If twice of A is equal to thrice of B and four times of B is equal to five times of C, then find the ratio of A to C.

Options

A. `4:5`
B. `2:3`
C. `8:15`
D. `15:8`

Answer

To solve the problem step by step, we need to express the relationships between A, B, and C mathematically.

### Step 1: Write the equations based on the problem statement.
We are given two relationships:
1. Twice of A is equal to thrice of B:
\( 2A = 3B \)
(Equation 1)

2. Four times of B is equal to five times of C:
\( 4B = 5C \)
(Equation 2)

### Step 2: Express B in terms of A using Equation 1.
From Equation 1, we can express B in terms of A:
\[ B = \frac{2A}{3} \]
(Equation 3)

### Step 3: Express C in terms of B using Equation 2.
From Equation 2, we can express C in terms of B:
\[ C = \frac{4B}{5} \]
(Equation 4)

### Step 4: Substitute Equation 3 into Equation 4 to express C in terms of A.
Now we will substitute Equation 3 into Equation 4:
\[ C = \frac{4 \left( \frac{2A}{3} \right)}{5} \]
\[ C = \frac{8A}{15} \]
(Equation 5)

### Step 5: Find the ratio of A to C.
Now we need to find the ratio of A to C:
\[ \text{Ratio of } A \text{ to } C = \frac{A}{C} \]
Substituting Equation 5 into this ratio gives:
\[ \frac{A}{C} = \frac{A}{\frac{8A}{15}} \]
\[ = \frac{A \times 15}{8A} \]
\[ = \frac{15}{8} \]

### Final Answer
Thus, the ratio of A to C is:
\[ \text{Ratio of } A : C = 15 : 8 \]