If ` ( a - b) /( b ) = ( 3 ) /(7)` , which of the following must also be true ?

Options

A. ` ( a )/(b) = - ( 4 ) /(7) `
B. ` ( a ) /(b) = (10)/(7) `
C. ` (a + b ) /(b ) = (10)/(7) `
D. ` ( a - 2b ) /(b ) = - (11 )/(7) `

Explanation

To solve the equation \(\frac{a - b}{b} = \frac{3}{7}\) and determine which of the given options must also be true, we can follow these steps:

### Step 1: Start with the given equation
We have:
\[ \frac{a - b}{b} = \frac{3}{7} \]

### Step 2: Rewrite the left-hand side
We can separate the terms in the numerator:
\[ \frac{a}{b} - \frac{b}{b} = \frac{3}{7} \]
This simplifies to:
\[ \frac{a}{b} - 1 = \frac{3}{7} \]

### Step 3: Isolate \(\frac{a}{b}\)
Now, we can add 1 to both sides of the equation:
\[ \frac{a}{b} = \frac{3}{7} + 1 \]

### Step 4: Convert 1 to a fraction
To add the fractions, we convert 1 to have a common denominator of 7:
\[ 1 = \frac{7}{7} \]
Thus, we have:
\[ \frac{a}{b} = \frac{3}{7} + \frac{7}{7} \]

### Step 5: Add the fractions
Now we can add the two fractions:
\[ \frac{a}{b} = \frac{3 + 7}{7} = \frac{10}{7} \]

### Conclusion
From our calculations, we find that:
\[ \frac{a}{b} = \frac{10}{7} \]
Thus, the correct option from the given choices is:
**Option 2: \(\frac{a}{b} = \frac{10}{7}\)**.