Help India Foundation and People for people Organistion decided to distribute the blankets among 22 men and 28 women who are Tsunami victims. When HIF and PPO distributed their repective blankets evenly among 28 women they were left with 24 and 16 blankets respectively. If they distributed their blankets evenly among 22 men they were left with 12 blankets each. So finally they dicided to combine all their blankets and then distributed among 22 men and 28 women altogether then no any blanket remained undistributed. : the ratio of blankets between HIF and PPO is:

Options

A. 43/157
B. 147/179
C. 59/101
D. can't be determined

Answer

To solve the problem, we need to find the ratio of blankets between Help India Foundation (HIF) and People for People Organization (PPO). Let's denote the total number of blankets with HIF as \( M \) and with PPO as \( N \).

### Step 1: Set up equations based on the information given

1. When HIF distributed their blankets among 28 women, they were left with 24 blankets. This means:
\[ M = 28k + 24 \quad \text{(for some integer } k\text{)} \]
where \( k \) is the number of blankets each woman received.

2. When PPO distributed their blankets among 28 women, they were left with 16 blankets. This means:
\[ N = 28j + 16 \quad \text{(for some integer } j\text{)} \]

### Step 2: Set up equations for distribution among men

1. When HIF distributed their blankets among 22 men, they were left with 12 blankets. This means:
\[ M = 22p + 12 \quad \text{(for some integer } p\text{)} \]

2. When PPO distributed their blankets among 22 men, they were left with 12 blankets. This means:
\[ N = 22q + 12 \quad \text{(for some integer } q\text{)} \]

### Step 3: Equate the two expressions for \( M \) and \( N \)

From the equations for \( M \):
1. \( 28k + 24 = 22p + 12 \)
Rearranging gives:
\[ 28k - 22p = -12 \quad \text{(Equation 1)} \]

From the equations for \( N \):
2. \( 28j + 16 = 22q + 12 \)
Rearranging gives:
\[ 28j - 22q = -4 \quad \text{(Equation 2)} \]

### Step 4: Solve the equations

**For Equation 1:**
\[ 28k - 22p = -12 \implies 14k - 11p = -6 \]
This can be rewritten as:
\[ 14k = 11p - 6 \]

**For Equation 2:**
\[ 28j - 22q = -4 \implies 14j - 11q = -2 \]
This can be rewritten as:
\[ 14j = 11q - 2 \]

### Step 5: Find integer solutions

To find integer solutions for \( k \) and \( j \), we can express \( p \) and \( q \) in terms of \( k \) and \( j \) respectively.

From \( 14k = 11p - 6 \):
\[ p = \frac{14k + 6}{11} \]

From \( 14j = 11q - 2 \):
\[ q = \frac{14j + 2}{11} \]

### Step 6: Combine the blankets and find the ratio

Now, we need to combine the total number of blankets:
\[ M + N = (28k + 24) + (28j + 16) = 28(k + j) + 40 \]

When they distribute the combined blankets among 22 men and 28 women:
\[ M + N = 22r + 0 \quad \text{(since no blankets remain)} \]

### Step 7: Find the ratio of \( M \) to \( N \)

To find the ratio of \( M \) to \( N \), we can use the equations we derived for \( M \) and \( N \):
\[ \frac{M}{N} = \frac{28k + 24}{28j + 16} \]

### Conclusion

After simplifying, we can find the ratio \( \frac{M}{N} \).