​If (a + b) ∶ (b + c) ∶ (c + a) = 5 ∶ 7 ∶ 6, then what is the value of (a - b + c) ∶ (a + b - c)?​

Correct option is C

Given:

If (a + b) ∶ (b + c) ∶ (c + a) = 5 ∶ 7 ∶ 6

Concept Used:

We use the concept of ratio and proportion.

Solution:

Let (a + b) = 5k, (b + c) = 7k, and (c + a) = 6k

We know that:

=> (a + b) + (b + c) + (c + a) = 5k + 7k + 6k

=> 2(a + b + c) = 18k

=> a + b + c = 9k … (1)

From equation (1):

=> a = 9k - (b + c)

=> a = 9k - 7k = 2k … (2)

=> b = 9k - (c + a)

=> b = 9k - 6k = 3k … (3)

=> c = 9k - (a + b)

=> c = 9k - 5k = 4k … (4)

Now, we need to find (a - b + c) ∶ (a + b - c)

Using equations (2), (3), and (4):

=> (a - b + c) = 2k - 3k + 4k = 3k

=> (a + b - c) = 2k + 3k - 4k = 1k

=> (a - b + c) ∶ (a + b - c) = 3k ∶ 1k

=> 3 ∶ 1

Hence, the value of (a - b + c) ∶ (a + b - c) is 3 ∶ 1.