If a : b = 3 : 4 and b : c = 8 : 9, find a : b : c

Answer

To find the ratio \( a : b : c \) given the ratios \( a : b = 3 : 4 \) and \( b : c = 8 : 9 \), we can follow these steps:

### Step 1: Write down the given ratios
We have:
- \( a : b = 3 : 4 \)
- \( b : c = 8 : 9 \)

### Step 2: Express the ratios in terms of a common variable
Let’s express \( b \) in terms of a common variable. From the first ratio \( a : b = 3 : 4 \), we can write:
- \( a = 3k \)
- \( b = 4k \) (where \( k \) is a common multiplier)

From the second ratio \( b : c = 8 : 9 \), we can express \( b \) as:
- \( b = 8m \)
- \( c = 9m \) (where \( m \) is another common multiplier)

### Step 3: Equate the two expressions for \( b \)
Since both expressions represent \( b \), we can set them equal to each other:
\[ 4k = 8m \]

### Step 4: Solve for one variable in terms of the other
From the equation \( 4k = 8m \), we can simplify it:
\[ k = 2m \]

### Step 5: Substitute back to find \( a \) and \( c \)
Now we can substitute \( k \) back into the expressions for \( a \) and \( c \):
- For \( a \):
\[ a = 3k = 3(2m) = 6m \]
- For \( c \):
\[ c = 9m \]

### Step 6: Write the final ratio
Now we have:
- \( a = 6m \)
- \( b = 8m \)
- \( c = 9m \)

Thus, the ratio \( a : b : c \) can be expressed as:
\[ a : b : c = 6m : 8m : 9m \]

### Step 7: Remove the common factor \( m \)
Since \( m \) is a common factor, we can simplify the ratio:
\[ a : b : c = 6 : 8 : 9 \]

### Final Answer
The final ratio \( a : b : c \) is:
\[ \boxed{6 : 8 : 9} \]