If P: Q =8.15 and Q:R =3:2 , then find P:Q:R

Options

A. `8:15`
B. `7:15:8`
C. `8:15:10`
D. `10:15:8`

Answer

To solve the problem of finding the ratio \( P:Q:R \) given \( P:Q = 8:15 \) and \( Q:R = 3:2 \), we can follow these steps:

### Step 1: Write the given ratios in fractional form.
We have:
- \( P:Q = 8:15 \) can be written as \( \frac{P}{Q} = \frac{8}{15} \)
- \( Q:R = 3:2 \) can be written as \( \frac{Q}{R} = \frac{3}{2} \)

### Step 2: Express \( Q \) in terms of a common variable.
From the ratio \( Q:R = 3:2 \), we can express \( Q \) in terms of \( R \):
Let \( R = 2x \), then \( Q = 3x \).

### Step 3: Substitute \( Q \) in the first ratio.
Now substitute \( Q = 3x \) into the first ratio \( P:Q = 8:15 \):
\[ \frac{P}{3x} = \frac{8}{15} \]
Cross-multiplying gives:
\[ 15P = 8 \cdot 3x \]
\[ 15P = 24x \implies P = \frac{24x}{15} = \frac{8x}{5} \]

### Step 4: Now we have expressions for \( P \), \( Q \), and \( R \).
- \( P = \frac{8x}{5} \)
- \( Q = 3x \)
- \( R = 2x \)

### Step 5: Write the ratios \( P:Q:R \).
Now we can express \( P:Q:R \) as:
\[ P:Q:R = \frac{8x}{5} : 3x : 2x \]

### Step 6: Eliminate \( x \) and find a common denominator.
To eliminate \( x \), we can multiply each term by 5 to get rid of the fraction:
\[ P:Q:R = 8x : 15x : 10x \]
Now we can simplify this by dividing each term by \( x \):
\[ P:Q:R = 8 : 15 : 10 \]

### Final Answer:
Thus, the ratio \( P:Q:R \) is \( 8:15:10 \).

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