(b) A group of children bought some apples. Each apple is cut into 4 equal pieces and divided by 1 piece, leaving 3 pieces. And when each apple was cut into 3 equal pieces, two pieces were not enough. Find the number of children and the number of apples by writing a pair of simultaneous equations, taking the number of children as x and the number of apples as y.

To solve the problem, we need to establish two equations based on the information provided.

Let's denote:
- \( x \) as the number of children.
- \( y \) as the number of apples.

### Step 1: Formulate the first equation
The first scenario states that each apple is cut into 4 equal pieces. Each child receives 1 piece, leaving 3 pieces.

So, the total number of pieces obtained from \( y \) apples is \( 4y \). If each of the \( x \) children receives 1 piece, the equation will be:
\[
x = 4y - 3
\]

### Step 2: Formulate the second equation
The second scenario states that each apple is cut into 3 equal pieces. Two pieces were not enough for all the children.

So, the total number of pieces obtained from \( y \) apples is \( 3y \). Since two pieces were not enough, it implies that the total number of children is two less than the total number of pieces:
\[
x = 3y + 2
\]

### Step 3: Set up the simultaneous equations
The two equations we derived are:
\[
x = 4y - 3 \quad \text{(Equation 1)}
\]
\[
x = 3y + 2 \quad \text{(Equation 2)}
\]

### Step 4: Solve the simultaneous equations
Set the equations equal to each other to eliminate \( x \):
\[
4y - 3 = 3y + 2
\]
Subtract \( 3y \) from both sides:
\[
y - 3 = 2
\]
Add 3 to both sides:
\[
y = 5
\]

Now that we have \( y \), substitute it into one of the original equations to find \( x \). Using Equation 2:
\[
x = 3(5) + 2 = 15 + 2 = 17
\]

### Final Answer:
- The number of children (\( x \)) is **17**.
- The number of apples (\( y \)) is **5**.

Would you like more details or have any questions?

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Here are 5 related questions that might interest you:

1. What happens if each apple is cut into 5 equal pieces?
2. How would the equations change if two pieces were left in the first scenario?
3. Can you solve a similar problem where each apple is divided into 6 pieces?
4. What is the solution if instead of apples, oranges are divided into 4 and 3 pieces?
5. How would the solution differ if the number of leftover pieces in the first scenario was 4 instead of 3?

**Tip:** Always double-check your equations by substituting your final values back into the original conditions to ensure accuracy.

Learn how to solve a simultaneous equations word problem involving children and apples. Find the number of children and apples using mathematical equations and logical reasoning. Suitable for Grades 7-9 students.

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