The key issue is that a box plot only gives us the values for the minimum, Q1, median (Q2), Q3, and maximum. It tells us that $25 %$ of the data lies between Q1 and Q2, and another $25 \%$ lies between Q2 and Q3. However, it does not tell us how the data is distributed within those quartiles.

Let's illustrate with an example:
We know:
- Q2 (50th percentile) $=142$ grams
- Q3 (75th percentile) $=160$ grams

We are looking for the 62.5th percentile. This value must be between 142 and 160 grams.
Consider two possible distributions of data between 142 and 160, both of which would result in the same box plot:

Scenario 1 (Data skewed towards Q2):
Imagine the $25 \%$ of data points between 142 and 160 are heavily clustered towards 142.
For example, if the values were:
$142,142,142, \ldots$, (many values around 142) ..., 150, 155, 160.
In this case, the 62.5th percentile (which is halfway through this 25\% segment) would be closer to 142. It could be, for example, 145 grams.

If it's 145 grams, then Quantity A (145) < Quantity B (152).

Scenario 2 (Data skewed towards Q3):
Imagine the $25 %$ of data points between 142 and 160 are heavily clustered towards 160 . For example, if the values were: $142,145,150, \ldots$, (many values around 155) ..., 160, 160, 160.
In this case, the 62.5th percentile would be closer to 160. It could be, for example, 155 grams. If it's 155 grams, then Quantity A (155) > Quantity B (152).

Since we can construct scenarios where Quantity A is less than Quantity B and scenarios where Quantity A is greater than Quantity B, based only on the information provided by the box plot, we cannot definitively determine the relationship.

The box plot only tells us the values at the 25th, 50th, and 75th percentiles (and $\mathrm{min} / \mathrm{max}$ ). It does not provide enough detail about the distribution within those quartiles to pinpoint an exact percentile value like the 62.5th percentile.

Therefore, the relationship cannot be determined from the information given.

The final answer is The relationship cannot be determined from the information given .