`A:B=2:3`, `B:C=4:5`
Find `A:B:C=?`

Answer

To find the combined ratio \( A : B : C \) given the ratios \( A : B = 2 : 3 \) and \( B : C = 4 : 5 \), we can follow these steps:

### Step 1: Write down the given ratios
We have:
- \( A : B = 2 : 3 \)
- \( B : C = 4 : 5 \)

### Step 2: Express the ratios in terms of a common variable
To combine these ratios, we need to express them with the same value for \( B \).

From \( A : B = 2 : 3 \), we can express \( A \) and \( B \) in terms of a variable \( k \):
- Let \( A = 2k \)
- Let \( B = 3k \)

From \( B : C = 4 : 5 \), we can express \( B \) and \( C \) in terms of another variable \( m \):
- Let \( B = 4m \)
- Let \( C = 5m \)

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

### Step 4: Solve for one variable in terms of the other
To find a relationship between \( k \) and \( m \), we can rearrange the equation:
\[ k = \frac{4m}{3} \]

### Step 5: Substitute back to find \( A \) and \( C \)
Now we can substitute \( k \) back into the expressions for \( A \) and \( C \):
- For \( A \):
\[ A = 2k = 2 \left(\frac{4m}{3}\right) = \frac{8m}{3} \]
- For \( C \):
\[ C = 5m \]

### Step 6: Write the combined ratio \( A : B : C \)
Now we have all parts in terms of \( m \):
- \( A = \frac{8m}{3} \)
- \( B = 4m \)
- \( C = 5m \)

To express the ratio \( A : B : C \):
\[ A : B : C = \frac{8m}{3} : 4m : 5m \]

### Step 7: Eliminate the variable \( m \) and simplify
To simplify, we can multiply through by 3 to eliminate the fraction:
\[ 8m : 12m : 15m \]

This simplifies to:
\[ 8 : 12 : 15 \]

### Final Answer
Thus, the combined ratio \( A : B : C \) is:
\[ A : B : C = 8 : 12 : 15 \]

---