`A:B:C=2:4:7`
`A:B:D=4:8:9`
Find `A:B:C:D=?`

Answer

To find the ratio \( A:B:C:D \) given the ratios \( A:B:C = 2:4:7 \) and \( A:B:D = 4:8:9 \), we can follow these steps:

### Step 1: Write down the given ratios
We have:
1. \( A:B:C = 2:4:7 \)
2. \( A:B:D = 4:8:9 \)

### Step 2: Express the ratios in terms of a common variable
From the first ratio \( A:B:C = 2:4:7 \), we can express \( A, B, \) and \( C \) in terms of a variable \( k \):
- Let \( A = 2k \)
- Let \( B = 4k \)
- Let \( C = 7k \)

From the second ratio \( A:B:D = 4:8:9 \), we can express \( A, B, \) and \( D \) in terms of another variable \( m \):
- Let \( A = 4m \)
- Let \( B = 8m \)
- Let \( D = 9m \)

### Step 3: Equate the values of \( A \) and \( B \) from both ratios
Since both expressions represent \( A \) and \( B \), we can set them equal:
1. From \( A \):
\[ 2k = 4m \quad \Rightarrow \quad k = 2m \]
2. From \( B \):
\[ 4k = 8m \quad \Rightarrow \quad k = 2m \]
(This confirms our previous equation.)

### Step 4: Substitute \( k \) back into the expressions for \( A, B, C, \) and \( D \)
Now we can substitute \( k = 2m \) into the expressions for \( A, B, C, \) and \( D \):
- \( A = 2k = 2(2m) = 4m \)
- \( B = 4k = 4(2m) = 8m \)
- \( C = 7k = 7(2m) = 14m \)
- \( D = 9m \)

### Step 5: Write the final ratio
Now we can express the ratio \( A:B:C:D \) in terms of \( m \):
\[ A:B:C:D = 4m : 8m : 14m : 9m \]
We can divide each term by \( m \) (assuming \( m \neq 0 \)):
\[ A:B:C:D = 4 : 8 : 14 : 9 \]

### Final Answer
Thus, the ratio \( A:B:C:D \) is \( 4:8:14:9 \).

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