If `A : B = 2 : 3` and `B : C = 4 : 5` then `C : A` = ?

Options

A. `15 : 8`
B. `6 : 5`
C. `8 : 5`
D. `8 : 15`

Answer

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

### Step 1: Express the ratios as fractions
From the given ratios, we can express them in fractional form:
- \( A : B = 2 : 3 \) can be written as:
\[ \frac{A}{B} = \frac{2}{3} \]
- \( B : C = 4 : 5 \) can be written as:
\[ \frac{B}{C} = \frac{4}{5} \]

### Step 2: Find \( \frac{A}{C} \)
To find \( \frac{A}{C} \), we can use the relationship between \( A \), \( B \), and \( C \):
\[ \frac{A}{C} = \frac{A}{B} \times \frac{B}{C} \]
Substituting the values we have:
\[ \frac{A}{C} = \frac{2}{3} \times \frac{4}{5} \]

### Step 3: Multiply the fractions
Now, we multiply the two fractions:
\[ \frac{A}{C} = \frac{2 \times 4}{3 \times 5} = \frac{8}{15} \]

### Step 4: Find \( C : A \)
Since we have \( \frac{A}{C} = \frac{8}{15} \), we can find \( C : A \) by taking the reciprocal:
\[ C : A = \frac{1}{\frac{8}{15}} = \frac{15}{8} \]
Thus, we can express this as:
\[ C : A = 15 : 8 \]

### Final Answer
Therefore, the ratio \( C : A \) is:
\[ \boxed{15 : 8} \]