If (2/3)P = (4/5)Q = (3/2)R, then what is P : Q : R?

Options

A. `18 : 15 : 8 `
B. `15 : 8 : 18`
C. `2 : 4 : 3 `
D. `3 : 5 : 2

Answer

To solve the problem, we need to find the ratio \( P : Q : R \) given that \( \frac{2}{3}P = \frac{4}{5}Q = \frac{3}{2}R \).

### Step-by-Step Solution:

1. **Set a common variable**:
Let \( k \) be the common value such that:
\[ \frac{2}{3}P = k, \quad \frac{4}{5}Q = k, \quad \frac{3}{2}R = k \]

2. **Express \( P \), \( Q \), and \( R \) in terms of \( k \)**:
- From \( \frac{2}{3}P = k \):
\[ P = \frac{3}{2}k \]
- From \( \frac{4}{5}Q = k \):
\[ Q = \frac{5}{4}k \]
- From \( \frac{3}{2}R = k \):
\[ R = \frac{2}{3}k \]

3. **Write the ratio \( P : Q : R \)**:
Now we can express the ratio:
\[ P : Q : R = \frac{3}{2}k : \frac{5}{4}k : \frac{2}{3}k \]
Since \( k \) is common, we can cancel it out:
\[ P : Q : R = \frac{3}{2} : \frac{5}{4} : \frac{2}{3} \]

4. **Convert to a common denominator**:
The denominators are 2, 4, and 3. The least common multiple (LCM) of these numbers is 12. We will convert each fraction to have a denominator of 12:
- For \( \frac{3}{2} \):
\[ \frac{3}{2} = \frac{3 \times 6}{2 \times 6} = \frac{18}{12} \]
- For \( \frac{5}{4} \):
\[ \frac{5}{4} = \frac{5 \times 3}{4 \times 3} = \frac{15}{12} \]
- For \( \frac{2}{3} \):
\[ \frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12} \]

5. **Combine the ratios**:
Now we can write the ratio:
\[ P : Q : R = 18 : 15 : 8 \]

### Final Answer:
Thus, the ratio \( P : Q : R \) is:
\[ \boxed{18 : 15 : 8} \]