The ratio of A to B is `a:8`, and the ratio of B to C is `12:c`. If the ratio of A to C is 2:1, what is the ratio of a to c?

Options

A. `2:3`
B. `3:2`
C. `4:3`
D. `3:4`

Answer

To solve the problem step by step, we will use the given ratios and manipulate them to find the ratio of \( a \) to \( c \).

### Step 1: Write down the given ratios.
We have the following ratios:
1. \( A : B = a : 8 \) which can be expressed as \( \frac{A}{B} = \frac{a}{8} \)
2. \( B : C = 12 : c \) which can be expressed as \( \frac{B}{C} = \frac{12}{c} \)
3. \( A : C = 2 : 1 \) which can be expressed as \( \frac{A}{C} = \frac{2}{1} \)

### Step 2: Express \( A \) in terms of \( B \) and \( C \).
From the first ratio, we can express \( A \) in terms of \( B \):
\[ A = \frac{a}{8} B \]

From the second ratio, we can express \( B \) in terms of \( C \):
\[ B = \frac{12}{c} C \]

### Step 3: Substitute \( B \) into the expression for \( A \).
Now, substitute the expression for \( B \) into the equation for \( A \):
\[ A = \frac{a}{8} \left( \frac{12}{c} C \right) = \frac{12a}{8c} C = \frac{3a}{2c} C \]

### Step 4: Set up the equation using the ratio of \( A \) to \( C \).
We know from the problem that:
\[ \frac{A}{C} = \frac{2}{1} \]
Substituting the expression we found for \( A \):
\[ \frac{\frac{3a}{2c} C}{C} = \frac{2}{1} \]
This simplifies to:
\[ \frac{3a}{2c} = 2 \]

### Step 5: Solve for \( a \) in terms of \( c \).
Cross-multiplying gives:
\[ 3a = 4c \]
Now, we can express \( a \) in terms of \( c \):
\[ a = \frac{4}{3} c \]

### Step 6: Find the ratio of \( a \) to \( c \).
Now, we can find the ratio of \( a \) to \( c \):
\[ \frac{a}{c} = \frac{\frac{4}{3}c}{c} = \frac{4}{3} \]

Thus, the ratio of \( a \) to \( c \) is \( 4 : 3 \).

### Final Answer:
The ratio of \( a \) to \( c \) is \( 4 : 3 \).

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