If `(3)/(5) P= (7)/(9) Q= (4)/(5)R`, then what is the ratio of P Q and R respectively?

Options

A. `35:36:15`
B. `140: 108: 105`
C. `135: 140: 105`
D. `70: 45: 63`

Answer

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

\[ \frac{3}{5} P = \frac{7}{9} Q = \frac{4}{5} R \]

Let us denote this common value as \( k \). Therefore, we can express \( P \), \( Q \), and \( R \) in terms of \( k \):

1. From the first equation:
\[ \frac{3}{5} P = k \implies P = \frac{5k}{3} \]

2. From the second equation:
\[ \frac{7}{9} Q = k \implies Q = \frac{9k}{7} \]

3. From the third equation:
\[ \frac{4}{5} R = k \implies R = \frac{5k}{4} \]

Now we have expressed \( P \), \( Q \), and \( R \) in terms of \( k \):
- \( P = \frac{5k}{3} \)
- \( Q = \frac{9k}{7} \)
- \( R = \frac{5k}{4} \)

Next, we need to find the ratio \( P : Q : R \). This can be written as:

\[ P : Q : R = \frac{5k}{3} : \frac{9k}{7} : \frac{5k}{4} \]

To simplify this ratio, we can eliminate \( k \) from all terms:

\[ = \frac{5}{3} : \frac{9}{7} : \frac{5}{4} \]

Now, to combine these fractions into a single ratio, we need to find a common denominator. The denominators are \( 3 \), \( 7 \), and \( 4 \). The least common multiple (LCM) of these numbers is \( 84 \).

Now we convert each term to have the common denominator of \( 84 \):

1. For \( \frac{5}{3} \):
\[ \frac{5}{3} = \frac{5 \times 28}{3 \times 28} = \frac{140}{84} \]

2. For \( \frac{9}{7} \):
\[ \frac{9}{7} = \frac{9 \times 12}{7 \times 12} = \frac{108}{84} \]

3. For \( \frac{5}{4} \):
\[ \frac{5}{4} = \frac{5 \times 21}{4 \times 21} = \frac{105}{84} \]

Now we can write the ratio:

\[ P : Q : R = 140 : 108 : 105 \]

Thus, the final answer is:

\[ \text{The ratio of } P : Q : R \text{ is } 140 : 108 : 105. \]