Question on Fraction Cancellation

Question

Not able to get a proper explanation for this question anywhere, please help!

Answer 1

Sure:$$\frac{26}{65}=\frac25.$$You can get this testing the fractions of the type$$\frac{20+a}{10a+5}$$with $a=1,2,\ldots$

Answer 2

The question is asking if there is an $a,b$ such that $$\frac{2a}{b5}=\frac 25$$ Consider it this way: $$b5=\frac{5\cdot2a}{2}$$

If $a$ is odd, the $RHS$ becomes a fraction and fails.

If $a$ is even, and a multiple of four, we get a number ending in $0$ (You've multiplied by $5\cdot2^2$, then divided by $2$, resulting in $5\cdot2$).

However, if it is only an even number and not also a multiple of four, the result ends in $5$ (Because it is a multiple of $5\cdot2$, then divided by $2$, resulting in just $5$). So the possible answers are $a=2, 6$. We get: $$\frac{22}{55}=\frac 25$$ $$\frac{26}{65}=\frac 25$$