a) To find the numbers A, B, and C, we can set up the following equations based on the given ratios:
A/B = 1/3 B/C = 2/3 A/C = 3/4
We can solve these equations simultaneously to find the values of A, B, and C.
From the first equation, we can rewrite it as A = (1/3)B.
Substituting this into the second equation, we get (1/3)B/C = 2/3. Simplifying, we have B/C = 2.
From the third equation, we can rewrite it as A = (3/4)C.
Substituting this into the first equation, we get (3/4)C/B = 1/3. Simplifying, we have C/B = 4/9.
Now we have two equations: B/C = 2 and C/B = 4/9.
We can solve these equations simultaneously by cross-multiplying: 2B = C 4C = 9B
Substituting the value of C from the first equation into the second equation, we get: 4(2B) = 9B 8B = 9B B = 8B/9
Now we can substitute the value of B back into the first equation to find C: 2(8B/9) = C 16B/9 = C
Finally, we can substitute the values of B and C into the equation A + B + C = 276 to find A: A + 8B/9 + 16B/9 = 276 9A + 8B + 16B = 9 * 276 9A + 24B = 9 * 276 9A + 24B = 2484
Now we have a system of two equations: 2B = C 9A + 24B = 2484
We can solve this system of equations to find the values of A, B, and C.
Using the first equation, we can substitute the value of C into the second equation: 9A + 24B = 2484 9A + 24(2B) = 2484 9A + 48B = 2484
Now we have a single equation with two variables. We can solve for one variable in terms of the other and substitute it back into the equation to find the value of the remaining variable.
From the equation 9A + 48B = 2484, we can solve for A: 9A = 2484 - 48B A = (2484 - 48B)/9
Substituting this into the equation A + 8B + 16B = 276, we get: (2484 - 48B)/9 + 8B + 16B = 276 (2484 - 48B + 9(8B) + 9(16B))/9 = 276 (2484 - 48B + 72B + 144B)/9 = 276 (168B + 2484)/9 = 276 168B + 2484 = 9 * 276 168B + 2484 = 2484 168B = 0 B = 0
Substituting the value of B back into the equation A = (2484 - 48B)/9, we get: A = (2484 - 48(0))/9 A = 276
Finally, we can substitute the values of A and B into the equation C = 2B to find C: C = 2(0) C = 0
Therefore, the numbers A, B, and C are 276, 0, and 0, respectively.