A group of children bought some apples. If each apple is divided into 4 equal pieces and 1 piece is given to each child, there are 3 pieces left over. If each apple is divided into 3 equal pieces, 2 pieces are needed. Letting $x$ be the number of children and $y$ be the number of apples, write a system of simultaneous equations and find the number of children and the number of apples.

1. Problem Description

A group of children bought some apples. If each apple is divided into 4 equal pieces and 1 piece is given to each child, there are 3 pieces left over. If each apple is divided into 3 equal pieces, 2 pieces are needed. Letting $x$ be the number of children and $y$ be the number of apples, write a system of simultaneous equations and find the number of children and the number of apples.

2. Solution Steps

Let $x$ be the number of children and $y$ be the number of apples.

When each apple is divided into 4 equal pieces, the total number of pieces is $4y$. If each child gets 1 piece, and there are 3 pieces left over, then we have:

$$4y = x + 3$$

When each apple is divided into 3 equal pieces, the total number of pieces is $3y$. If 2 pieces are needed to give each child one piece, then we have:

$$3y = x - 2$$

We now have a system of two simultaneous equations with two unknowns:

$$4y = x + 3 \hspace{1cm} (1)$$

$$3y = x - 2 \hspace{1cm} (2)$$

We can solve this system by substitution or elimination. Let's use elimination. Subtract equation (2) from equation (1):

$$4y - 3y = (x+3) - (x-2)$$

$$y = x + 3 - x + 2$$

$$y = 5$$

Substitute $y=5$ into equation (1):

$$4(5) = x + 3$$

$$20 = x + 3$$

$$x = 20 - 3$$

$$x = 17$$

So, there are 17 children and 5 apples.

3. Final Answer

The number of children is 17 and the number of apples is

5.