1. ## Rational Number Proof

Hello all,

While practicing proofs I came across one that I'm having trouble with. Here it is:

Let x and y be non-zero rational numbers. Prove that (4x + 7y) / 6y is a rational number.

Let:
x = a/b . . . where b cannot equal 0
y = c/d . . . where d cannot equal 0

Then:
(4(a/b) + 7(c/d)) / 6(c/d)

Wouldn't this last statement be a proof in it self since I have expressed (4x + 7y) / 6y as a ratio of two integers, namely 4(a/b) + 7(c/d) and 6(c/d), where 6(c/d) cannot equal 0? Doesn't this show that (4x + 7y) / 6y is rational?

Thanks for any suggestions or pointers to similar problems.

2. ## Re: Rational Number Proof

I would take it a step further:

$\frac{4 \cdot \frac{a}{b}+7 \cdot \frac{c}{d}}{6 \cdot \frac{c}{d}} \cdot \frac{bd}{bd}= \frac{4ad+7bc}{6bc}$

3. ## Re: Rational Number Proof

Nice. I didn't see that. Thanks.

4. ## Re: Rational Number Proof

Although, wouldn't it be possible for c to equal 0, making 6bc = 0?

5. ## Re: Rational Number Proof

If x and y are non-zero, then none of a,b,c,d can be zero.

6. ## Re: Rational Number Proof

Oh yeah. Thanks again.