Question

1. jonathan subtracted 2 numbers that had a difference of 12.67. he asked 2 numbers that had a sum of 12.67. what numbers might he have subtracted and what numbers might he have added to arrive at the number 12.67?

Explanation:

Step1: Define Variables

Let the two numbers be \( x \) and \( y \). We know two cases:

• Case 1 (Subtraction): \( |x - y| = 12.67 \)
• Case 2 (Addition): \( x + y = 12.67 \)

Step2: Solve the System of Equations

We can set up the system:

1. \( x - y = 12.67 \) (assuming \( x > y \))
2. \( x + y = 12.67 \)

Add the two equations:
\( (x - y) + (x + y) = 12.67 + 12.67 \)
\( 2x = 25.34 \)
\( x = \frac{25.34}{2} = 12.67 \)

Substitute \( x = 12.67 \) into \( x + y = 12.67 \):
\( 12.67 + y = 12.67 \)
\( y = 0 \)

Another possibility for subtraction: If \( y - x = 12.67 \) and \( x + y = 12.67 \), solving gives \( y = 12.67 \), \( x = 0 \).

For other pairs, we can choose any \( x \) and solve for \( y \):

• Subtraction: Let \( x = 13.67 \), then \( y = 13.67 - 12.67 = 1 \) (since \( 13.67 - 1 = 12.67 \))
• Addition: Let \( x = 1 \), then \( y = 12.67 - 1 = 11.67 \) (since \( 1 + 11.67 = 12.67 \))

Step3: Verify

• For subtraction: \( 13.67 - 1 = 12.67 \) (correct)
• For addition: \( 1 + 11.67 = 12.67 \) (correct)

Answer:

For subtraction, possible pairs: \((12.67, 0)\), \((13.67, 1)\), etc. For addition, possible pairs: \((12.67, 0)\), \((1, 11.67)\), etc. (Any pair satisfying \(|x - y| = 12.67\) or \(x + y = 12.67\) is valid.)