If twice A is three times B and 5 times B is equal to 6 times C, then A: C is

Options

A. `4:5`
B. `5:9`
C. `9:5`
D. `5:4`

Answer

To solve the problem, we need to express A, B, and C in terms of a common variable and then find the ratio A:C.

1. **Set up the equations based on the problem statement:**
- From the first statement, "twice A is three times B":
\[ 2A = 3B \quad \text{(Equation 1)} \]
- From the second statement, "5 times B is equal to 6 times C":
\[ 5B = 6C \quad \text{(Equation 2)} \]

2. **Rearrange Equation 1 to express B in terms of A:**
\[ B = \frac{2A}{3} \]

3. **Substitute B from Equation 1 into Equation 2:**
- Replace B in Equation 2:
\[ 5\left(\frac{2A}{3}\right) = 6C \]
- Simplifying this gives:
\[ \frac{10A}{3} = 6C \]

4. **Rearrange the equation to express C in terms of A:**
\[ C = \frac{10A}{18} = \frac{5A}{9} \]

5. **Now, find the ratio A:C:**
- We have:
\[ A:C = A : \frac{5A}{9} \]
- This can be simplified:
\[ A:C = 1 : \frac{5}{9} = 9 : 5 \]

Thus, the ratio \( A:C \) is \( 9:5 \).