The roots of $3x^2-10x+k=0$ are in the ratio 2 : 3. Then $k$ equals:

Write the roots as $2t$ and $3t$. Their sum is $5t$ and must equal 10/3, so $t=2/3$ and the roots are 4/3 and 2. Their product is 8/3, and the product of the roots is also $k/3$, so $k=8$. (6 takes the roots to be literally 2 and 3, which would need the sum to be 5; 8/3 forgets that the product equals $k/3$ rather than $k$; 24 multiplies by the leading coefficient a second time.) Check: $3x^2-10x+8=(3x-4)(x-2)$, whose roots 4/3 and 2 are in the ratio 4 : 6 = 2 : 3.