those relationships most directly.


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