Context: Consider the following number trick: 1. Think of a number. 2. Double it. 3. Add four. 4. Divide by two. 5. Subtract the original number you thought of. (This trick always results in 2). Q. Can you come up with more complicated steps that always lead to the same final value?

Algebra helps us show a number trick always gives the same result. Step 1 — Understanding the original trick Let us use a variable to represent the number. Let be the number you think of. We will follow the steps of the original trick. First, we double the number. Next, we add four to this result. Then, we divide the entire expression by two. Finally, we subtract the original number . > Step 2 — Designing new complicated steps We want to create a new trick. It must also always result in 2. Let us make the steps more complex. Here are the new steps we designed: 1. Think of a number. 2. Multiply it by six. 3. Add twelve to the result. 4. Subtract two times your original number. 5. Divide the new result by four. 6. Subtract one from this value. 7. Subtract the original number you thought of. Step 3 — Verifying the new trick Let us check if our new trick works. Let be the number you think of. We will follow the new steps using . First, we multiply the number by six. Next, we add twelve to this result. Then, we subtract two times the original number . After that, we divide the new result by four. Next, we subtract one from this value. Finally, we subtract the original number . > Answer > (i) Here are more complicated steps that always lead to 2: > 1. Think of a number. > 2. Multiply it by six. > 3. Add twelve to the result. > 4. Subtract two times your original number. > 5. Divide the new result by four. > 6. Subtract one from this value. > 7. Subtract the original number you thought of.

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Question 3

Context: Consider the following number trick:

Think of a number.
Double it.
Add four.
Divide by two.
Subtract the original number you thought of. (This trick always results in 2).

Q. Can you come up with more complicated steps that always lead to the same final value?

Solution

Algebra helps us show a number trick always gives the same result.

Step 1 — Understanding the original trick

Let us use a variable to represent the number. Let $x$ be the number you think of. We will follow the steps of the original trick.

First, we double the number. $2x$ Next, we add four to this result. $2x + 4$ Then, we divide the entire expression by two. $\frac{2x + 4}{2}$ $= \frac{2x}{2} + \frac{4}{2}$ $= x + 2$ Finally, we subtract the original number $x$ . $(x + 2) - x$ $= 2$

$\boxed{2}$

Step 2 — Designing new complicated steps

We want to create a new trick. It must also always result in 2. Let us make the steps more complex. Here are the new steps we designed:

Think of a number.
Multiply it by six.
Add twelve to the result.
Subtract two times your original number.
Divide the new result by four.
Subtract one from this value.
Subtract the original number you thought of.

Step 3 — Verifying the new trick

Let us check if our new trick works. Let $x$ be the number you think of. We will follow the new steps using $x$ .

First, we multiply the number by six. $6x$ Next, we add twelve to this result. $6x + 12$ Then, we subtract two times the original number $x$ . $(6x + 12) - 2x$ $= 4x + 12$ After that, we divide the new result by four. $\frac{4x + 12}{4}$ $= \frac{4x}{4} + \frac{12}{4}$ $= x + 3$ Next, we subtract one from this value. $(x + 3) - 1$ $= x + 2$ Finally, we subtract the original number $x$ . $(x + 2) - x$ $= 2$

$\boxed{2}$

Answer

(i) Here are more complicated steps that always lead to 2:

Think of a number.

Multiply it by six.

Add twelve to the result.

Subtract two times your original number.

Divide the new result by four.

Subtract one from this value.

Subtract the original number you thought of.

Question 3

Answer

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