OG Quant Review Q. 175

[akshatgupta87]

Q.) A square wooden plaque has a square brass inlay in the center, leaving a wooden strip of uniform width around the brass square.If the ratio of the brass area to the wooden area is 25 to 39, which of the following could be the width,in inches, of the wooden strip?
I. 1
II. 3
III. 4

A)I
B)II
C)I,II
D)I,III
E)I,II,III

Someone pls solve..

[therealtomrose]

First a stolen answer from BTG member: manpsingh87, then thoughts about how you might approach this in a quicker way.

-------START stolen answer
let length of square wooden plaque be x, and length of square brass be y;
as per question, are of brass/area of wooden= 25/39;
hence y^2/x^2-y^2=25/39;
39y^2+25y^2=25x^2
x/y=8/5;
width of wooden strip would be x-y=x-5/8x=3/8x;
if x=8/3, then width=1,
if x=8, width=3;
if x=32/3, width =4;

as all values are possible, hence E
-------END stolen answer

I think this is really a computation trap. That is, the test wants to trap you into doing the above calculation even though you don't need to. How can you solve this faster?

I instead use a mental exercise with extremes to shortcut the process:

Imagine a fixed width wooden border (1" wide, or any other number) around a brass inlay.

If the width of the brass inlay is 0", the ratio of Brass:Wood is [area brass]/[area wood] = [0] / [some number] = 0
If the width of the brass inlay is HUGE", the ratio of Brass:Wood is [area brass]/[area wood] = [HUGE] / [some number] = HUGE

Therefore, the potential ratios range from 0 to huge, for any width of wood. In other words, a strip of any width can be accomodated and the answer is E. The answer is all of the above for any group of positive numbers. It's a trivial solution that takes about 20 seconds for a problem that otherwise looks very complicated.