An amount of money was divided Between some people in such a way that if there had been 4 more people,everyone would have got tk 16 less. But if there had been 4 less people,everyone would have got tk 24 more. How many people were there in the group?

Let the total amount of money be \( T \), and let the number of people be \( n \). We are given two conditions:

1. If there had been 4 more people, everyone would have received Tk 16 less.
2. If there had been 4 fewer people, everyone would have received Tk 24 more.

Let's express these conditions mathematically:

1. The amount each person receives normally is \( \frac{T}{n} \).
2. If there were 4 more people, each person would receive \( \frac{T}{n+4} \). According to the first condition, this is Tk 16 less than \( \frac{T}{n} \), so we can write the equation:
   \[
   \frac{T}{n} - \frac{T}{n+4} = 16
   \]

3. If there were 4 fewer people, each person would receive \( \frac{T}{n-4} \). According to the second condition, this is Tk 24 more than \( \frac{T}{n} \), so we can write the equation:
   \[
   \frac{T}{n-4} - \frac{T}{n} = 24
   \]

Now we solve this system of equations step by step.

### Step 1: Solve the first equation
The first equation is:
\[
\frac{T}{n} - \frac{T}{n+4} = 16
\]
Multiply both sides by \( n(n+4) \) to eliminate the denominators:
\[
T(n+4) - Tn = 16n(n+4)
\]
Simplify:
\[
Tn + 4T - Tn = 16n(n+4)
\]
\[
4T = 16n(n+4)
\]
Solve for \( T \):
\[
T = 4n(n+4)
\]
So, \( T = 4n^2 + 16n \).

### Step 2: Solve the second equation
The second equation is:
\[
\frac{T}{n-4} - \frac{T}{n} = 24
\]
Multiply both sides by \( n(n-4) \) to eliminate the denominators:
\[
Tn - T(n-4) = 24n(n-4)
\]
Simplify:
\[
Tn - Tn + 4T = 24n(n-4)
\]
\[
4T = 24n(n-4)
\]
Solve for \( T \):
\[
T = 6n(n-4)
\]
So, \( T = 6n^2 - 24n \).

### Step 3: Set the two expressions for \( T \) equal
Now we have two expressions for \( T \):
\[
4n^2 + 16n = 6n^2 - 24n
\]
Simplify this equation:
\[
0 = 6n^2 - 24n - 4n^2 - 16n
\]
\[
0 = 2n^2 - 40n
\]
Factor out \( 2n \):
\[
0 = 2n(n - 20)
\]
So, \( n = 0 \) or \( n = 20 \).

Since the number of people cannot be zero, we have \( n = 20 \).

### Conclusion:
There were 20 people in the group.

Would you like more details or explanations on any step? Here are some related questions you might find interesting:

1. How would the solution change if the amount of money received was different?
2. What if there were 5 fewer or 5 more people instead of 4?
3. Can you apply a similar method for different amounts of money reductions or increases?
4. How would this problem change if we knew the total amount \( T \) upfront?
5. How can we generalize this problem for any change in the number of people?

**Tip:** Always start by expressing each condition as a mathematical equation to simplify solving problems with multiple unknowns.

An amount of money was divided between some people in such a way that if there had been 4 more people, everyone would have got Tk 16 less. But if there had been 4 fewer people, everyone would have got Tk 24 more. How many people were there in the group?

Learn how to solve a money distribution problem using quadratic equations. The problem involves determining the number of people in a group where changes in group size affect individual shares by Tk 16 or Tk 24. Suitable for Grades 10-12, with step-by-step explanations.

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