Average weight of a family of X members is 18 kg. When three men of this family get married, average weight increases by 1kg and when an old man having weight 18kg more than total weight of wife of these three men died, average weight become 16 kg. Find the value of X?

Options

A. A)24
B. B)28
C. C)26
D. D)25

Answer

To solve the problem step by step, we will follow the logical progression of the information given in the question.

### Step 1: Calculate the initial total weight of the family
The average weight of the family is given as 18 kg, and the number of members in the family is X. Therefore, the total weight of the family can be calculated as:

\[ \text{Total Weight} = \text{Average Weight} \times \text{Number of Members} = 18 \times X \]

### Step 2: Calculate the new total weight after three men get married
When three men from the family get married, the average weight increases by 1 kg to 19 kg. The total number of family members now becomes \(X + 3\). Let the total weight of the three wives be \(Y\) kg. The new total weight can be expressed as:

\[ \text{New Total Weight} = 18X + Y \]

Since the average weight is now 19 kg, we can set up the equation:

\[ \frac{18X + Y}{X + 3} = 19 \]

Multiplying both sides by \(X + 3\):

\[ 18X + Y = 19(X + 3) \]

Expanding the right side:

\[ 18X + Y = 19X + 57 \]

Rearranging gives:

\[ Y = 19X + 57 - 18X \]
\[ Y = X + 57 \]

### Step 3: Calculate the new average weight after the old man dies
When the old man, who weighs 18 kg more than the total weight of the wives, dies, the average weight of the family decreases to 16 kg. The weight of the old man can be expressed as:

\[ \text{Weight of Old Man} = Y + 18 = (X + 57) + 18 = X + 75 \]

After the old man dies, the total number of family members becomes \(X + 2\), and the new total weight is:

\[ \text{New Total Weight} = 18X + Y - (X + 75) \]

Substituting \(Y\) into the equation:

\[ \text{New Total Weight} = 18X + (X + 57) - (X + 75) \]
\[ = 18X + X + 57 - X - 75 \]
\[ = 18X + 57 - 75 \]
\[ = 18X - 18 \]

Now, we set up the equation for the new average weight:

\[ \frac{18X - 18}{X + 2} = 16 \]

Multiplying both sides by \(X + 2\):

\[ 18X - 18 = 16(X + 2) \]

Expanding the right side:

\[ 18X - 18 = 16X + 32 \]

Rearranging gives:

\[ 18X - 16X = 32 + 18 \]
\[ 2X = 50 \]

### Step 4: Solve for X
Dividing both sides by 2:

\[ X = 25 \]

### Final Answer
The value of \(X\) is 25.

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