If `X: Y= 7:5 and Y: Z= 7:11`, then what is the ratio of `X: Y : Z` ?

Options

A. `30:35: 49`
B. `49:35:55`
C. `7:5:1`
D. `49:30:55`

Answer

To find the ratio of \(X : Y : Z\) given the ratios \(X : Y = 7 : 5\) and \(Y : Z = 7 : 11\), we can follow these steps:

### Step 1: Write down the given ratios
We have:
- \(X : Y = 7 : 5\)
- \(Y : Z = 7 : 11\)

### Step 2: Express \(Y\) in terms of a common value
From the first ratio \(X : Y = 7 : 5\), we can express \(X\) and \(Y\) in terms of a variable \(k\):
- Let \(X = 7k\)
- Let \(Y = 5k\)

### Step 3: Express \(Z\) in terms of \(Y\)
From the second ratio \(Y : Z = 7 : 11\), we can express \(Z\) in terms of \(Y\):
- Since \(Y = 5k\), we can set up the ratio:
\[ \frac{Y}{Z} = \frac{7}{11} \]
Substituting \(Y\):
\[ \frac{5k}{Z} = \frac{7}{11} \]

### Step 4: Cross-multiply to find \(Z\)
Cross-multiplying gives:
\[ 5k \cdot 11 = 7 \cdot Z \]
So,
\[ Z = \frac{55k}{7} \]

### Step 5: Write the ratios together
Now we have:
- \(X = 7k\)
- \(Y = 5k\)
- \(Z = \frac{55k}{7}\)

### Step 6: Find a common denominator
To combine these ratios, we can express all terms with a common denominator. The common denominator for \(Z\) is \(7\):
- \(X = 7k = \frac{49k}{7}\)
- \(Y = 5k = \frac{35k}{7}\)
- \(Z = \frac{55k}{7}\)

### Step 7: Combine the ratios
Now we can write:
\[ X : Y : Z = \frac{49k}{7} : \frac{35k}{7} : \frac{55k}{7} \]
This simplifies to:
\[ X : Y : Z = 49 : 35 : 55 \]

### Final Answer
Thus, the ratio of \(X : Y : Z\) is:
\[ \boxed{49 : 35 : 55} \]