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MathNet 1964 — Part 01 Question 5

Math Olympiad Vault · MathNet 1964 — Part 01

Question 5

It is well known that if $\frac{p}{q} = \frac{r}{s}$, both reasons are equal to $\frac{p - r}{q - s}$. We now write the equality $$
\frac{3x - b}{3x - 5b} = \frac{3a - 4b}{3a - 8b}
$$ by the previous property, both fractions must be equal to $$
\frac{3x - 5b - 3a + 8b}{3x - b - 3a + 4b} = \frac{3x - 3a + 3b}{3x - 3a + 3b} = 1
$$ while the proposals are usually different from the unit. clearly explain what this result is due to.