If `(ab-1)/(b)=(cb-1)/(c )=(ac-1)/(a)`, then find the value of `((a)/(c )+(b)/(a) +(c )/(b))` .

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the problem, we start with the given equation:

\[ \frac{ab - 1}{b} = \frac{cb - 1}{c} = \frac{ac - 1}{a} \]

Let’s denote this common value as \( k \). Therefore, we can write:

1. \(\frac{ab - 1}{b} = k\)
2. \(\frac{cb - 1}{c} = k\)
3. \(\frac{ac - 1}{a} = k\)

From these equations, we can express \( ab - 1 \), \( cb - 1 \), and \( ac - 1 \) in terms of \( k \):

1. \( ab - 1 = kb \) → \( ab = kb + 1 \) → \( ab - kb = 1 \) → \( b(a - k) = 1 \) → \( b = \frac{1}{a - k} \) (1)
2. \( cb - 1 = kc \) → \( cb = kc + 1 \) → \( cb - kc = 1 \) → \( c(b - k) = 1 \) → \( c = \frac{1}{b - k} \) (2)
3. \( ac - 1 = ka \) → \( ac = ka + 1 \) → \( ac - ka = 1 \) → \( a(c - k) = 1 \) → \( a = \frac{1}{c - k} \) (3)

Next, we can substitute the expressions for \( b \) and \( c \) from equations (1) and (2) into equation (3) to find relationships among \( a \), \( b \), and \( c \).

However, a simpler approach is to assume specific values for \( a \), \( b \), and \( c \) that satisfy the original equation. Let's try \( a = 1 \), \( b = 1 \), and \( c = 1 \):

Substituting these values into the original equations:

1. \(\frac{1 \cdot 1 - 1}{1} = 0\)
2. \(\frac{1 \cdot 1 - 1}{1} = 0\)
3. \(\frac{1 \cdot 1 - 1}{1} = 0\)

Since all three expressions equal 0, the assumption holds true.

Now, we need to find the value of:

\[ \frac{a}{c} + \frac{b}{a} + \frac{c}{b} \]

Substituting \( a = 1 \), \( b = 1 \), and \( c = 1 \):

\[ \frac{1}{1} + \frac{1}{1} + \frac{1}{1} = 1 + 1 + 1 = 3 \]

Thus, the value of \( \frac{a}{c} + \frac{b}{a} + \frac{c}{b} \) is:

\[ \boxed{3} \]