Given information:
- Two waves have lengths of 50 cm and 51 cm.
- The two waves produce 12 beats per second.

To find the velocity of sound, we can use the formula:

Velocity of sound (v) = Frequency (f) x Wavelength (λ)

To use this formula, we need to find the frequency of the beats.

Finding the Frequency of Beats:
Beats are produced when two waves of slightly different frequencies interfere with each other. The number of beats per second is equal to the difference between the frequencies of the two waves.

In this case, the beats are produced by waves with lengths of 50 cm and 51 cm. Since the speed of sound is the same for both waves, the ratio of their wavelengths will be the same as the ratio of their frequencies.

Let the frequency of the wave with length 50 cm be f1 and the frequency of the wave with length 51 cm be f2.

We can write the following equation:

f2 - f1 = 12 beats/second

Since the ratio of wavelengths is the same as the ratio of frequencies, we can write:

51 cm / 50 cm = f2 / f1

Simplifying this equation, we get:

f2 = (51/50) x f1

Substituting this value of f2 in the first equation, we get:

(51/50) x f1 - f1 = 12

Simplifying further, we get:

f1/50 = 12

f1 = 600

Substituting this value of f1 in the equation for f2, we get:

f2 = (51/50) x 600 = 612

Finding the Velocity of Sound:
Now that we have the frequencies of the two waves, we can use the formula to find the velocity of sound.

Velocity of sound (v) = Frequency (f) x Wavelength (λ)

Let's use the frequency of the wave with length 50 cm to calculate the velocity of sound:

v = f1 x λ1 = 600 x 50 cm = 30000 cm/s

Converting cm/s to m/s, we get:

v = 30000 cm/s = 300 m/s

Therefore, the velocity of sound is 300 m/s.