If A:b=2:3 and B:C=4:5 then C:A is equal to

Options

A. 15 : 8
B. 12 : 10
C. 5 : 8
D. 8 : 5

Answer

To solve the problem where \( A : B = 2 : 3 \) and \( B : C = 4 : 5 \), we want to find the ratio \( C : A \).

### Step-by-Step Solution:

1. **Express A and B in terms of a common variable:**
Given \( A : B = 2 : 3 \), we can express \( A \) and \( B \) as:
\[ A = 2x \quad \text{and} \quad B = 3x \]
where \( x \) is a common multiplier.

**Hint:** Use a variable to represent the ratio parts for easier calculations.

2. **Express B and C in terms of another common variable:**
Given \( B : C = 4 : 5 \), we can express \( B \) and \( C \) as:
\[ B = 4y \quad \text{and} \quad C = 5y \]
where \( y \) is another common multiplier.

**Hint:** Again, use a variable for the second ratio to maintain clarity in the calculations.

3. **Set the expressions for B equal to each other:**
Since both expressions represent \( B \), we can equate them:
\[ 3x = 4y \]

**Hint:** This step helps to find a relationship between the two variables \( x \) and \( y \).

4. **Solve for one variable in terms of the other:**
Rearranging the equation gives:
\[ y = \frac{3}{4}x \]

**Hint:** Isolate one variable to substitute it back into the expressions for A and C.

5. **Substitute y back into the expression for C:**
Now, substitute \( y \) into the expression for \( C \):
\[ C = 5y = 5\left(\frac{3}{4}x\right) = \frac{15}{4}x \]

**Hint:** This substitution allows you to express \( C \) in terms of \( x \), making it easier to find the ratio.

6. **Now we have A and C in terms of x:**
We have:
\[ A = 2x \quad \text{and} \quad C = \frac{15}{4}x \]

**Hint:** Ensure both variables are expressed in terms of the same base variable for direct comparison.

7. **Find the ratio C : A:**
Now, we can find the ratio \( C : A \):
\[ C : A = \frac{15}{4}x : 2x \]
Simplifying this gives:
\[ C : A = \frac{15}{4} : 2 = \frac{15}{4} : \frac{8}{4} = 15 : 8 \]

**Hint:** Simplifying the ratio involves finding a common denominator.

### Final Answer:
Thus, the ratio \( C : A \) is equal to \( 15 : 8 \).