After disturbing the sweets equally 25 childres, 8 sweets remain. Had the number of children been 28, 22 sweets would have been left after equally distributing. What was the total number of sweets ?

Options

A. 328
B. 343
C. 358
D. Data inadequate

Answer

To solve the problem step by step, we will define the total number of sweets and use the information given about the distribution among children.

### Step 1: Define the total number of sweets
Let the total number of sweets be \( S \).

### Step 2: Set up the first equation
According to the problem, when the sweets are distributed among 25 children, 8 sweets remain. This can be expressed as:
\[ S = 25x + 8 \]
where \( x \) is the number of sweets each child receives.

### Step 3: Set up the second equation
If the number of children were 28, then 22 sweets would remain. This can be expressed as:
\[ S = 28y + 22 \]
where \( y \) is the number of sweets each child receives in this scenario.

### Step 4: Equate the two expressions for \( S \)
Since both expressions represent the total number of sweets, we can set them equal to each other:
\[ 25x + 8 = 28y + 22 \]

### Step 5: Rearrange the equation
Rearranging the equation gives us:
\[ 25x - 28y = 22 - 8 \]
\[ 25x - 28y = 14 \]

### Step 6: Solve for one variable in terms of the other
From the equation \( 25x - 28y = 14 \), we can express \( x \) in terms of \( y \):
\[ 25x = 28y + 14 \]
\[ x = \frac{28y + 14}{25} \]

### Step 7: Substitute \( x \) back into the first equation
Now, substitute \( x \) back into the first equation \( S = 25x + 8 \):
\[ S = 25\left(\frac{28y + 14}{25}\right) + 8 \]
\[ S = 28y + 14 + 8 \]
\[ S = 28y + 22 \]

### Step 8: Set \( S \) from both equations
We know \( S = 28y + 22 \) and \( S = 25x + 8 \). We can also express \( y \) in terms of \( x \) from the equation \( 25x - 28y = 14 \):
\[ y = \frac{25x - 14}{28} \]

### Step 9: Substitute \( y \) into the equation for \( S \)
Now substitute \( y \) back into \( S = 28y + 22 \):
\[ S = 28\left(\frac{25x - 14}{28}\right) + 22 \]
\[ S = 25x - 14 + 22 \]
\[ S = 25x + 8 \]

### Step 10: Solve for \( S \)
Now we have two equations that are equivalent. We can find integer values for \( x \) and \( y \) that satisfy both equations. We can try different values for \( x \) to find a suitable integer solution.

### Step 11: Finding integer values
Let’s try \( x = 14 \):
\[ S = 25(14) + 8 = 350 + 8 = 358 \]

### Step 12: Check with the second condition
Now check if this value satisfies the second condition:
\[ S = 28y + 22 \Rightarrow 358 = 28y + 22 \]
\[ 336 = 28y \Rightarrow y = \frac{336}{28} = 12 \]

### Conclusion
Both conditions are satisfied with \( x = 14 \) and \( y = 12 \). Thus, the total number of sweets is:
\[ \boxed{358} \]