There are three numbers A, B, C such that twice A Is equal to thrice B and four times B is equal to five times C. Then the ratio between A and C is

Options

A. `3:4`
B. `8:15`
C. `15:8`
D. `4:3`

Answer

To solve the problem, we need to express the relationships between A, B, and C using equations and then find the ratio between A and C.

### Step-by-Step Solution:

1. **Set Up the Equations:**
- From the problem, we have two relationships:
1. Twice A is equal to thrice B:
\[ 2A = 3B \quad \text{(Equation 1)} \]
2. Four times B is equal to five times C:
\[ 4B = 5C \quad \text{(Equation 2)} \]

2. **Express B in Terms of A:**
- From Equation 1, we can express B in terms of A:
\[ B = \frac{2A}{3} \]

3. **Substitute B in Equation 2:**
- Now, substitute the expression for B into Equation 2:
\[ 4\left(\frac{2A}{3}\right) = 5C \]
- Simplifying this gives:
\[ \frac{8A}{3} = 5C \]

4. **Express C in Terms of A:**
- Rearranging the equation to express C in terms of A:
\[ C = \frac{8A}{15} \]

5. **Find the Ratio A to C:**
- Now, we need to find the ratio of A to C:
\[ \text{Ratio of } A \text{ to } C = \frac{A}{C} = \frac{A}{\frac{8A}{15}} = \frac{A \cdot 15}{8A} = \frac{15}{8} \]

6. **Final Result:**
- Therefore, the ratio of A to C is:
\[ A : C = 15 : 8 \]