If `x : y = 3:4` and `y : z = 3:4`, then `(x + y+z)/(3z)` is equal to:

Options

A. `13/27`
B. `1/2`
C. `73/84`
D. `37/48`

Answer

To solve the problem, we need to find the value of \((x + y + z) / (3z)\) given the ratios \(x : y = 3 : 4\) and \(y : z = 3 : 4\).

### Step-by-Step Solution:

1. **Express the Ratios in Terms of a Common Variable:**
- From the ratio \(x : y = 3 : 4\), we can express \(x\) and \(y\) in terms of a variable \(k\):
\[ x = 3k \quad \text{and} \quad y = 4k \]

2. **Express \(z\) in Terms of \(y\):**
- From the ratio \(y : z = 3 : 4\), we can express \(y\) and \(z\) in terms of another variable \(m\):
\[ y = 3m \quad \text{and} \quad z = 4m \]
- Since \(y\) is common in both ratios, we can equate the two expressions for \(y\):
\[ 4k = 3m \]
- Solving for \(m\) in terms of \(k\):
\[ m = \frac{4k}{3} \]

3. **Substituting \(m\) to Find \(z\):**
- Now substitute \(m\) back into the expression for \(z\):
\[ z = 4m = 4 \left(\frac{4k}{3}\right) = \frac{16k}{3} \]

4. **Calculate \(x + y + z\):**
- Now we can find \(x + y + z\):
\[ x + y + z = 3k + 4k + \frac{16k}{3} \]
- To combine these, we need a common denominator (which is 3):
\[ x + y + z = \frac{9k}{3} + \frac{12k}{3} + \frac{16k}{3} = \frac{9k + 12k + 16k}{3} = \frac{37k}{3} \]

5. **Calculate \(3z\):**
- Now, calculate \(3z\):
\[ 3z = 3 \left(\frac{16k}{3}\right) = 16k \]

6. **Formulate the Final Expression:**
- Now we can find \(\frac{x + y + z}{3z}\):
\[ \frac{x + y + z}{3z} = \frac{\frac{37k}{3}}{16k} = \frac{37}{3 \cdot 16} = \frac{37}{48} \]

### Final Answer:
\[ \frac{x + y + z}{3z} = \frac{37}{48} \]