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Algebra Difficulty 4.4 AIME Find the answer Slovenia

In the equality of the three fractions $\frac{2}{6} = \frac{3}{9} = \frac{58}{174}$ each digit from $1$ to $9$ occurs exactly once and the value of all three fractions is $\frac{1}{3}$ . Here is another example of such equality, $\frac{*}{*} = \frac{*}{*} = \frac{7*}{15*}$ , where some of the digits have been replaced by an asterisk. What is the value of all three fractions in this case?

Pick one

Solution

Sorting the available answers we get $\frac{1}{3} < \frac{1}{2} < \frac{3}{5} < \frac{2}{3} < \frac{3}{4}$ . The last of the three fractions in the equality can be estimated as
$\frac{1}{3} = \frac{60}{180} < \frac{7*}{15*} < \frac{90}{150} = \frac{3}{5}.$
So, the only possible answer is (A). After some quick trial and error we get $\frac{2}{4} = \frac{3}{6} = \frac{79}{158}$ , so (A) is indeed the correct answer.

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.