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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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