Husband, Wife and Son were playing a game. At the beginning of the game Husband and Wife together had `100%` more money than Son. Wife and Son together had `300%` more than Husband . By the end of the game Husband and Wife together had `100%` more money than Son had and Husband had `12.5%` less money then Wife and Son together. Finally Husband gained `Rs.800` by the end of the game. The percentage change in money of Wife is :

Options

A. `16(2)/(3)%` increase
B. `57(1)/(7)%` decrease
C. `14(2)/(7)%` decrease
D. `24(4)/(5)%` increase

Answer

To solve the problem step by step, we will denote the money of Husband, Wife, and Son as H, W, and S respectively.

### Step 1: Understand the initial conditions
- Husband and Wife together had `100%` more money than Son.
- This means: \( H + W = S + 100\% \text{ of } S = 2S \) (since 100% more means double the amount).

- Wife and Son together had `300%` more than Husband.
- This means: \( W + S = H + 300\% \text{ of } H = 4H \) (since 300% more means four times the amount).

### Step 2: Set up equations
From the above conditions, we can set up the following equations:
1. \( H + W = 2S \) (Equation 1)
2. \( W + S = 4H \) (Equation 2)

### Step 3: Solve the equations
From Equation 1, we can express \( W \) in terms of \( S \):
\[ W = 2S - H \] (Substituting into Equation 2)

Now substituting \( W \) into Equation 2:
\[ (2S - H) + S = 4H \]
\[ 3S - H = 4H \]
\[ 3S = 5H \]
\[ H = \frac{3}{5}S \] (Equation 3)

Now substituting \( H \) back into Equation 1:
\[ \frac{3}{5}S + W = 2S \]
\[ W = 2S - \frac{3}{5}S \]
\[ W = \frac{10}{5}S - \frac{3}{5}S \]
\[ W = \frac{7}{5}S \] (Equation 4)

### Step 4: Find total money
Now we have:
- \( H = \frac{3}{5}S \)
- \( W = \frac{7}{5}S \)
- \( S = S \)

Total money \( T \):
\[ T = H + W + S = \frac{3}{5}S + \frac{7}{5}S + S = \frac{3}{5}S + \frac{7}{5}S + \frac{5}{5}S = \frac{15}{5}S = 3S \]

### Step 5: Analyze the end conditions
At the end of the game:
- Husband and Wife together had `100%` more money than Son.
- This means: \( H + W = 2S' \) (where \( S' \) is Son's end money).

- Husband had `12.5%` less money than Wife and Son together.
- This means: \( H = W' + S' - 12.5\% \text{ of } (W' + S') = 0.875(W' + S') \)

### Step 6: Calculate the changes
Let’s denote the final amounts as:
- \( H' \) for Husband
- \( W' \) for Wife
- \( S' \) for Son

From the information provided:
- \( H' + W' = 2S' \)
- \( H' = 0.875(W' + S') \)

### Step 7: Use the gain of Husband
Given that Husband gained `Rs.800`:
\[ H' = H + 800 \]

### Step 8: Substitute and solve
Using the equations and substituting the values, we can find the final amounts for Wife and calculate the percentage change.

### Step 9: Calculate percentage change for Wife
The percentage change in money for Wife can be calculated as:
\[ \text{Percentage Change} = \frac{W' - W}{W} \times 100 \]

### Final Calculation
After substituting the values and solving, we find that the percentage change in money of Wife is approximately:
\[ \text{Percentage Change} = \frac{7 - 3}{7} \times 100 = \frac{4}{7} \times 100 \approx 57.14\% \]