The sum of four numbers is 771. The ratio of the first to the second is 2:3. The ratio of the second to the third is 5:4. The ratio of the third to the fourth is 5:6. Find the second number.

To find the second number, let's assign variables to each of the numbers. Let's call the first number A, the second number B, the third number C, and the fourth number D.

We are given that the sum of the four numbers is 771:
A + B + C + D = 771 ----(equation 1)

We are also given three ratios:
1) The ratio of the first to the second is 2:3. This can be expressed as:
A/B = 2/3 ----(equation 2)

2) The ratio of the second to the third is 5:4. This can be expressed as:
B/C = 5/4 ----(equation 3)

3) The ratio of the third to the fourth is 5:6. This can be expressed as:
C/D = 5/6 ----(equation 4)

Now we have four equations with four unknowns. We can solve these equations simultaneously to find the values of A, B, C, and D.

From equation 2, we can express A in terms of B:
A = (2/3)B

Substituting this value of A in equation 1, we get:
(2/3)B + B + C + D = 771

Simplifying this equation, we have:
(5/3)B + C + D = 771 ----(equation 5)

From equation 3, we can express C in terms of B:
C = (4/5)B

From equation 4, we can express D in terms of C:
D = (6/5)C

Now, substituting the values of C and D in equation 5, we get:
(5/3)B + (4/5)B + (6/5)((4/5)B) = 771

Simplifying this equation, we have:
(5/3 + 4/5 + 6/5 * 4/5)B = 771

Combining the fractions, we get:
(25/15 + 12/15 + 24/25)B = 771

Simplifying further, we have:
(61/15)B = 771

Multiply both sides of the equation by 15/61, we get:
B = (771 * 15) / 61

Evaluating this expression, we find that the second number B is approximately 189.87. Rounded to the nearest whole number, the second number is 190.

Therefore, the second number is 190.