VIC- why isn't plugging in numbers working here?

[tomslawsky]

OK, after a year off, I started studying again. Now, after reading (for the 4th time) word translations, I became pretty confident with VICS. Then I came across this problem- its an official ETS (SAT 1992) question that could just as easy appear on the GMAT as the SAT from a published SAT. Its a VIC and I plug numbers in, but cant replicate the correct answer. Help, please:


If
X + Y = t
X - Y = V

Then, in terms of T and V, what is XY

Answer: (T^2 - V^2) / 4

Help me get there, please!

Thanx

[jnelson0612]

Sure! Let's say:
X=5
Y=3
Thus, T=8 and V=2.

What is XY? 5*3=15. Our target number is 15.

Answer: (T^2 - V^2) / 4

Sub in T=8 and V=2: (8^2 - 2^2)/4 is (64 - 4)/4 =15. Bingo!

[arindam.mastermind]

There was this formula...

4xy = (x+y)^2 - (x-y)^2

Plug in T and V on the rhs and you will get the result.

[neelabhthakur84]

Here's an algebraic approach:

Step 1. Square both sides of both equations:

(x+y)^2= t^2
or, x^2 + y^2 + 2xy= t^2.... (1)

(x-y)^2= v^2
or, x^2 + y^2- 2xy= v^2... (2)

Now, subtract (2) from (1), you get:

4xy= t^2 - v^2

Therefore, xy= (t^2 - v^2)/4. Answer.

[jnelson0612]

neelab, very nice. This is a great thread as it shows that many strategies lead to the right answer. I'm a huge fan of picking numbers, but others prefer algebra and that is good too.

[terence.battle1]

I just saw a question very similar to this in our course. It was slightly reworded but the result was the same. Plugging in small prime factors for X and Y to get your target answer usually helps. Once you have what you are looking for just plug in the values.

[jnelson0612]

terence, yes. Plugging in small primes will usually get you where you need to go very quickly.