The average weight of all children in a group is 45 kg. If 10 children of average weight 42 kg leave the group and 2 children of average weight 55 kg join the group, then the average weight of 1 the children in the group increases by 1 kg. The number of children in the group initially was: options are 38 48 52 and 42

Let's analyze the problem using variables and equations.

Let:
- \( N \) be the initial number of children in the group.
- Since the average weight of all children in the group initially is \( 45 \) kg, the total weight of the group is \( 45N \) kg.

### Step 1: After 10 children leave
If 10 children with an average weight of 42 kg leave, their total weight is:
\[
10 \times 42 = 420 \text{ kg}
\]
So, the new total weight after these 10 children leave is:
\[
45N - 420
\]
And the new number of children is:
\[
N - 10
\]

### Step 2: After 2 new children join
When 2 children, each with an average weight of 55 kg, join, their total weight is:
\[
2 \times 55 = 110 \text{ kg}
\]
So, the updated total weight is:
\[
45N - 420 + 110 = 45N - 310
\]
And the updated number of children is:
\[
N - 10 + 2 = N - 8
\]

### Step 3: New Average Weight Equation
We are given that the new average weight of the group increases by 1 kg, which makes it \( 45 + 1 = 46 \) kg. Therefore:
\[
\frac{45N - 310}{N - 8} = 46
\]

### Step 4: Solve for \( N \)
Now we solve the equation:
\[
45N - 310 = 46(N - 8)
\]
Expanding the terms:
\[
45N - 310 = 46N - 368
\]
Rearrange to isolate \( N \):
\[
45N - 46N = -368 + 310
\]
\[
-N = -58
\]
\[
N = 58
\]

Thus, the initial number of children in the group was **58**.

**Final Answer: None of the options match, as the answer is 58.**

Would you like further details on the steps? 

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Here are some related questions:

1. What if 5 children of average weight 40 kg left and 4 joined with an average weight of 50 kg?
2. How would the average change if more than 2 children joined?
3. How would the equation adjust if the initial average weight were different?
4. Can we set up similar problems using proportions instead of averages?
5. What if the question provided the final average instead of the increase?

**Tip:** When working with weighted averages, systematically use equations to simplify group changes.

The average weight of all children in a group is 45 kg. If 10 children of average weight 42 kg leave the group and 2 children of average weight 55 kg join the group, then the average weight of the children in the group increases by 1 kg. The number of children in the group initially was: options are 38, 48, 52, and 42.

Find the initial number of children in a group where the average weight is impacted by some children leaving and others joining. This problem involves algebraic calculations with weighted averages and equations, suitable for students in Grades 8-10.

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