If `A : B = 2.3` and `B : C = 4 : 5` then `A : B : C` is

Options

A. `2 : 3 : 5`
B. `5 : 4 : 6`
C. `8 : 12 : 15`
D. `6 : 4 : 5`

Answer

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

### Step 1: Express the ratios in terms of a common variable
Given:
- \( A : B = 2 : 3 \)
- \( B : C = 4 : 5 \)

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

### Step 2: Express \( C \) in terms of \( B \)
From the second ratio \( B : C = 4 : 5 \), we can express \( C \) in terms of \( B \):
- Let \( B = 4m \) and \( C = 5m \)

### Step 3: Equate the two expressions for \( B \)
Now we have two expressions for \( B \):
1. \( B = 3k \)
2. \( B = 4m \)

Setting these equal to each other:
\[ 3k = 4m \]

### Step 4: Solve for one variable in terms of the other
From \( 3k = 4m \), we can express \( k \) in terms of \( m \):
\[ k = \frac{4m}{3} \]

### Step 5: Substitute \( k \) back into the expressions for \( A \) and \( C \)
Now, substitute \( k \) back into the expressions for \( A \) and \( C \):
- \( A = 2k = 2 \left(\frac{4m}{3}\right) = \frac{8m}{3} \)
- \( C = 5m \)

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

To express the ratio \( A : B : C \) in whole numbers, we can multiply all terms by 3 to eliminate the fraction:
\[ A : B : C = 8m : 12m : 15m \]

### Step 7: Simplify the ratio
Since \( m \) is a common factor, we can simplify the ratio:
\[ A : B : C = 8 : 12 : 15 \]

Thus, the final answer is:
\[ \boxed{8 : 12 : 15} \]

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