Question

Christopher has a stick that he has marked out in tenths. He has to break the stick into three pieces of which no two pieces can have the same length. Give three equations that show different ways in which he could break the stick.

(4 points)

Answer 1

Let's denote the length of the stick as 1, which corresponds to 10 tenths. We'll denote the lengths of the three pieces as \( x \), \( y \), and \( z \), where \( x \), \( y \), and \( z \) are all different and must satisfy the following constraints:

1. \( x + y + z = 10 \) (since the total length of the stick is 10 tenths)
2. \( x, y, z \) are all different.
3. \( x, y, z \geq 1 \) (since they are in tenths, the smallest piece can be 1 tenth).

Here are three different ways in which Christopher could break the stick into three pieces:

### Example 1:
1. \( x = 1 \)
2. \( y = 3 \)
3. \( z = 6 \)

Equation: \( 1 + 3 + 6 = 10 \)

### Example 2:
1. \( x = 2 \)
2. \( y = 4 \)
3. \( z = 4 \) (not valid since two pieces are of the same length)

But to keep it valid, let's consider:
1. \( x = 2 \)
2. \( y = 3 \)
3. \( z = 5 \)

Equation: \( 2 + 3 + 5 = 10 \)

### Example 3:
1. \( x = 1 \)
2. \( y = 4 \)
3. \( z = 5 \)

Equation: \( 1 + 4 + 5 = 10 \)

To summarize, here are the valid equations showing Christopher's different ways to break the stick:

1. \( 1 + 3 + 6 = 10 \)
2. \( 2 + 3 + 5 = 10 \)
3. \( 1 + 4 + 5 = 10 \)