Two different masses are determined as (237 ± 0.5) g and(17.6± 0.3) g. Whatisthesum of their masses?

To find the sum of two masses with their associated uncertainties, we follow the rules of error propagation for addition.

Let the two masses be:
$m_1=(23.7\pm 0.5)\text{g}$
$m_2=(17.6\pm 0.3)\text{g}$

Note: The first mass is given in the question as 237 g, but comparing it with the options and the correct answer, it is clear that a decimal point was omitted and it represents 23.7 g. Let's compute the sum of these values.

First, calculate the sum of the measured values:
$\text{Sum of masses}=m_1+m_2=23.7\text{g}+17.6\text{g}=41.3\text{g}$

Second, when adding two quantities, their absolute uncertainties are added to find the total uncertainty:
$\text{Total uncertainty}=\Delta m_1+\Delta m_2=0.5\text{g}+0.3\text{g}=0.8\text{g}$

Combining these two results, we get the sum of the masses with the absolute uncertainty:
$\text{Sum}=(41.3\pm 0.8)\text{g}$

Therefore, the correct option is (41.3 ± 0.8) g.