If $0

Question

The options are:

$c/3b$
$a/c$
$b/3c$

According to the test rules, there should be one corrected answer only.

I think both the first and second options are correct because:

$a/b > c/3b$

$a > c/3$ (I can do this because I know $b > 0$)

$3a > c$ which is true considering the hypothesis.
$a/b > a/c$

$1/b > 1/c$ (always because $a > 0$)

$c > b$

So my first question is: is my procedure wrong? Or are the two options actually both true?

My second question is: what about the third option? I don't know how to demonstrate whether it is true or false.

Assuming $c/3b=c/(3b)$, you are correct that since $0<c/3<a$, we know $(c/3)/b<a/b$. Also, $0<b<c$ implies $a/b>a/c$. — Joe, Sep 20 '21 at 15:50
Option 3 also appears to be always true, as $b/3c < a/c < a/b$. — eyeballfrog, Sep 20 '21 at 15:51
Indeed, $0<b<c$ implies that $b/(3c)<1/3$, but $0<b<3a$ implies that $a/b>1/3$. — Joe, Sep 20 '21 at 15:54

score 1 · Answer 1 · answered Sep 20 '21 at 15:54

1

You are correct. If $a>0$ and $b<c$ then $a/c<a/b$. Also we have $0<1<c/b<3a/b$, so $c/3b < a/b$. Lastly as $b<3a$ we have $$ b/3c < 3a/3c=a/c < a/b$$

So all three are true.

Edit: We also get the last one by $a/b > 1/3$ and $b/3c < c/3c=1/3$.

answered Sep 20 '21 at 15:54

Lazy

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If $0

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