If `A : B = 1, 4 and B: C= 2: 3`, then what is `A : B : C`?

Options

A. `1: 4: 6`
B. `2: 8: 6`
C. `2: 8: 5`
D. `1: 4: 5`

Answer

To find the ratio \( A : B : C \) given \( A : B = 1 : 4 \) and \( B : C = 2 : 3 \), we can follow these steps:

### Step 1: Express the ratios in terms of a common variable
From the first ratio \( A : B = 1 : 4 \), we can express:
- \( A = 1x \)
- \( B = 4x \)

From the second ratio \( B : C = 2 : 3 \), we can express:
- \( B = 2y \)
- \( C = 3y \)

### Step 2: Equate the two expressions for \( B \)
Since both expressions represent \( B \), we can set them equal to each other:
\[ 4x = 2y \]

### Step 3: Solve for one variable in terms of the other
To express \( y \) in terms of \( x \), we can rearrange the equation:
\[ y = \frac{4x}{2} = 2x \]

### Step 4: Substitute back to find \( A \) and \( C \)
Now, we can substitute \( y \) back into the expressions for \( A \) and \( C \):
- \( A = 1x \)
- \( B = 4x \)
- \( C = 3y = 3(2x) = 6x \)

### Step 5: Write the final ratio
Now we have:
- \( A = 1x \)
- \( B = 4x \)
- \( C = 6x \)

Thus, the ratio \( A : B : C \) can be expressed as:
\[ A : B : C = 1x : 4x : 6x \]

### Step 6: Simplify the ratio
Since \( x \) is a common factor, we can simplify the ratio:
\[ A : B : C = 1 : 4 : 6 \]

### Final Answer
The ratio \( A : B : C \) is \( 1 : 4 : 6 \).

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