# Solved to be 7 after arithmetic

I recently made a blunder while trying to explain a question asked to me in an interview, The question was

• Think of $X$
• Add $X$ to itself ($X+X = y$)
• Times the result by $3$ ($y\times 3 = z$)
• Divide that result by the original answer ($z/x = q$)
• Add 1 to that result ($q + 1 = w$)
• The result is $7$

I was unable to articulate the reason for this and I hope someone could just break it down for me.

• That is pretty easy! You should have tried before asking here.
– hola
Commented Oct 17, 2014 at 14:07
• It didn't work for me. Since the first number I come of with happens to be$~0$. Commented Oct 17, 2014 at 16:47
• You've actually already posted half of the solution! Start with your second to last point ($q + 1 = w$) and continue replacing the variables until the second point. $w = q + 1 = \frac{z}{x} + 1 = \frac{3y}{x} + 1 = \frac{3(x+x)}{x} + 1 = \frac{6x}{x} + 1 = 6 + 1 = 7$. Commented Oct 17, 2014 at 17:35
• Wording nitpick, because this is something I see all the time and it's my pet peeve: you don't "times" something; rather, you "multiply" something. That is, you would "multiply the result by $3$," not "times the result by $3$." Commented Oct 17, 2014 at 19:13
• @pushpen.paul I'd say having this question asked in an interview but not being able to come up with a solution is trying.
– JiK
Commented Oct 18, 2014 at 6:55

The outcomes of the calculations are, in order

$X$

$2X$

$6X$

$6X/X=6$

$6+1=7$

• Thanks for your speedy reply Mark, is there a deeper explanation beyond the calculations are in order? I answered that the reason why you get 7 is because its a procedural operation. Commented Oct 17, 2014 at 13:12
• @kabuto178 Well it is the step of dividing by $X$ which eliminates the value of the original choice - if you had chosen zero you would have been stuck. Once this is out of the picture, the result doesn't depend on what you first chose. There are other forms of this kind of question - for example, you might do some calculations before subtracting the number you first thought of. Answering the question is generally a matter of following through what happened to the number you first thought of - better to do that than to use more and more letters. Commented Oct 17, 2014 at 13:17
• Mark's explanation is very simple,neat and clean. Commented Oct 17, 2014 at 13:24
• @MarkBennet sure, I understand your answer was just asking for a more direct explanation if possible but I do understand the process. As you said it is the step with dividing X. Thanks again, I will surely try to remember it this time haha Commented Oct 17, 2014 at 13:40

Add X to itself: $$x+x=2x$$ Times the result by 3: $$3\times2x = 6x$$ Divide that result by the original answer: $$6x\div x = 6 \quad with \quad(x\neq 0)\\$$ As you see x is cancelled out leaving 6 alone Add 1 to that result: $$6+1 = 7$$

You can write the whole operation as one expression: $3\times (x + x)\div x + 1$. $x+x$ is just $2\times x$, so you have $3\times 2\times x\div x + 1$. The only thing you do with $x$ is multiple and divide the same value by it, which are inverse operations, so the result is independent of the value of $x$.

I think it is easier to intuit the answer if you see the whole computation at once:

$$\frac{3(x+x)}{x}+1$$ The numerator is easily factored:

$$3(x+x)=3(2x)=6x$$ We then have $\frac{6x}{x}+1$ which, if you're familiar with the rules of division in algebra, simplifies to $6+1$, or $7$.