Use Einstein’s photoelectric equation

For a photon of energy $E$, the maximum kinetic energy of emitted photoelectrons is $K_{\max} = E - \phi$ where $\phi$ is the work function of the metal.

First stream: photon energy is twice the work function

$E_1 = 2\phi$ So, $K_1 = E_1 - \phi = 2\phi - \phi = \phi$

If $v_1$ is the maximum speed, then $\frac{1}{2}mv_1^2 = \phi$ Hence, $v_1 = \sqrt{\frac{2\phi}{m}}$

Second stream: photon energy is ten times the work function

$E_2 = 10\phi$ So, $K_2 = E_2 - \phi = 10\phi - \phi = 9\phi$

If $v_2$ is the maximum speed, then $\frac{1}{2}mv_2^2 = 9\phi$ Hence, $v_2 = \sqrt{\frac{18\phi}{m}} = 3\sqrt{\frac{2\phi}{m}}$

Find the ratio of maximum velocities

$v_1 : v_2 = \sqrt{\frac{2\phi}{m}} : 3\sqrt{\frac{2\phi}{m}} = 1:3$

Therefore, in the ratio $x:y = 1:3$, we get $x = 1$

Comparison with stored answer

Derived answer: $1$

Stored correct answer: $1$

They match.