Subraction of Rational Numbers

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Subraction of Rational Numbers

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Let's do some problems that involve subtracting fractions,
or instead of saying fractions, we could say
rational numbers, either way.
I'll do every other of these up here.
Let's start with b.
We have 2/3 minus 1/4.
Just as in the case with adding fractions, you need to
have a common denominator here.
You have to know how many pieces of the pie we're
dealing with.
So the common denominator of 3 and 4, or the least common
multiple of 3 and 4, the smallest number that's
divisible by both of them, that's 12.
So let me write it this way.
Let me do it this way.
This will be more fun.
2/3, if we write 12 in the denominator, 3 times 4 is 12,
so 2 times 4 is 8.
These are equivalent.
We multiplied the numerator and the denominator by 4.
And then 1/4, if we put 12 in the denominator,
4 times 3 is 12.
1 times 3 is 3.
3/12 and 1/4 are the same thing.
And we have a minus sign right there.
So this is going to be equal to 8 minus 3 over 12, now that
we have our common denominator,
which is equal to 5/12.
Now let's do d.
Let me do it in a different color.
15 over 11, 15/11 minus 9/7.
Let's get a common denominator.
Let's see, between 11 and 7, I think 77 is going to be the
first number you're going to find that is
divisible by both.
You just multiply the two.
So you have 77.
To go from 11 to 77, you have to multiply it by 7.
So 15 times 7 is 70 plus 35.
That is 105.
And then we have minus over 77.
7 times 11 is 77.
9 times 11 is 99.
So 105 minus-- this is going to be 105 minus 99 over 77.
What's 105 minus 99?
It is 6, and we are done.
You can't reduce this any more.
6 and 77 aren't divisible by-- don't have any common factors.
Let's do f.
Let me do it in a new color.
So f has us doing 7/27 minus 9/39.
All right, so what is the common denominator here?
And actually, there's something we
might be able to simplify.
We might be able to simplify this a little bit.
9/39, they're both divisible by 3.
So I can rewrite 9/39 as divide 9 by 3, it becomes 3.
You divide 39 by 3, it becomes 13.
So this becomes 7/27 minus 3/13.
So what's the common denominator between 27 and 13?
Well, they don't have any common factors, so we're just
going to have to multiply the two.
So let's see, 27 times 13-- let me do it over on
the side over here.
So 27 times 13.
3 times 7 is 21, carry the 2, 3 times 2 is 6 plus 2 is 8.
Let's put the 0 down here.
1 times 7 is 7.
1 times 2 is 2.
And we add.
1 plus 0 is 1.
8 plus 7 is 15.
1 plus 2 is 3.
So our common denominator is going to be 351 minus
something over 351.
To go from 27 to 351, we have to multiply by 13.
So to go from 7 to whatever numerator here, we're going to
have to multiply by 13.
7 times 13 is 70, plus 21 is 91.
This is 91/351.
And to go from this 13 to 351, we have to multiply by 27.
So on the numerator, we have to multiply by 27.
3 times 27, that is 60 plus 21, that is 81.
And so our answer is going to be over 351.
91 minus 81, well, that's just 10.
And then we are done.
And then let's do one more.
These are getting a little hairy, but we can power
through them.
h says 13/64 minus 7/40.
So let's get a common denominator between 64 and 40.
So 80 won't work.
We could actually try to-- actually, let's do this the
old-fashioned way.
Let's look at the factors.
Let's look at the prime factorization of the two and
then we could find the least common multiple.
So this'll be interesting.
So 64-- let me do it in a different color here-- 64, we
can factor that as 2 times 32, which is the same thing as 2
times 16, which is the same thing as 2 times 8, which is
the same thing as 2 times 4, which is the same
thing as 2 times 2.
So this is 2 times itself, one, two, three,
four, five, six times.
So we could say 64 is the same thing as 2 times 2 times 2
times 2 times 2 times 2.
If you know exponents, this is the same
thing as 2 to the sixth.
That's 64.
Now, 40 you do the same thing.
40-- let me do that in a different color.
40 is 2 times 20, which is 2 times 10, which is 2 times 5.
So 40 is 2 times 2 times 2 times 2 times 2 times 5.
This is the prime factorization of 40.
Now, if we compare the two, this 2 times 2 times 2, this
is inside of 64.
So in order to find the least common multiple, we don't have
a 5 in the prime factorization of 64.
We have the 2 times 2 times 2.
So to get there, the least common multiple of these will
be if we just multiply this times 5.
So we take 64 and we multiply it times 5 because this number
now is going to have 64 in it, that's 64, and it's also going
to have 40 in it right there.
So 64 times 5.
So what is 64 times 5?
So if we take 64-- I'm running out of real estate; let me
scroll over to the right-- so 64 times 5 is 320.
So it's 320, so that is times 5.
So if we multiply 64 times 5, we have to
multiply 13 times 5.
So that's 50 plus 15.
13 times 5, so it's 65, so that is 65.
And then to go from 40 to 320, we have to multiply by 8.
And you could see that there.
You have this other three 2's, 2 to the third power.
So we have to multiply it by 8 over here.
So 7 times 8 is 56.
So we have a minus 56 is equal to 65 minus 56 is 9, is equal
to 9 over 320.
That was a little bit of a hairy problem, but we were
able to power through it.
Let's do this last problem down here.
Consider the equation 2/3x plus 1/2.
All right, y is equal to 2/3x plus 1/2.
Determine the change in y between x is equal to 1 and x
is equal to 2.
All right.
When x-- let's make little table: x and y.
When x is equal to 1, what is y?
It's 2/3 times 1 plus 1/2, which is equal to 2/3 plus
1/2, which is equal to a common denominator of 6.
2/3 is 4/6-- multiply both of them by 2-- plus 1/2 is 3/6.
So this is equal to 7/6.
All right.
When x is equal to 2.
When x is equal to 2, you have 2/3 times 2 plus 1/2, so it's
going to be equal to 4/3 plus 1/2, which is equal to a
common denominator of 6.
This is equal to 8/6-- 3 times 2 is 6; 4 times 2 is 8-- plus
3/6, which is equal to 11/6.
And they say determine the change in y between x equal 1
and x equal to 2.
So how much larger is this-- this is our new y when x is
equal to 2-- relative to this?
So we could take our 11/6 minus 7/6, and our change in y
between x is equal to 1 and x is equal to 2, 11 minus 7 is 4
over 6, which is equal to 2/3.
And that makes complete sense because every time you
increase x by 1, you're increasing this expression by
another 2/3.
When you go from 1 to 2, you should increase by 2/3.

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