I had a question today on the GMAT Prep test that stumped me.

There was one equalateral triangle - all three sides = t, and a square - all four sides = s. The question asked,
"If the two regions above have the same area, what is the ratio of t:s?
Answer choices:
A- 2:3
B- 16:3
C- 4: radical3
D- 2: little 4 radical 3
E- 4: little 4 radical 3"

The answer is D, but i tried multiple times and never got that answer. Can someone help?

Thanks, Kate

Find the area of the square:
s * s = s^2


Find the area of the triangle:
(1/2)*b*h

b = t
h = (t/2)(sqrt 3) using the principles of a 30-60-90 right triangle

(1/2)*(t)*(t/2)(sqrt 3) = (1/4)*(sqrt 3)*(t^2)


Make them equal to each other:
s^2 = (1/4)*(sqrt 3)*(t^2)


Solve for t/s:
[4/(sqrt3)] = t^2 / s^2

Square root both sides
2/(fourth root 3) = t/s

Answer:
D) 2 : (fourth root 3)

i can't improve upon this impressive solution (thanks tmmyc), except to say the following:
memorize the formula for the area of an equilateral triangle.
area = (side^2)*(root3) / 4 - ugh, these things look terrible on the forum.
it's well worth your time to memorize this humble formula, which will save you a lot of time if the area of an equilateral triangle is ever at issue in a problem.

here's another powerful method:
draw an accurate picture and guess.
it might seem as though i'm being facetious, but i'm not at all: one of the best weapons in your geometry arsenal is the combination of your pen, paper, and mathematical intuition.

if you draw an accurate picture of an equilateral triangle and a square with the same area, two things will be clear:
* the side of the triangle is longer
* ... but not by that much (your diagram should show that t is more than s, but less than 2s).

these two observations kill choice a (in which t < s) and choice b (in which t >>>> s) immediately.

as for the others, you can guess if you need to make up some time; guessing from 3 choices isn't bad.

if you've memorized the approximation root3 - 1.7, then you can eliminate choice c (a ratio that's greater than 2:1, because 4 is more than twice as big as 1.7); you're thus left with the two final choices.

the fourth root of 3 is a slippery fish - in fact, from the way you notated it in your post it's unclear whether you understand what it is - but you can approximate it by realizing that it's the square root of the square root of 3. since the square root of 3 is about 1.7, the fourth root of 3 must be roughly halfway between that number and 1; let's say 1.3.

in that case e is way too big, so go with d.

--

note that i'm not saying you have to be able to approximate the fourth root of 3 - most students wouldn't be able to do that - but even basic eyeballing can get you down to 2-3 choices in a very short time.

i never would have gotten this without your help. thanks very much you guys.

Ron, you're absolutely right about certain problems/formulas looking ugly on the forum.
I've noticed this many times previously.
Is there any chance we can get some symbol choices inserted above as we already have for bold, etc.?
I think it would make the forum A LOT easier to use.

Thanks again, K

as my level of tech sophistication lies somewhere between 'neanderthal' and 'caveman', i'm afraid i can't tell you for sure.

i will tell you, though, that this forum is hosted on a standard forum template, so the answer is probably 'no'.

if you have trouble with the notation on the forum, my best suggestion is to whip out a pad of paper and a pen, and write the formulas out for yourself. granted, that requires you to reduce the laziness factor a bit, but it's well worth it. :) (especially because, when you take the gmat, you'll have to write on a real, live pad with a real, live pen... so may as well get used to it)

IN THE LAST STEP UP THE PROBLEM WHERE DOES FOURTH ROOT 3 COME FROM? HOW DO YOU ATTAIN IT? WHEN I SOLVE IT I GET 2:SQRT 3 WITHOUT THE LITTLE 4.

THANKS

When you take a square root, you have to take the square root of each thing - so the 4 does become a 2, as you found, the the square root of 3 then becomes the fourth root of 3. (Taking the square root of a square root gives you the 4th root.)

Try writing this out using the exponent 1/2 instead of a square root symbol. You've got 3^(1/2) and then you raise that to (1/2) again to take the square root. When you raise an exponent to an exponent, you multiply the two exponents. 1/2 * 1/2 = 1/4, so now you've got 3^ (1/4), or the 4th root of 3.

Also, in future, please use normal lower case when you are typing - all caps is the equivalent of SHOUTING on the Internet, and I'm sure you didn't mean to do that. :) Thanks!