Find the least common multiple (LCM) of the ringing intervals.

Bell 1 rings every 30 minutes.
Bell 2 rings every 40 minutes.
Prime factorization:
$30 = 2 \times 3 \times 5$
$40 = 2^3 \times 5$
LCM$(30, 40) = 2^3 \times 3 \times 5 = 8 \times 3 \times 5 = 120$ minutes.
The bells will ring together every 120 minutes, which is 2 hours.

Start time: 7:50 a.m.
End time: 3:00 p.m.
Duration from 7:50 a.m. to 3:00 p.m.:
From 7:50 a.m. to 8:00 a.m. = 10 minutes.
From 8:00 a.m. to 3:00 p.m. = 7 hours.
Total duration = 7 hours and 10 minutes.
Convert to minutes: $7 \times 60 + 10 = 420 + 10 = 430$ minutes.

The bells ring together every 120 minutes.
Number of times = $\frac{\text{Total duration}}{\text{LCM}}$

Since the bells must ring simultaneously, we only count the full intervals. This means they will ring together 3 more times after the initial 7:50 a.m. ring.

The times they will ring together are:
1st time: 7:50 a.m. (initial ring)
2nd time: 7:50 a.m. + 2 hours = 9:50 a.m.
3rd time: 9:50 a.m. + 2 hours = 11:50 a.m.
4th time: 11:50 a.m. + 2 hours = 1:50 p.m.
The next ring would be at 3:50 p.m., which is after 3:00 p.m.

So, between 7:50 a.m. and 3:00 p.m., they will ring together 3 more times.
The bells will ring together again between 7:50 a.m. and 3 p.m. $\boxed{\text{3 times}}$.