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The maximum tension a rope can withstand is $60 \mathrm{~kg}$-wt. The ratio of maximum acceleration with which two boys of masses $20 \mathrm{~kg}$ and $30 \mathrm{~kg}$ can climb up the rope at the same time is
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3 : 2
$\begin{aligned} 60 \mathrm{~kg}-\mathrm{wt} & =20\left(g+a_1\right)+30\left(g+a_2\right) \\ = & 20 \mathrm{~kg}-\mathrm{wt}+30 \mathrm{~kg}-\mathrm{wt}+20 a_1+30 a_2\end{aligned}$
$\begin{aligned} & \Rightarrow \quad 20 a_1+30 a_2=10 \mathrm{~kg}-\mathrm{wt} \\ & 2 a_1+3 a_2=1 \\ & a_1=\frac{1}{4} \mathrm{~g}, \quad a_2=\frac{1}{6} \mathrm{~g} \\ & a_1: a_2=6: 4=3: 2\end{aligned}$
$\begin{aligned} & \Rightarrow \quad 20 a_1+30 a_2=10 \mathrm{~kg}-\mathrm{wt} \\ & 2 a_1+3 a_2=1 \\ & a_1=\frac{1}{4} \mathrm{~g}, \quad a_2=\frac{1}{6} \mathrm{~g} \\ & a_1: a_2=6: 4=3: 2\end{aligned}$
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