   SEARCH HOME Math Central Quandaries & Queries  Question from Emily, a student: A square pen and a pen shaped like an equilateral triangle have equal perimeters. Find the length of the sides of each pen if the sides of the triangular pen are fifteen less than twice a side of the square pen. Please help. I don't even know how to get started on this problem, I don't actually need the answer, just the formula for solving it. Thank you. Hi Emily.

I would proceed like this:

There are two things that are unknown in this problem: the length of the side of the square pen and the length of a side of the triangular pen. That's what you are asked to find. So start by defining some
variables:

Let S = the length of one side of the square pen and
let T = the length of one side of the triangular pen.

Now interpret the words into mathematical equations:

"A square pen and a pen shaped like an equilateral triangle have equal perimeters."

The perimeter is the distance around, so add up the sides:
S + S + S + S = T + T + T
this reduces to:
4S = 3T

That's our first equation.

"...the sides of the triangular pen are fifteen less than twice a side of the square pen."

T = 2S - 15.

You see how I got that? The side of the triangular pen is T, so I wrote T. Then we have the verb To Be, which usually means an equal sign. Then I have 15 less than twice a side of the square pen. I could have written it -15 + 2S, but that is more awkward and means the same thing mathematically as 2S - 15.

So here are the two equations:
4S = 3T and T = 2S - 15.

Now you can use either the substitution method (this is the one I would use for this problem) or the elimination method to solve this "system of equations" (also called simultaneous equations).

If you need examples of that, then type substitution method or elimination method into the search on the main Quandaries and Queries page.

Cheers,
Stephen La Rocque.     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.