Word problem

Word problems can kill you if you aren't good at rewriting them in the form of an equation that can be solved. Here's an example:

Given three equally spaced candles, the middle candle is 3 times the height of the other two. From left to right, the candles burn down in 9, 6, and 18 hours respectively. After t hours, all of the tops of the candles are collinear. Determine the smallest possible value for t.


Is collinear spelled right? My spell checker says it is, but for some reason it looks odd to me. Oh well.

A couple of things you should acknowledge off the bat:

We are not given specific heights, just the relation between the heights of the candles. Just write the heights in terms of an unknown x, and when we solve for t, it will most likely divide out.

They are equally spaced. This helps, because we know all that's required for them to be collinear is the same difference in height (I'll explain this better soon. In fact right now).

Collinear means all of the points lay on the same line. Since a line is, well, a line, its slope is constant. That means, for the points to be collinear (for the sake of explanation, I'm going to start calling the points from left to right A, B, and C), the difference between the height of A and B should be the same as the difference between the height of B and C. We already know that their width is constant. A much more challenging problem would have something like slanted candles so the width AND height is changing. Why somebody would set up slanted candles, I don't know. It sounds like a fire hazard.

So now that we know what equality must exist for the points to be collinear, we have an objective: write this equality in terms of t, our variable for time. To make it even clearer, this equality is:

Difference between height of B and height of A = Difference between height of C and height of A.

Just to confirm: Does it matter if we write the difference as A - B = B - C, as opposed to B - A = C - B ? Not really. The equality holds up either way. If you aren't convinced, draw a line, and pick three points with the same distance horizontally between them, and see for yourself.

We can rewrite this again:

Height of B - Height of A = Height of C - Height of B

So we set out to find these heights in terms of t. If a candle is burning at a rate r, then the candle loses height rt after time t. OR, the height of the candle, in terms of its original height H, becomes:

H - rt

We have an original height for each, and a rate of burning, so we can write each of the heights of the candles in terms of t (and our unknown height x, but this will divide out, since both sides of the equation can factor out a X).

Height of A:
Original height: X
Rate: x / 9

If you're confused how we got the rate as x / 9, I'll explain. Using our formula
Distance = Rate * Time
And the fact that the entire candle of "distance" x (Think of it as how fast the top of the candle moves downwards), plus the fact the entire candle burns in 9 hours:

X = 9r
r = x / 9

Alright, there we go.

Plus these back in:



I was hoping that would look prettier.

The process for the others is exactly the same, except that the height for the middle one is 3x, because it's 3 times the height of the others.




Now substitute all of these back into our original equality:





Just like a promised, the x goes away. Divide each side by x:




Get rid of those parentheses and distribute negatives where needed:





Combine the constant terms:




Multiply by 18:




Like terms:




Now just solve for t:






And there's our answer.