Select the correct answerIf (a+b) : (b+c) : (c+a) = 5 : 7 : 6 and 2a -3b + 4c = 66 then value of c will be ?Options1224 13(27)18

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. We can express a and b in terms of c and x from the equations in step 1 and substitute them into this equation.

Step 4: From a+b = 5x, we get a = 5x - b. From b+c = 7x, we get b = 7x - c. Substituting these into the equation 2a - 3b + 4c = 66, we get 2(5x - (7x - c)) - 3(7x - c) + 4c = 66.

Step 5: Simplifying this equation, we get -6x + 10c = 66.

Step 6: We also know from step 2 that c = 9x - a - b. Substituting a = 5x - b and b = 7x - c into this, we get c = 9x - (5x - (7x - c)) - (7x - c).

Step 7: Simplifying this equation, we get 4c = 6x, which gives c = 1.5x.

Step 8: Substituting c = 1.5x into the equation from step 5, we get -6x + 15x = 66, which simplifies to 9x = 66, giving x = 66/9 = 7.33.

Step 9: Substituting x = 7.33 into c = 1.5x, we get c = 1.5 * 7.33 = 11 approximately.

So, none of the given options are correct. The value of c should be approximately 11.