If `A:B=4:5,B:C=3:4,C:D=7:11`, then A:D is

Options

A. `3:4`
B. `21:55`
C. `21:44`
D. `7:5`

Answer

To find the ratio \( A:D \) given the ratios \( A:B = 4:5 \), \( B:C = 3:4 \), and \( C:D = 7:11 \), we can follow these steps:

### Step 1: Express \( B \) in terms of \( A \)
From the ratio \( A:B = 4:5 \), we can express \( B \) in terms of \( A \):
\[ B = \frac{5}{4} A \]

**Hint:** Use the ratio to express one variable in terms of another.

### Step 2: Express \( C \) in terms of \( B \)
From the ratio \( B:C = 3:4 \), we can express \( C \) in terms of \( B \):
\[ C = \frac{4}{3} B \]

**Hint:** Again, use the ratio to express the next variable in terms of the previous one.

### Step 3: Substitute \( B \) into the equation for \( C \)
Now substitute \( B = \frac{5}{4} A \) into the equation for \( C \):
\[ C = \frac{4}{3} \left(\frac{5}{4} A\right) = \frac{5}{3} A \]

**Hint:** Make sure to replace \( B \) with its expression in terms of \( A \).

### Step 4: Express \( D \) in terms of \( C \)
From the ratio \( C:D = 7:11 \), we can express \( D \) in terms of \( C \):
\[ D = \frac{11}{7} C \]

**Hint:** Continue using the ratios to express each variable in terms of the previous one.

### Step 5: Substitute \( C \) into the equation for \( D \)
Now substitute \( C = \frac{5}{3} A \) into the equation for \( D \):
\[ D = \frac{11}{7} \left(\frac{5}{3} A\right) = \frac{55}{21} A \]

**Hint:** Replace \( C \) with its expression in terms of \( A \).

### Step 6: Find the ratio \( A:D \)
Now we have \( D \) in terms of \( A \):
\[ A:D = A : \frac{55}{21} A \]
This simplifies to:
\[ A:D = 1 : \frac{55}{21} = 21 : 55 \]

### Final Answer
Thus, the ratio \( A:D \) is:
\[ \boxed{21:55} \]

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