# Basic question :)

#### Nakani

##### New member
So, I have a fairly easy problem-solving question. But I don't fully get the 'best' way to solve it, other than guessing. It goes like this:
"How much is X*Y if X+Y=45, X-Y=27, and X/Y=4?"
Anyone know how to efficiently solve it? Maybe I'm just stupid, but I would appreciate the help!

#### Dr.Peterson

##### Elite Member
You put this under arithmetic, even though you used variables to state it. I'll assume you don't want to use algebra to solve it (though that might be the best way).

You want two numbers, X and Y, which differ by 27 and sum to 45. If you reduced the larger one (X) by 27, the sum would be reduced by 27: 45 - 27 = 18. But that would be the sum of two copies of Y; so Y must be 9. (You could do the same thinking by drawing pictures.)

What is X, then?

And is it true that X/Y = 4? If not, then this would be an invalid problem!

Finally, what is the product XY?

Since there is more information than you need, there are several ways to solve this, with or without algebra, depending on which facts you choose to start with. I wouldn't say any of them is "best".

#### Mr. Bland

##### Junior Member
It looks like algebra was probably the intended way to solve this, since it works out so cleanly. It is a system of equations.
Take two equations:
$$\displaystyle x + y = 45$$
$$\displaystyle x - y = 27$$​

Solve for $$\displaystyle x$$ by adding both equations to eliminate $$\displaystyle y$$:
$$\displaystyle 2x = 72$$
$$\displaystyle x = 36$$​

Solve for $$\displaystyle y$$ by substituting $$\displaystyle x$$ in one of the equations:
$$\displaystyle 36 + y = 45$$
$$\displaystyle y = 9$$​

Verify:
$$\displaystyle x / y = 4 = 36 / 9$$​

#### Jomo

##### Elite Member
x/y = 4 means that x=4y. That is whatever y equals, x is 4 times that.
So pick values for y
y=1, then x=4. The sum is 5, NOT 45.
y=2, then x=8. The sum is 10, not 45. We are moving in the right direction but still far away from 45. So don't try y=3, but maybe try y=7.
y=7, then x=28. The sum is 35, still too low.
y=8, x=32. Sum is 40.
y=9, x= 36. The sum is 45. Now we check to see if the difference is 27. 36-9 = 27.
The answer is y=9 and x=36.

#### MarkFL

##### Super Moderator
Staff member
We could square the first two equations to get:

$$\displaystyle X^2+2XY+Y^2=45^2$$

$$\displaystyle X^2-2XY+Y^2=27^2$$

Subtract the latter from the former:

$$\displaystyle 4XY=45^2-27^2=72\cdot18$$

Divide by 4 to get:

$$\displaystyle XY=18^2=324$$

#### toponlinetool

##### New member
I think if you follow the following steps, you can easily get answer-
1) Add 2 equations X+Y=45 & X-Y=27, Y is eliminated & you will get 2X=72, & X=36.
2) Now consider, X/Y=4. Here, replace the value of X=36, you will get Y=9.
3)Thus, you got both the values, X=36 & Y=9. Now, it's easy to calculate X*Y=36*9=324.