If `P = 3[x] + 5` and `Q = 4[x + 3] + 6 and P : Q = 1 : 2`, where [x] is the greatest integer less than or equal to x then the value of `[P + Q/2]` is :

Options

A. 34
B. 51
C. 40
D. data insufficient

Answer

To solve the problem step by step, we need to find the values of \( P \) and \( Q \) based on the given equations and the ratio \( P : Q = 1 : 2 \).

### Step 1: Write down the equations for \( P \) and \( Q \)
Given:
\[ P = 3[x] + 5 \]
\[ Q = 4[x + 3] + 6 \]

### Step 2: Substitute \( [x] \) with \( n \) (where \( n \) is the greatest integer less than or equal to \( x \))
Let \( n = [x] \). Then we can rewrite \( P \) and \( Q \) as:
\[ P = 3n + 5 \]
\[ Q = 4(n + 3) + 6 = 4n + 12 + 6 = 4n + 18 \]

### Step 3: Set up the ratio \( P : Q = 1 : 2 \)
From the ratio, we have:
\[ \frac{P}{Q} = \frac{1}{2} \]
Substituting the expressions for \( P \) and \( Q \):
\[ \frac{3n + 5}{4n + 18} = \frac{1}{2} \]

### Step 4: Cross-multiply to solve for \( n \)
Cross-multiplying gives:
\[ 2(3n + 5) = 1(4n + 18) \]
Expanding both sides:
\[ 6n + 10 = 4n + 18 \]

### Step 5: Rearrange the equation to isolate \( n \)
Subtract \( 4n \) from both sides:
\[ 6n - 4n + 10 = 18 \]
This simplifies to:
\[ 2n + 10 = 18 \]
Now, subtract 10 from both sides:
\[ 2n = 8 \]
Dividing by 2:
\[ n = 4 \]

### Step 6: Calculate \( P \) and \( Q \) using \( n = 4 \)
Substituting \( n = 4 \) back into the equations for \( P \) and \( Q \):
\[ P = 3(4) + 5 = 12 + 5 = 17 \]
\[ Q = 4(4) + 18 = 16 + 18 = 34 \]

### Step 7: Calculate \( P + \frac{Q}{2} \)
Now we need to find \( P + \frac{Q}{2} \):
\[ P + \frac{Q}{2} = 17 + \frac{34}{2} = 17 + 17 = 34 \]

### Step 8: Find the value of \( [P + \frac{Q}{2}] \)
Since \( P + \frac{Q}{2} = 34 \), we have:
\[ [P + \frac{Q}{2}] = [34] = 34 \]

### Final Answer
The value of \( [P + \frac{Q}{2}] \) is \( 34 \).

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