Transposing the two digits of A's age gives B's age. The difference between their ages is twice C's age and В is ten times as old as С. What are the three ages?
Let A's age be $ab=10a+b$ so B's age is $ba=10b+a$
$$10b+a=10 \times( \text{C's age}), \frac{10b+a}{5}=2 \times (\text{ C's age})$$
$$|10(a-b)+(b-a)|=\frac{10b+a}{5}$$
If $a>b$ then $10(a-b)+(a-b)=\frac{10b+a}{5}$ gives $54a-65b=0$
If $b>a$ then $10(b-a)+(b-a)=\frac{10b+a}{5}$ gives $45b-56a=0$
None of the last two equations give $a$ and $b$ as single digit.