If `a,b,c gt 0` then `(a)/( a + b + c) = (b)/( a + b + c) = (c )/(a + b + c) = ? `

Options

A. `5/9`
B. `1/3`
C. `7/3`
D. `2/3`

Answer

To solve the equation \(\frac{a}{a + b + c} = \frac{b}{a + b + c} = \frac{c}{a + b + c}\), we can follow these steps:

### Step 1: Understand the Equation
We have three fractions that are all equal to each other. This means that the ratios of \(a\), \(b\), and \(c\) to the total \(a + b + c\) are the same.

### Step 2: Set the Common Value
Let’s denote the common value of these fractions as \(k\). Therefore, we can write:
\[ \frac{a}{a + b + c} = k \]
\[ \frac{b}{a + b + c} = k \]
\[ \frac{c}{a + b + c} = k \]

### Step 3: Express \(a\), \(b\), and \(c\) in Terms of \(k\)
From the first equation, we can express \(a\) as:
\[ a = k(a + b + c) \]
Similarly, from the second and third equations:
\[ b = k(a + b + c) \]
\[ c = k(a + b + c) \]

### Step 4: Sum the Equations
Adding all three equations gives:
\[ a + b + c = k(a + b + c) + k(a + b + c) + k(a + b + c) \]
This simplifies to:
\[ a + b + c = 3k(a + b + c) \]

### Step 5: Solve for \(k\)
If we assume \(a + b + c \neq 0\) (which is true since \(a, b, c > 0\)), we can divide both sides by \(a + b + c\):
\[ 1 = 3k \]
Thus, we find:
\[ k = \frac{1}{3} \]

### Conclusion
Therefore, the common value is:
\[ \frac{a}{a + b + c} = \frac{b}{a + b + c} = \frac{c}{a + b + c} = \frac{1}{3} \]