If `A : B = 5 : 6` and `B : C = 4 : 7` find `A : B : C`

Answer

To solve the problem of finding the ratio \( A : B : C \) given \( A : B = 5 : 6 \) and \( B : C = 4 : 7 \), we can follow these steps:

### Step-by-Step Solution:

1. **Express A in terms of B**:
From the ratio \( A : B = 5 : 6 \), we can express \( A \) as:
\[ A = \frac{5}{6} B \]

2. **Express C in terms of B**:
From the ratio \( B : C = 4 : 7 \), we can express \( C \) as:
\[ C = \frac{7}{4} B \]

3. **Substituting B**:
To find a common ratio, we can assume a value for \( B \). Let's set \( B = 24 \) (this is a convenient choice as it is a common multiple of 6 and 4).

4. **Calculate A**:
Substitute \( B = 24 \) into the expression for \( A \):
\[ A = \frac{5}{6} \times 24 = 20 \]

5. **Calculate C**:
Substitute \( B = 24 \) into the expression for \( C \):
\[ C = \frac{7}{4} \times 24 = 42 \]

6. **Write the ratio A : B : C**:
Now we have:
\[ A = 20, \quad B = 24, \quad C = 42 \]
Therefore, the ratio \( A : B : C \) can be written as:
\[ A : B : C = 20 : 24 : 42 \]

7. **Simplify the ratio**:
To simplify \( 20 : 24 : 42 \), we can divide each term by their greatest common divisor (GCD). The GCD of 20, 24, and 42 is 2.
\[ \frac{20}{2} : \frac{24}{2} : \frac{42}{2} = 10 : 12 : 21 \]

### Final Answer:
Thus, the ratio \( A : B : C \) is:
\[ \boxed{10 : 12 : 21} \]