If `(3/2)x = (5/7)y = (6/5)z`, what is `x:y:z` ?

Options

A. `105: 50 : 84`
B. `24 : 25 : 32`
C. `15 : 21 : 25`
D. `20 : 42 : 25`

Answer

To find the ratio \( x:y:z \) given that \( \frac{3}{2}x = \frac{5}{7}y = \frac{6}{5}z \), we can follow these steps:

### Step 1: Set a common variable
Let \( k \) be the common value such that:
\[ \frac{3}{2}x = k, \quad \frac{5}{7}y = k, \quad \frac{6}{5}z = k \]

### Step 2: Express \( x \), \( y \), and \( z \) in terms of \( k \)
From the first equation:
\[ x = \frac{2k}{3} \]

From the second equation:
\[ y = \frac{7k}{5} \]

From the third equation:
\[ z = \frac{5k}{6} \]

### Step 3: Find a common denominator
To find the ratio \( x:y:z \), we need to express \( x \), \( y \), and \( z \) with a common denominator. The denominators are 3, 5, and 6. The least common multiple (LCM) of these numbers is 30.

### Step 4: Convert each variable to have the common denominator
Now we convert each expression to have a denominator of 30:
- For \( x \):
\[ x = \frac{2k}{3} = \frac{2k \cdot 10}{3 \cdot 10} = \frac{20k}{30} \]

- For \( y \):
\[ y = \frac{7k}{5} = \frac{7k \cdot 6}{5 \cdot 6} = \frac{42k}{30} \]

- For \( z \):
\[ z = \frac{5k}{6} = \frac{5k \cdot 5}{6 \cdot 5} = \frac{25k}{30} \]

### Step 5: Write the ratio
Now we can write the ratio \( x:y:z \):
\[ x:y:z = 20k:42k:25k \]

### Step 6: Simplify the ratio
Since \( k \) is common in all terms, we can simplify:
\[ x:y:z = 20:42:25 \]

### Step 7: Further simplify if possible
To simplify \( 20:42:25 \), we can find the GCD of the numbers:
- The GCD of 20 and 42 is 2.
- The GCD of 20, 42, and 25 is 1 (since 25 is not divisible by 2).

Thus, the simplest form remains:
\[ x:y:z = 20:42:25 \]

### Final Answer
The ratio \( x:y:z \) is \( 20:42:25 \).

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