Find a : b : c, if 6a = 9b = 10c.

Options

A. `12 : 10 : 8`
B. ` 15 : 4 : 3`
C. ` 15 : 18 : 9`
D. ` 15 : 10 : 9`

Answer

To find the ratio \( a : b : c \) given that \( 6a = 9b = 10c \), we can follow these steps:

### Step 1: Set a common variable
Let \( k \) be the common value such that:
\[ 6a = 9b = 10c = k \]

### Step 2: Express \( a \), \( b \), and \( c \) in terms of \( k \)
From the equation \( 6a = k \):
\[ a = \frac{k}{6} \]

From the equation \( 9b = k \):
\[ b = \frac{k}{9} \]

From the equation \( 10c = k \):
\[ c = \frac{k}{10} \]

### Step 3: Write the ratio \( a : b : c \)
Now we can express the ratio \( a : b : c \):
\[ a : b : c = \frac{k}{6} : \frac{k}{9} : \frac{k}{10} \]

### Step 4: Eliminate \( k \) from the ratio
Since \( k \) is common in all terms, we can simplify the ratio by removing \( k \):
\[ a : b : c = \frac{1}{6} : \frac{1}{9} : \frac{1}{10} \]

### Step 5: Find a common denominator
To simplify the ratio further, we need to find the least common multiple (LCM) of the denominators 6, 9, and 10. The LCM of 6, 9, and 10 is 90.

### Step 6: Convert each term to have the common denominator
Now we convert each term in the ratio to have the common denominator of 90:
\[ \frac{1}{6} = \frac{15}{90}, \quad \frac{1}{9} = \frac{10}{90}, \quad \frac{1}{10} = \frac{9}{90} \]

### Step 7: Write the final ratio
Thus, the ratio \( a : b : c \) becomes:
\[ a : b : c = 15 : 10 : 9 \]

### Conclusion
The final answer is:
\[ \boxed{15 : 10 : 9} \]

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