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?

We can use algebra to represent the steps of the number trick and find out how to change it. Step 1 — Understand the original trick Let us use a variable to represent the number we think of. Let be the number you think of. The first step is to double the number. Next, we add four to this result. Then, we divide the whole expression by two. We can simplify this expression. Finally, we subtract the original number, . This simplifies to just 2. > This confirms the trick always results in 2. Step 2 — Change for final answer 3 We want the trick to result in 3 instead of 2. Let us change the "add four" step to "add ". Let be the number you think of. First, we double the number. Next, we add to this result. Then, we divide the whole expression by two. We simplify this expression. Finally, we subtract the original number, . This simplifies to just . We want this final answer to be 3. So, we set the expression equal to 3. To find , we multiply both sides by 2. > So, to get a final answer of 3, we should change "Add four" to "Add six". Step 3 — Change for final answer 5 Now, we want the trick to result in 5. We use the same modified step from before. The final answer of the modified trick is . We want this final answer to be 5. So, we set the expression equal to 5. To find , we multiply both sides by 2. > So, to get a final answer of 5, we should change "Add four" to "Add ten". Answer > (i) To make the final answer 3, change "Add four" to "Add six". > (ii) To make the final answer 5, change "Add four" to "Add ten".

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Question 2

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

Solution

We can use algebra to represent the steps of the number trick and find out how to change it.

Step 1 — Understand the original trick

Let us use a variable to represent the number we think of. Let $x$ be the number you think of.

The first step is to double the number. $2x$

Next, we add four to this result. $2x + 4$

Then, we divide the whole expression by two. $\frac{2x + 4}{2}$

We can simplify this expression. $= \frac{2x}{2} + \frac{4}{2}$ $= x + 2$

Finally, we subtract the original number, $x$ . $(x + 2) - x$

This simplifies to just 2. $= 2$

$\boxed{\text{The final answer is } 2}$

This confirms the trick always results in 2.

Step 2 — Change for final answer 3

We want the trick to result in 3 instead of 2. Let us change the "add four" step to "add $A$ ". Let $x$ be the number you think of.

First, we double the number. $2x$

Next, we add $A$ to this result. $2x + A$

Then, we divide the whole expression by two. $\frac{2x + A}{2}$

We simplify this expression. $= \frac{2x}{2} + \frac{A}{2}$ $= x + \frac{A}{2}$

Finally, we subtract the original number, $x$ . $(x + \frac{A}{2}) - x$

This simplifies to just $A/2$ . $= \frac{A}{2}$

We want this final answer to be 3. So, we set the expression equal to 3. $\frac{A}{2} = 3$

To find $A$ , we multiply both sides by 2. $A = 3 \times 2$ $A = 6$

$\boxed{\text{We should add 6}}$

So, to get a final answer of 3, we should change "Add four" to "Add six".

Step 3 — Change for final answer 5

Now, we want the trick to result in 5. We use the same modified step from before. The final answer of the modified trick is $A/2$ .

We want this final answer to be 5. So, we set the expression equal to 5. $\frac{A}{2} = 5$

To find $A$ , we multiply both sides by 2. $A = 5 \times 2$ $A = 10$

$\boxed{\text{We should add 10}}$

So, to get a final answer of 5, we should change "Add four" to "Add ten".

Answer

(i) To make the final answer 3, change "Add four" to "Add six". (ii) To make the final answer 5, change "Add four" to "Add ten".

Question 2

Answer

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