Average of present age of P & R is 33 years. Q is 18 years older than Rand Q is 6 years younger than S. If present age of P is `33(1)/3` % less than present age of S, then find average of present-age of Q, R & S.

Options

A. 44 years
B. 42 years
C. 45 years
D. 41 years

Answer

To solve the problem step by step, we will break it down into manageable parts:

### Step 1: Establish the relationship between ages
We know that the average age of P and R is 33 years. Therefore, we can express this mathematically:
\[ \frac{P + R}{2} = 33 \]
Multiplying both sides by 2 gives us:
\[ P + R = 66 \quad \text{(Equation 1)} \]

### Step 2: Express Q and S in terms of R
According to the problem:
- Q is 18 years older than R, which can be expressed as:
\[ Q = R + 18 \quad \text{(Equation 2)} \]
- Q is also 6 years younger than S, which gives us:
\[ S = Q + 6 \quad \text{(Equation 3)} \]

Substituting Equation 2 into Equation 3:
\[ S = (R + 18) + 6 = R + 24 \quad \text{(Equation 4)} \]

### Step 3: Relate P to S
The problem states that P is \(33 \frac{1}{3}\%\) less than S. This percentage can be converted to a fraction:
\[ 33 \frac{1}{3}\% = \frac{1}{3} \]
Thus, \(P\) is \(\frac{2}{3}\) of \(S\):
\[ P = S - \frac{1}{3}S = \frac{2}{3}S \quad \text{(Equation 5)} \]

### Step 4: Substitute S into the equation for P
Using Equation 4 in Equation 5:
\[ P = \frac{2}{3}(R + 24) \]

### Step 5: Substitute P into Equation 1
Now we can substitute \(P\) from Equation 5 into Equation 1:
\[ \frac{2}{3}(R + 24) + R = 66 \]

### Step 6: Solve for R
To eliminate the fraction, multiply the entire equation by 3:
\[ 2(R + 24) + 3R = 198 \]
Expanding gives:
\[ 2R + 48 + 3R = 198 \]
Combining like terms:
\[ 5R + 48 = 198 \]
Subtracting 48 from both sides:
\[ 5R = 150 \]
Dividing by 5:
\[ R = 30 \]

### Step 7: Find Q and S
Now that we have \(R\), we can find \(Q\) and \(S\):
Using Equation 2:
\[ Q = R + 18 = 30 + 18 = 48 \]
Using Equation 4:
\[ S = R + 24 = 30 + 24 = 54 \]

### Step 8: Calculate the average of Q, R, and S
Now we can find the average of the present ages of Q, R, and S:
\[ \text{Average} = \frac{Q + R + S}{3} = \frac{48 + 30 + 54}{3} \]
Calculating the sum:
\[ 48 + 30 + 54 = 132 \]
Now divide by 3:
\[ \text{Average} = \frac{132}{3} = 44 \]

Thus, the average of the present ages of Q, R, and S is **44 years**.