If (a+b) : (b+c) : (c+a) = 5 : 7 : 6 and 2a -3b + 4c = 66 then value of c will be ?

To solve this problem, we need to follow these steps:

Step 1: From the given ratio (a+b) : (b+c) : (c+a) = 5 : 7 : 6, we can infer that the sum of the ratios is equal to the sum of the variables. Therefore, we can write:

5x + 7x + 6x = 2(a + b + c)

where x is a common multiplier. Simplifying this gives:

18x = 2(a + b + c)

or

9x = a + b + c ---- (Equation 1)

Step 2: We are also given the equation 2a - 3b + 4c = 66. We can rearrange this equation to express it in terms of a, b, and c:

a = (66 + 3b - 4c) / 2 ---- (Equation 2)

Step 3: Substitute Equation 2 into Equation 1:

9x = (66 + 3b - 4c) / 2 + b + c

Solving this equation for c will give us the value of c.

Please note that without the specific values of x, b, and c, we cannot solve this equation. The problem seems to be missing some information.

The problem is a combination of ratio and algebraic equations. Here are the steps to solve it:

Step 1: Given that (a+b) : (b+c) : (c+a) = 5 : 7 : 6. This means that a+b, b+c, and c+a are in the ratio 5:7:6 respectively. Let's assume that a+b = 5x, b+c = 7x, and c+a = 6x for some real number x.

Step 2: Adding these three equations, we get 2a + 2b + 2c = 18x, which simplifies to a + b + c = 9x.

Step 3: We also know that 2a - 3b + 4c = 66.

Step 4: We need to find the value of c. To do this, we can express a and b in terms of c and x from the equations in step 1 and substitute them into the equation in step 3.

From a+b = 5x, we get a = 5x - b. From b+c = 7x, we get b = 7x - c.

Step 5: Substituting these values into the equation 2a - 3b + 4c = 66, we get:

2(5x - (7x - c)) - 3(7x - c) + 4c = 66 10x - 14x + 2c - 21x + 3c + 4c = 66 -25x + 9c = 66

Step 6: We also know that a + b + c = 9x. Substituting the values of a and b from step 4, we get:

(5x - (7x - c)) + (7x - c) + c = 9x 5x - 7x + c + 7x - c + c = 9x 5x + c = 9x c = 4x

Step 7: Substituting c = 4x into the equation from step 5, we get:

-25x + 9(4x) = 66 -25x + 36x = 66 11x = 66 x = 66 / 11 x = 6

Step 8: Substituting x = 6 into c = 4x, we get:

c = 4 * 6 c = 24

So, the value of c is 24.