If P : Q : R = 2:3:4, then find `P/Q : Q/R : R/P`

Options

A. a)`8 : 9 : 24`
B. b)`9 : 8 : 24`
C. c)`24 : 8 : 9`
D. d)`8 : 24 : 9`

Answer

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

### Step 1: Express P, Q, and R in terms of a common variable
Given the ratio \( P : Q : R = 2 : 3 : 4 \), we can express \( P \), \( Q \), and \( R \) in terms of a variable \( k \):
- Let \( P = 2k \)
- Let \( Q = 3k \)
- Let \( R = 4k \)

### Step 2: Calculate \( \frac{P}{Q} \)
Now, we calculate \( \frac{P}{Q} \):
\[ \frac{P}{Q} = \frac{2k}{3k} = \frac{2}{3} \]

### Step 3: Calculate \( \frac{Q}{R} \)
Next, we calculate \( \frac{Q}{R} \):
\[ \frac{Q}{R} = \frac{3k}{4k} = \frac{3}{4} \]

### Step 4: Calculate \( \frac{R}{P} \)
Now, we calculate \( \frac{R}{P} \):
\[ \frac{R}{P} = \frac{4k}{2k} = \frac{4}{2} = 2 \]

### Step 5: Write the ratios together
Now we have:
- \( \frac{P}{Q} = \frac{2}{3} \)
- \( \frac{Q}{R} = \frac{3}{4} \)
- \( \frac{R}{P} = 2 \)

We can express the ratios as:
\[ \frac{P}{Q} : \frac{Q}{R} : \frac{R}{P} = \frac{2}{3} : \frac{3}{4} : 2 \]

### Step 6: Convert to a common denominator
To combine these ratios, we can convert them to a common denominator. The least common multiple of the denominators (3, 4, and 1) is 12.

- For \( \frac{2}{3} \):
\[ \frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12} \]

- For \( \frac{3}{4} \):
\[ \frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12} \]

- For \( 2 \):
\[ 2 = \frac{2 \times 12}{1 \times 12} = \frac{24}{12} \]

### Step 7: Write the final ratio
Now we can express the ratios:
\[ \frac{P}{Q} : \frac{Q}{R} : \frac{R}{P} = 8 : 9 : 24 \]

Thus, the final answer is:
\[ P/Q : Q/R : R/P = 8 : 9 : 24 \]