Finding the Unknown | A

Question 1

IT'S PUZZLE TIME!

A Magic Trick

Think of any number. Now multiply it by 2. Add 10. Divide by 2. Now subtract the original number you thought of. Finally, add 3.

I predict that you now have 8. Am I correct? Try the trick on your friends and family! Can you explain why the trick works? [Hint: Denote the first number thought of by $x$ .] Can you make your own such tricks?

Solution

Understand the Question

Let the initial chosen number be represented by a variable $x$ .
By writing out the given instructions as algebraic expressions, we can trace how the value changes at each step.
Subtracting the original number eliminates the variable $x$ , which ensures that the final result is always a fixed constant number regardless of what number is chosen.

Step 1 · Trace the Operations Algebraically

Let the number thought of be $x$ .

Think of any number: $x$
Multiply it by 2:

\begin{aligned} x \times 2 = 2x \end{aligned}

Add 10: $2x + 10$
Divide by 2:

\begin{aligned} \dfrac{2x + 10}{2} &= \dfrac{2x}{2} + \dfrac{10}{2} \\[0.6em] &= x + 5 \end{aligned}

Subtract the original number:

\begin{aligned} (x + 5) - x = 5 \end{aligned}

Add 3:

\begin{aligned} 5 + 3 = 8 \end{aligned}

Step 2 · Explain Why the Trick Works

The prediction is correct because the final value is $8$ .

The trick works because the variable $x$ cancels out completely when the original number is subtracted: $(x + 5) - x = 5$

Since $x$ is eliminated, the outcome depends only on the constant arithmetic operations and will always equal $8$ regardless of the chosen number.

Step 3 · Create a New Magic Trick

We can create a new trick using the same principle where the chosen variable cancels out to give a fixed result, such as $10$ :

Think of any number: $x$
Multiply by 3: $3x$
Add 15: $3x + 15$
Divide by 3:

\begin{aligned} \dfrac{3x + 15}{3} = x + 5 \end{aligned}

Subtract the original number:

\begin{aligned} (x + 5) - x = 5 \end{aligned}

Add 5:

\begin{aligned} 5 + 5 = 10 \end{aligned}

The result is always $10$ .

Answer

Yes, the prediction of $8$ is correct. The trick works because the chosen number $x$ cancels out during the subtraction step, leaving a fixed constant.

Common Mistakes

Distributive Division Error: Incorrectly dividing only the first term by $2$ in $\dfrac{2x + 10}{2}$ , leading to $x + 10$ instead of $x + 5$ .
Omitting the Subtraction Step: Forgetting that subtracting the original number $x$ is the key step that eliminates the variable and makes the trick work for any input.