If `(4/5)A = 3B = (2/3)C`, then what is `A : B : C`?

Options

A. `12 : 3 : 16`
B. `12 : 45 : 10`
C. `15 : 4 : 18`
D. `9 : 12 : 16`

Answer

To solve the problem \( \frac{4}{5}A = 3B = \frac{2}{3}C \), we will find the ratio \( A : B : C \) step by step.

### Step 1: Set up the equations
We can set a common variable \( k \) such that:
\[ \frac{4}{5}A = 3B = \frac{2}{3}C = k \]

### Step 2: Express A, B, and C in terms of k
From the equation \( \frac{4}{5}A = k \):
\[ A = \frac{5}{4}k \]

From the equation \( 3B = k \):
\[ B = \frac{k}{3} \]

From the equation \( \frac{2}{3}C = k \):
\[ C = \frac{3}{2}k \]

### Step 3: Substitute A, B, and C
Now we have:
- \( A = \frac{5}{4}k \)
- \( B = \frac{k}{3} \)
- \( C = \frac{3}{2}k \)

### Step 4: Find a common denominator
To find the ratio \( A : B : C \), we will express all terms with a common denominator. The denominators are 4, 3, and 2. The least common multiple (LCM) of these numbers is 12.

### Step 5: Convert each term to have the common denominator
- For \( A = \frac{5}{4}k \):
\[ A = \frac{5 \times 3}{4 \times 3}k = \frac{15}{12}k \]

- For \( B = \frac{k}{3} \):
\[ B = \frac{1 \times 4}{3 \times 4}k = \frac{4}{12}k \]

- For \( C = \frac{3}{2}k \):
\[ C = \frac{3 \times 6}{2 \times 6}k = \frac{18}{12}k \]

### Step 6: Write the ratio
Now we can write the ratio \( A : B : C \):
\[ A : B : C = \frac{15}{12}k : \frac{4}{12}k : \frac{18}{12}k \]
This simplifies to:
\[ A : B : C = 15 : 4 : 18 \]

### Step 7: Simplify the ratio if necessary
To simplify \( 15 : 4 : 18 \), we can check for any common factors. Since there are no common factors, the ratio remains:
\[ A : B : C = 15 : 4 : 18 \]

### Final Answer
Thus, the final answer is:
\[ A : B : C = 15 : 4 : 18 \]