If 3P = 2Q and 2Q = 3R, then what is P : Q : R?

Options

A. `6:9:1`
B. `2:3:2`
C. `4:6:9`
D. `4:6:1`

Answer

To solve the problem, we start with the given equations:

1. \( 3P = 2Q \)
2. \( 2Q = 3R \)

We need to find the ratio \( P : Q : R \).

### Step 1: Express Q in terms of P
From the first equation \( 3P = 2Q \), we can express \( Q \) in terms of \( P \):

\[ Q = \frac{3P}{2} \]

**Hint:** Rearranging equations can help express one variable in terms of another.

### Step 2: Express R in terms of Q
From the second equation \( 2Q = 3R \), we can express \( R \) in terms of \( Q \):

\[ R = \frac{2Q}{3} \]

**Hint:** Similarly, rearranging the second equation helps to express another variable in terms of the previous one.

### Step 3: Substitute Q in terms of P into the equation for R
Now, we substitute \( Q \) from Step 1 into the equation for \( R \):

\[ R = \frac{2 \left(\frac{3P}{2}\right)}{3} \]

This simplifies to:

\[ R = \frac{3P}{3} = P \]

**Hint:** Substituting known values into equations can simplify the problem further.

### Step 4: Now we have expressions for Q and R in terms of P
We have:
- \( Q = \frac{3P}{2} \)
- \( R = P \)

### Step 5: Write the ratio P : Q : R
Now we can express the ratio \( P : Q : R \):

\[ P : Q : R = P : \frac{3P}{2} : P \]

To simplify this, we can factor out \( P \):

\[ = 1 : \frac{3}{2} : 1 \]

### Step 6: Convert the ratio to whole numbers
To convert this ratio into whole numbers, we can multiply each part by 2:

\[ = 2 : 3 : 2 \]

### Final Answer
Thus, the ratio \( P : Q : R \) is:

\[ \boxed{2 : 3 : 2} \]