Answer
The potential energy of the child will increase by $444J$.
Work Step by Step
We have $$\Delta PE=mg(h_f-h_0)$$
The shape of the staircase is not important, as $PE$ is a conservative force. We only care about the height of the staircase, which is the same for the 2 staircases here.
- For the adult climbing the staircase, $$\Delta PE_{adult}=m_{adult}g(h_f-h_0)$$
- For the child climbing the staircase, $$\Delta PE_{child}=m_{child}g(h_f-h_0)$$
We have $$\frac{\Delta PE_{child}}{\Delta PE_{adult}}=\frac{m_{child}g(h_f-h_0)}{m_{adult}g(h_f-h_0)}=\frac{m_{child}}{m_{adult}}=0.222$$ $$\Delta PE_{child}=0.222\Delta PE_{adult}=444J$$
