The average age (in years) of a group of people is twice the number of people in the group. A person, X, leaves the group and the average age is still twice the number of people in the group. Now another person, Y, leaves the group and the average age is still twice the number of people in the group. If the ratio of the ages of X and Y is 19: 17, then find the average age of the group, if one more person, Z, of age 16 years, leaves the group.

Options

A. A)10
B. B)15
C. C)16
D. D)18

Answer

To solve the problem step by step, we will follow the logical flow based on the information given in the question.

### Step 1: Define Variables
Let:
- \( n \) = number of people in the group initially
- \( A \) = average age of the group

According to the question, the average age is twice the number of people in the group:
\[ A = 2n \]

### Step 2: Calculate Initial Sum of Ages
The total sum of ages of the group can be calculated as:
\[ \text{Sum} = A \times n = 2n \times n = 2n^2 \]

### Step 3: After Person X Leaves
When person X leaves, the number of people becomes \( n - 1 \) and the average age remains twice the number of people:
\[ A' = 2(n - 1) = 2n - 2 \]

The new sum of ages can be expressed as:
\[ \text{New Sum} = A' \times (n - 1) = (2n - 2)(n - 1) = 2n^2 - 2n - 2n + 2 = 2n^2 - 4n + 2 \]

Since the sum of ages remains the same, we can set up the equation:
\[ 2n^2 = 2n^2 - 4n + 2 + \text{Age of X} \]
This simplifies to:
\[ 0 = -4n + 2 + \text{Age of X} \quad \Rightarrow \quad \text{Age of X} = 4n - 2 \]

### Step 4: After Person Y Leaves
Now, when person Y leaves, the number of people becomes \( n - 2 \) and the average age is still twice the number of people:
\[ A'' = 2(n - 2) = 2n - 4 \]

The new sum of ages is:
\[ \text{New Sum} = A'' \times (n - 2) = (2n - 4)(n - 2) = 2n^2 - 4n - 4n + 8 = 2n^2 - 8n + 8 \]

Setting this equal to the previous sum of ages:
\[ 2n^2 = 2n^2 - 8n + 8 + \text{Age of X} + \text{Age of Y} \]
Substituting for Age of X:
\[ 2n^2 = 2n^2 - 8n + 8 + (4n - 2) + \text{Age of Y} \]
This simplifies to:
\[ 0 = -8n + 8 + 4n - 2 + \text{Age of Y} \quad \Rightarrow \quad \text{Age of Y} = 4n - 6 \]

### Step 5: Age Ratio of X and Y
Given the ratio of ages of X and Y is \( 19:17 \):
\[ \frac{\text{Age of X}}{\text{Age of Y}} = \frac{4n - 2}{4n - 6} = \frac{19}{17} \]

Cross-multiplying gives:
\[ 17(4n - 2) = 19(4n - 6) \]
Expanding both sides:
\[ 68n - 34 = 76n - 114 \]
Rearranging gives:
\[ 76n - 68n = 114 - 34 \quad \Rightarrow \quad 8n = 80 \quad \Rightarrow \quad n = 10 \]

### Step 6: Calculate Average Age
Now substituting \( n \) back to find the average age:
\[ A = 2n = 2 \times 10 = 20 \]

### Step 7: After Person Z Leaves
When person Z, aged 16, leaves the group, the number of people becomes \( n - 3 = 10 - 3 = 7 \). The new sum of ages is:
\[ \text{New Sum} = 2n^2 - 4n + 8 - 16 = 2(10)^2 - 4(10) + 8 - 16 = 200 - 40 + 8 - 16 = 152 \]

The new average age is:
\[ \text{New Average} = \frac{\text{New Sum}}{\text{Number of People}} = \frac{152}{7} \approx 21.71 \]

### Conclusion
Thus, the average age of the group after person Z leaves is approximately \( 21.71 \) years.

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