Two pendulums begin to swing simultaneously. The first pendulum makes nine full oscillations when the other makes seven. The ratio of the lengths of the two pendulum is

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As two pendulums begin two swings simultaneously, then
$n_{1} T_{1} = n_{2} T_{2}$
Where, $n_{1}$ and $n_{2}$ are the number of oscillations of first and second pendulum respectively and $T_{1}$ and $T_{2}$ be their respective time periods.
The time period of simple pendulum is given by
$T = 2 π \sqrt{\frac{l}{g}}$
where, l =length of pendulum
and g = acceleration due to gravity
$\Rightarrow T^{2} \propto 1$ …. (ii)
So, from Eqs. (i) And (ii), we get
$\frac{l_{1}}{l_{2}} = \frac{T_{1}^{2}}{T_{2}^{2}} = \frac{n_{2}^{2}}{n_{1}^{2}}$
Here, $n_{1} = 9, n_{2} = 7$
$\Rightarrow \frac{l_{1}}{l_{2}} = \frac{{(7)}^{2}}{{(9)}^{2}} \frac{49}{81}$
Hence, the ratio of pendulum lengths
$l_{1} : l_{2} = 49 : 81$

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