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Solution

Watch gains 10s in 5 min
$\text{Gain rate} = \dfrac{10}{5} = 2 \text{ s per minute}$
$\text{Every } 60 \text{ s real time} \Rightarrow \text{watch shows } 62 \text{ s}$

Watch time elapsed:
$9:00 \text{ am to } 7:20 \text{ pm} = 10 \text{ hr } 20 \text{ min} = 620 \text{ min}$

Real time formula:
$\text{Real time} = \text{Watch time} \times \dfrac{60}{62}$

$= 620 \times \dfrac{60}{62}$

$= \dfrac{37200}{62}$

$= 600 \text{ minutes}$

$= 10 \text{ hours}$

Correct Time:
$9:00 \text{ am} + 10 \text{ hours} = \boxed{7:00 \text{ pm}}$