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. How would you change this game to make the final answer 3? What about 5?

Algebra Play | IT

Question 2

Context: Consider the following number trick:

Q. How would you change this game to make the final answer 3? What about 5?

Solution

Understand the Question

Let the number chosen be $x$ .
Following the operations algebraically:
1. Think of a number: $x$
2. Double it: $2x$
3. Add a number $A$ : $2x + A$
4. Divide by two: $\dfrac{2x + A}{2} = x + \dfrac{A}{2}$
5. Subtract the original number: $\left(x + \dfrac{A}{2}\right) - x = \dfrac{A}{2}$
The final result is always half of the number added in Step 3 $\left(\dfrac{A}{2}\right)$ .
In the original game, adding $4$ results in $\dfrac{4}{2} = 2$ . To get any desired final answer $R$ , we must add $A = 2 \times R$ .

(i) How would you change this game to make the final answer 3?

Step 1 · Find the Number to Add for a Result of 3

Let $x$ be the number thought of, and replace "Add four" with "Add $A$ ":

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

Set the final result equal to $3$ : $\dfrac{A}{2} = 3$

\begin{aligned} A &= 3 \times 2 \\ &= 6 \end{aligned}

Therefore, change "Add four" to "Add six".

Answer

(i) Change "Add four" to "Add six".

(ii) How would you change this game to make the final answer 5?

Step 1 · Find the Number to Add for a Result of 5

Using the general formula where the final result is $\dfrac{A}{2}$ , set the expression equal to $5$ :

$\dfrac{A}{2} = 5$

\begin{aligned} A &= 5 \times 2 \\ &= 10 \end{aligned}

Therefore, change "Add four" to "Add ten".

Answer

(ii) Change "Add four" to "Add ten".

Common Mistakes

Adding the Target Directly: Incorrectly thinking that adding $3$ gives a final result of $3$ , forgetting that the sum is subsequently divided by $2$ .
Doubling the Subtraction: Forgetting to subtract the original number $x$ at the final step, leaving the answer dependent on $x$ rather than a constant number.