If `P: Q= 5:6 and R : Q = 25:9`, then what is the ratio of `P : R` ?

Options

A. `10:13`
B. `10:3`
C. `3:10`
D. `13:10`

Answer

To find the ratio of \( P : R \) given the ratios \( P : Q = 5 : 6 \) and \( R : Q = 25 : 9 \), we can follow these steps:

### Step 1: Write the ratios in fraction form
From the given ratios, we can express them as:
\[ \frac{P}{Q} = \frac{5}{6} \]
\[ \frac{R}{Q} = \frac{25}{9} \]

### Step 2: Express \( P \) and \( R \) in terms of \( Q \)
From the first ratio, we can express \( P \) in terms of \( Q \):
\[ P = \frac{5}{6}Q \]

From the second ratio, we can express \( R \) in terms of \( Q \):
\[ R = \frac{25}{9}Q \]

### Step 3: Find the ratio \( P : R \)
Now, we can substitute the expressions for \( P \) and \( R \) into the ratio \( P : R \):
\[ P : R = \frac{5}{6}Q : \frac{25}{9}Q \]

### Step 4: Simplify the ratio
Since \( Q \) is common in both terms, we can cancel it out:
\[ P : R = \frac{5}{6} : \frac{25}{9} \]

To simplify this ratio, we can multiply both sides by the least common multiple (LCM) of the denominators (6 and 9), which is 18:
\[ P : R = \left(\frac{5}{6} \times 18\right) : \left(\frac{25}{9} \times 18\right) \]
Calculating each term:
\[ P : R = 15 : 50 \]

### Step 5: Further simplify the ratio
Now, we can simplify \( 15 : 50 \) by dividing both terms by their greatest common divisor (GCD), which is 5:
\[ P : R = 3 : 10 \]

### Final Answer
Thus, the ratio \( P : R \) is \( 3 : 10 \).

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