If a+b : b+c : c+a = 6 : 7 : 8 and a + b + c = 14, then the value of c is : . Options :6872

The problem can be solved in the following steps:

1. Given that a+b : b+c : c+a = 6 : 7 : 8, we can say that a+b, b+c, and c+a are in the ratio 6:7:8.

2. We also know that a + b + c = 14.

3. We can add the three ratios together to get 2(a+b+c) = 6+7+8 = 21.

4. Substituting the value of a+b+c from step 2, we get 2*14 = 21.

5. Solving this equation, we get the value of 1 unit = 28/21 = 4/3.

6. Now, we can find the value of c by subtracting the value of a+b from the total, i.e., 14 - (6*4/3) = 14 - 8 = 6.

So, the value of c is 6.

The given ratios can be written as:

a+b : b+c : c+a = 6x : 7x : 8x

Adding all these, we get:

2(a+b+c) = 21x

Given that a+b+c = 14, we can substitute this into the equation:

2(14) = 21x

28 = 21x

Solving for x, we get:

x = 28/21 = 4/3

Now, we can substitute x = 4/3 into the ratios to find the values of a, b, and c:

a+b = 6x = 6*(4/3) = 8 b+c = 7x = 7*(4/3) = 28/3 c+a = 8x = 8*(4/3) = 32/3

We have two equations with b and c, and we know that a+b+c = 14, so we can solve for c:

From a+b = 8, we get b = 8 - a.

Substituting b into the equation b+c = 28/3, we get:

8 - a + c = 28/3

Multiplying through by 3 to clear the fraction, we get:

24 - 3a + 3c = 28

Rearranging, we get:

3a + 3c = -4

Dividing through by 3, we get:

a + c = -4/3

But we know that a + c = 14 - b, so:

14 - b = -4/3

Solving for b, we get:

b = 14 + 4/3 = 42/3 = 14

Substituting b = 14 into the equation a+b = 8, we get:

a = 8 - 14 = -6

Substituting a = -6 and b = 14 into the equation a+b+c = 14, we get:

-6 + 14 + c = 14

Solving for c, we get:

c = 14 - 8 = 6

So, the value of c is 6.