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

Options

A. `24 : 45 : 40`
B. `24 : 18 : 45`
C. `24 : 40 : 45`
D. `12 : 45 : 40`

Answer

To solve the problem, we need to find the ratio \( P : Q : R \) given the equations:

\[ \frac{5}{4} P = \frac{2}{3} Q = \frac{3}{4} R \]

Let's denote the common value as \( K \). Therefore, we can write:

1. \( \frac{5}{4} P = K \)
2. \( \frac{2}{3} Q = K \)
3. \( \frac{3}{4} R = K \)

Now, we can express \( P \), \( Q \), and \( R \) in terms of \( K \):

### Step 1: Solve for \( P \)

From the equation \( \frac{5}{4} P = K \):

\[ P = \frac{4}{5} K \]

### Step 2: Solve for \( Q \)

From the equation \( \frac{2}{3} Q = K \):

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

### Step 3: Solve for \( R \)

From the equation \( \frac{3}{4} R = K \):

\[ R = \frac{4}{3} K \]

### Step 4: Write the ratios

Now we have:

- \( P = \frac{4}{5} K \)
- \( Q = \frac{3}{2} K \)
- \( R = \frac{4}{3} K \)

To find the ratio \( P : Q : R \), we can express these in a common format. We will find a common denominator for the fractions.

### Step 5: Find a common denominator

The denominators are 5, 2, and 3. The least common multiple (LCM) of these numbers is 30.

Now, we can express each term with a denominator of 30:

- For \( P \):

\[ P = \frac{4}{5} K = \frac{4 \times 6}{5 \times 6} K = \frac{24}{30} K \]

- For \( Q \):

\[ Q = \frac{3}{2} K = \frac{3 \times 15}{2 \times 15} K = \frac{45}{30} K \]

- For \( R \):

\[ R = \frac{4}{3} K = \frac{4 \times 10}{3 \times 10} K = \frac{40}{30} K \]

### Step 6: Write the final ratio

Now we can write the ratio:

\[ P : Q : R = \frac{24}{30} K : \frac{45}{30} K : \frac{40}{30} K \]

This simplifies to:

\[ P : Q : R = 24 : 45 : 40 \]

### Step 7: Simplify the ratio

To simplify \( 24 : 45 : 40 \), we can divide each term by their greatest common divisor (GCD). The GCD of 24, 45, and 40 is 3.

Thus, we divide each term by 3:

\[ P : Q : R = \frac{24}{3} : \frac{45}{3} : \frac{40}{3} = 8 : 15 : \frac{40}{3} \]

### Final Answer

The final simplified ratio is:

\[ P : Q : R = 8 : 15 : 40 \]