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

Options

A. `6 : 15 : 10`
B. `3 : 8 : 6`
C. `6 : 10 : 15`
D. `3 : 7 : 3`

Answer

To solve the problem of finding the ratio A : B : C given that A : B = 3 : 5 and B : C = 2 : 3, we can follow these steps:

### Step-by-Step Solution:

1. **Write the given ratios**:
- A : B = 3 : 5
- B : C = 2 : 3

2. **Express B in terms of a common value**:
- From A : B = 3 : 5, we can express A and B in terms of a variable, say k:
- A = 3k
- B = 5k

3. **Express C in terms of B**:
- From B : C = 2 : 3, we can express B and C in terms of another variable, say m:
- B = 2m
- C = 3m

4. **Find a common value for B**:
- We have two expressions for B:
- From A : B, B = 5k
- From B : C, B = 2m
- To find a common value for B, we set them equal:
- 5k = 2m

5. **Find the least common multiple (LCM)**:
- We need to find the LCM of the coefficients of B (5 and 2) to express both in terms of a common variable.
- The LCM of 5 and 2 is 10.

6. **Express k and m in terms of the LCM**:
- For B = 10:
- From 5k = 10, we find k = 2.
- From 2m = 10, we find m = 5.

7. **Substitute k and m back to find A and C**:
- Now substitute k into the expression for A:
- A = 3k = 3 * 2 = 6
- Substitute m into the expression for C:
- C = 3m = 3 * 5 = 15

8. **Write the final ratio**:
- Now we have A = 6, B = 10, C = 15.
- Therefore, the ratio A : B : C = 6 : 10 : 15.

9. **Simplify the ratio**:
- To simplify, we can divide each term by the greatest common divisor (GCD), which is 1 in this case.
- Thus, the final simplified ratio remains 6 : 10 : 15.

### Final Answer:
A : B : C = 6 : 10 : 15