If `x = 10a, y = 3b , z = 7c and x : y : z = 10 : 3 : 7`, then `(7x + 2y + 5z)/(8a + b + 3c)` =

Options

A. `111//12`
B. `7//6`
C. `8//15`
D. `108//123`

Answer

To solve the problem step by step, we need to find the value of \((7x + 2y + 5z)/(8a + b + 3c)\) given the relationships between \(x\), \(y\), \(z\), \(a\), \(b\), and \(c\).

### Step 1: Express \(a\), \(b\), and \(c\) in terms of \(x\), \(y\), and \(z\)
We know:
- \(x = 10a \implies a = \frac{x}{10}\)
- \(y = 3b \implies b = \frac{y}{3}\)
- \(z = 7c \implies c = \frac{z}{7}\)

### Step 2: Substitute \(a\), \(b\), and \(c\) into the denominator
Now, substituting these into the denominator \(8a + b + 3c\):
\[ 8a + b + 3c = 8\left(\frac{x}{10}\right) + \frac{y}{3} + 3\left(\frac{z}{7}\right) \]
This simplifies to:
\[ = \frac{8x}{10} + \frac{y}{3} + \frac{3z}{7} \]
\[ = \frac{4x}{5} + \frac{y}{3} + \frac{3z}{7} \]

### Step 3: Find a common denominator for the denominator
The common denominator for \(5\), \(3\), and \(7\) is \(105\). We will convert each term:
\[ \frac{4x}{5} = \frac{4x \cdot 21}{105} = \frac{84x}{105} \]
\[ \frac{y}{3} = \frac{y \cdot 35}{105} = \frac{35y}{105} \]
\[ \frac{3z}{7} = \frac{3z \cdot 15}{105} = \frac{45z}{105} \]
So,
\[ 8a + b + 3c = \frac{84x + 35y + 45z}{105} \]

### Step 4: Substitute \(y\) and \(z\) in terms of \(x\)
From the ratio \(x : y : z = 10 : 3 : 7\), we can express \(y\) and \(z\) in terms of \(x\):
\[ y = \frac{3}{10}x \quad \text{and} \quad z = \frac{7}{10}x \]

### Step 5: Substitute \(y\) and \(z\) into the numerator
Now substituting \(y\) and \(z\) into the numerator \(7x + 2y + 5z\):
\[ 7x + 2y + 5z = 7x + 2\left(\frac{3}{10}x\right) + 5\left(\frac{7}{10}x\right) \]
This simplifies to:
\[ = 7x + \frac{6}{10}x + \frac{35}{10}x = 7x + \frac{41}{10}x = \frac{70x + 41x}{10} = \frac{111x}{10} \]

### Step 6: Combine the results
Now we have:
\[ \frac{7x + 2y + 5z}{8a + b + 3c} = \frac{\frac{111x}{10}}{\frac{84x + 35y + 45z}{105}} \]
Substituting \(y\) and \(z\) again:
\[ = \frac{\frac{111x}{10}}{\frac{84x + 35\left(\frac{3}{10}x\right) + 45\left(\frac{7}{10}x\right)}{105}} \]
Calculating the denominator:
\[ = \frac{84x + \frac{105}{10}x + \frac{315}{10}x}{105} = \frac{84x + 10.5x + 31.5x}{105} = \frac{126x}{105} \]
Thus, we have:
\[ \frac{111x/10}{126x/105} = \frac{111 \cdot 105}{10 \cdot 126} \]
This simplifies to:
\[ = \frac{11655}{1260} = \frac{111}{12} \]

### Final Answer
\[ \frac{(7x + 2y + 5z)}{(8a + b + 3c)} = \frac{111}{12} \]