If a:b= 2:3 and b:c= 5:6, then a:b:cis

Options

A. `2 : 3 : 6`
B. `10 : 15 : 18`
C. `2 : 5 : 6`
D. `6 : 5 : 12`

Answer

To solve the problem, we need to find the combined ratio \( a:b:c \) given the ratios \( a:b = 2:3 \) and \( b:c = 5:6 \).

### Step-by-Step Solution:

1. **Write the given ratios**:
- We have \( a:b = 2:3 \) and \( b:c = 5:6 \).

2. **Express the ratios in terms of a common variable**:
- Let \( a = 2x \) and \( b = 3x \) from the first ratio.
- From the second ratio \( b:c = 5:6 \), we can express \( b \) in terms of another variable. Let \( b = 5y \) and \( c = 6y \).

3. **Set the two expressions for \( b \) equal to each other**:
- From \( a:b \), we have \( b = 3x \).
- From \( b:c \), we have \( b = 5y \).
- Therefore, we can set \( 3x = 5y \).

4. **Solve for one variable in terms of the other**:
- Rearranging gives us \( y = \frac{3x}{5} \).

5. **Substitute \( y \) back to find \( c \)**:
- Now substitute \( y \) back into the expression for \( c \):
\[ c = 6y = 6 \left(\frac{3x}{5}\right) = \frac{18x}{5}. \]

6. **Now we have expressions for \( a \), \( b \), and \( c \)**:
- \( a = 2x \)
- \( b = 3x \)
- \( c = \frac{18x}{5} \)

7. **Convert all ratios to a common denominator**:
- To express \( a:b:c \) in a single ratio, we can multiply each term by 5 to eliminate the fraction:
\[ a = 2x \cdot 5 = 10x, \quad b = 3x \cdot 5 = 15x, \quad c = \frac{18x}{5} \cdot 5 = 18x. \]

8. **Write the final ratio**:
- Thus, we have \( a:b:c = 10x:15x:18x \).
- This simplifies to \( 10:15:18 \).

9. **Final simplification**:
- We can simplify \( 10:15:18 \) by dividing each term by 1 (since they have no common factors other than 1):
- Therefore, the final ratio is \( 10:15:18 \).

### Final Answer:
The combined ratio \( a:b:c \) is \( 10:15:18 \).