If 6a = 4b = 9C, find `A : B : C`

Options

A. `6 : 4:9`
B. ` 6 : 9 : 4`
C. `4 : 9 : 6 `
D. `9 : 6 : 4`

Answer

To solve the problem where \( 6a = 4b = 9c \), we will follow these steps:

### Step 1: Set a common variable
Let \( k \) be the common value such that:
\[ 6a = 4b = 9c = k \]

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

From the equation \( 4b = k \):
\[ b = \frac{k}{4} \]

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

### Step 3: Write the ratio \( A : B : C \)
Now we can express the ratio \( A : B : C \) as:
\[ A : B : C = \frac{k}{6} : \frac{k}{4} : \frac{k}{9} \]

### Step 4: Eliminate \( k \) from the ratio
Since \( k \) is common in all terms, we can simplify the ratio:
\[ A : B : C = \frac{1}{6} : \frac{1}{4} : \frac{1}{9} \]

### Step 5: Find a common denominator
To combine these fractions, we need to find the least common multiple (LCM) of the denominators 6, 4, and 9. The LCM of 6, 4, and 9 is 36.

### Step 6: Convert each term to have the common denominator
Now we convert each term:
- For \( \frac{1}{6} \):
\[ \frac{1}{6} = \frac{6}{36} \]
- For \( \frac{1}{4} \):
\[ \frac{1}{4} = \frac{9}{36} \]
- For \( \frac{1}{9} \):
\[ \frac{1}{9} = \frac{4}{36} \]

### Step 7: Write the final ratio
Now we can write the ratio as:
\[ A : B : C = 6 : 9 : 4 \]

### Conclusion
Thus, the final ratio \( A : B : C \) is:
\[ \boxed{6 : 9 : 4} \]