If A:B = 2:3 and B:C = 7:8 then A:C is

Options

A. `9:11`
B. `7:12`
C. `16:21`
D. none of these

Answer

To find the ratio of A to C given the ratios A:B = 2:3 and B:C = 7:8, we can follow these steps:

### Step 1: Write the given ratios in fractional form.
From the problem, we have:
- A:B = 2:3 can be written as \( \frac{A}{B} = \frac{2}{3} \) (Equation 1)
- B:C = 7:8 can be written as \( \frac{B}{C} = \frac{7}{8} \) (Equation 2)

### Step 2: Express B in terms of A using Equation 1.
From Equation 1, we can express B in terms of A:
\[ B = \frac{3}{2} A \]

### Step 3: Substitute B in Equation 2.
Now, substitute the expression for B into Equation 2:
\[ \frac{B}{C} = \frac{7}{8} \]
Substituting \( B = \frac{3}{2} A \):
\[ \frac{\frac{3}{2} A}{C} = \frac{7}{8} \]

### Step 4: Cross-multiply to solve for C.
Cross-multiplying gives us:
\[ 3A \cdot 8 = 7C \cdot 2 \]
This simplifies to:
\[ 24A = 14C \]

### Step 5: Rearrange to find the ratio of A to C.
Rearranging the equation gives us:
\[ \frac{A}{C} = \frac{14}{24} \]
This can be simplified by dividing both the numerator and the denominator by 2:
\[ \frac{A}{C} = \frac{7}{12} \]

### Final Answer:
Thus, the ratio of A to C is \( 7:12 \).

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