Question

Use the description shown below. "Choose any number Add 10 to the number. Multiply the result by 2 Subtract 18 from the result. Multiply the result by one half Subtract the original number." Write a number trick of your own that will give similar results. Explain why your trick works.

Short answer

The expression simplifies to 1. Reasoning: as the operations continue, the original number gets added and subtracted later, thereby getting rid of it completely. The result arises from the constants involved in the operations. Similar reasoning holds for the second sequence, which results in 0

Step-by-step solution

Choice of original number

Let the original number be x.

Perform addition

Add 10 to the original number. The resulting expression will be \( x + 10 \)

Perform multiplication

Multiply the result by 2. The resulting expression will be \( 2(x + 10) = 2x + 20 \)

Perform subtraction

Subtract 18 from the result. The resulting expression will be \( 2x + 20 - 18 = 2x + 2 \)

Perform second multiplication

Multiply the result by one half. The resulting expression will be \( (2x + 2) / 2 = x + 1 \)

Subtract original number

Subtract the original number from the result. The resulting expression will be \( x + 1 - x = 1 \)

Define new arithmetic sequence

Propose a new sequence. For instance: Choose any number, add 5, multiply by 3, subtract 15, divide by 3, subtract the original number. This will always result in 0 as the sequence simplifies to \( ((x+5)*3 - 15) / 3 - x = 0 \)