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We have our scale again,
with some masses on the left-hand side,

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and some masses on the right-hand side
and we see that our scale is balanced.

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We have the same total mass on the left-hand side
that we have on the right-hand side.

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Instead of labeling the mystery masses with a question mark,

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I have labeled them all "x."

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Since they all have an 'x' on them, we know
that each of these has the same mass.

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What I am curious about is:
what is that mass?

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What is the mass of each of these
mystery masses?

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I will let you think about this for a second:
how would you figure out what the mystery mass is,

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or what this x-value actually is?
How many kilograms is each of these things?

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What could you do to either one 
or both sides of this scale?

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I will give you a few seconds to think about that...

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You might be tempted to say,

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"If I could end up with just one mystery mass,
and if I keep my scale balanced,

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then that thing will be equal
to whatever is on the right-hand side."

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And that part would be a true statement.

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And to get to just one of these mystery masses,
you might say,

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"Why don't I just remove two of them?

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Why don't I just remove that one 
and that one?

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Then I will just be left with that single one."

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But if you just removed these two,

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the left-hand side would become lighter
than the right-hand side,

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So left will move up, 
and right will move down.

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Now you could say: "OK, I understand!
Whatever I do to the left-hand side,

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I have to do to the right-hand side
so I can keep my scale balanced."

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So you might say: "Why don't I remove 
2 of these mystery masses from the right-hand side?"

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But that is a problem too, because
you don't know what this mystery mass is.

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You could try to remove two,

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but how many of these blocks
represent a mystery mass?

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We actually do not know.

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But you might then say, "I have got
three of these things here,

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If I multiply what I have here by a third,

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If I leave a third of the stuff here,
and I leave a third of the stuff here,

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then the scales should be balanced."

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If this has the same total mass as this,

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then 1/3 of this total mass will be equal
to 1/3 of that total mass.

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So let's just keep only one-third
of this here.

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This is equivalent to multiplying
by one-third.

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If we are only going to keep 1/3 there,
we will be left with only one of the masses,

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And if we only keep 1/3 here, let's see,

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we have 1, 2, 3, 4, 5, 6, 7, 8, 9 masses.

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If we multiply this by one-third:
1/3 x 9 is only 3.

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So we remove these, and now we have 1/3 of 
what we had on the right-hand side,

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and 1/3 of what we originally had
on the left-hand side.

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And they will be balanced, because
we took 1/3 of the same total masses.

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And so, what you are left with is
just one of these mystery masses

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- this 'x' thing over here, 
whatever x might be -

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and you have 3 kilograms on the right-hand side.

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So you can make the conclusion,
as you kept this thing balanced the whole time,

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that x = 3.