A calorimeter holds 50 g of water at 50^ ^ ^ C , and its...

According to Newton's law of cooling, the time $t$ taken to cool through a given temperature range is directly proportional to the total heat capacity of the system.

$t \propto (m + W)$

where $m$ is the mass of water and $W$ is the water equivalent of the calorimeter.

For the two cases, we can write the ratio of the times taken as:

$\dfrac{t_1}{t_2} = \dfrac{m_1 + W}{m_2 + W}$

Substituting the given values:

$\dfrac{10}{18} = \dfrac{50 + W}{100 + W}$

$\dfrac{5}{9} = \dfrac{50 + W}{100 + W}$

$5(100 + W) = 9(50 + W)$

$500 + 5W = 450 + 9W$

$4W = 50$

$W = 12.5 \text{ g}$