0:00:00.000,0:00:00.330
0:00:00.330,0:00:02.900
Let's add some rational[br]numbers.
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And I'm using that word because[br]that's the word that
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this book uses, but in more[br]popular terminology we'll be
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adding fractions.
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So let's just go through all[br]of these, actually, just to
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see all of the examples.
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So first we're going to[br]have 3/7 plus 2/7.
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Our denominators are the same,[br]so we can just add the
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numerators.
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So our denominator is[br]7, 3 plus 2 is 5.
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That is a.
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Let me do every other.
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It would take forever[br]to do all of them.
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Not forever, but just more time[br]than I want to spend.
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So c is 5/16 plus 5/12.
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Our denominators are[br]not the same.
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We have to find a common[br]denominator, which has to be
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the least common-- it actually[br]could be any common multiple
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of these, but for simplicity[br]let's do the
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least common multiple.
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So what's the smallest number[br]that's a multiple
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of both 16 and 12?
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So let's see, 16 times 2[br]is 32, not there yet.
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Times 3, 48.
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That seems to work.
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12 goes into 48 four times.
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So let's use 48 as our[br]common denominator.
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So we had to multiply 16 times 3[br]to get to 48, so we're going
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to have to multiply[br]this 5 times 3.
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We're just multiplying the[br]numerator and the denominator
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by the same number, so we're[br]not really changing it.
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So 5 times 3 is 15.
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And then to get from this 12 to[br]this 48 right there, we had
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to multiply times 4.
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So then to get to 5 to this[br]numerator over here, we have
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to multiply times 4.
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5 times 4 is 20.
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Now we have the same[br]denominator.
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So this is going to be equal[br]to, our denominator is 48.
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And so we can add 15 plus[br]20, which is 35.
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And can we reduce this?
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Let's see, 5 does[br]not go into 48.
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7 does not go into 48.
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It looks like we're all done.
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Let's do e right there.
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8/25 plus 7 over 10.
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Once again, we don't have[br]a common denominator.
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But we can solve that.
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Let's make, let's see, 50 is the[br]smallest number that both
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of these go into.
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25 times 2, so that's 50.
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8 over 25, to go to 50[br]we multiply by 2.
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So the 8, we're going to[br]have to multiply by 2.
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So it's going to[br]be 16 over 50.
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And then the 7 over 10,[br]we're going to want
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to put it over 50.
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We multiply the 10 times[br]5, so we have to
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multiply the 7 times 5.
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So it's going to[br]be 35 over 50.
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Now that our denominators are[br]the same, we have it over 50.
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16 plus 35, what is that?
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10 plus 35 is 45,[br]plus 6 is 51.
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So it is 51 over 50.
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Problem g.
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Let me do it in a new color.
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Problem g.
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So here we have 7 over 15-- I'll[br]write the second one in a
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different color--[br]plus 2 over 9.
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Once again, the denominators[br]are different.
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Find a common denominator.
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What is the smallest number that[br]both 15 and 9 go into?
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Let's see, 15 times 2 is 30.
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Nope, not divisible by 9.
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15 times 3 is 45, that works.
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45 is divisible by 9.
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So we use 45.
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15 times 3 is 45, so[br]7 times 3 is 21.
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These two fractions[br]are equivalent.
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Plus we're going over 45.
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To get from 9 to 45, we have[br]to multiply times 5.
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So to get our numerator[br]over here, we have to
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multiply it times 5.
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So 2 times 5 is 10.
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2/9 is the same thing[br]as 10/45.
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So now we can add.
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We're adding fractions of 45.
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21 plus 10 is 31,[br]and we are done.
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Let's do one more problem down[br]here, a word problem.
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Nadia, Peter and Ian are pooling[br]their money to buy a
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gallon of ice cream.
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Nadia's the oldest and gets[br]the greatest allowance.
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She contributes 1/2 the cost.[br]So Nadia is contributing 1/2
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the cost. So that is[br]Nadia right there.
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Ian is next oldest and[br]contributes 1/3 of the cost.
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So Ian contributes 1/3.
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That is Ian.
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Peter, the youngest, gets the[br]smallest allowance and
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contributes 1/4 of the cost.[br]So Peter gives 1/4 of the
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cost. Peter contributes[br]1/4 of cost.
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They figure that this will[br]be enough money.
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When they get to the checkout,[br]they realize that they forgot
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about sales tax and[br]worry there will
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not be enough money.
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Amazingly, they have exactly[br]the right amount of money.
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What fraction of the cost of[br]ice cream was added as tax?
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Well, let's see, if we add 1/2[br]plus 1/3, plus 1/4 of the
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cost, let's see what we get.
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So we have to find a common[br]denominator, some number that
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is the least common multiple[br]of 2, 3, and 4.
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And let's see, 4, it would[br]have to be 12, right?
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12 is divisible by 2, it's[br]divisible by 3, and it's
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divisible by 4.
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So 1/2 is the same[br]thing as 6/12.
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2 times 6 is 12.
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1 times 6 is 6.
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These are equivalent.
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6 is 1/2 of 12.
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1/3, if we use 12 as a common[br]denominator, to go from 3 to
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12 you have to multiply by 4.
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So you take that 4 and[br]you multiply it by 1.
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4/12 is the same thing as 1/3.
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And then 1/4, if you use your[br]denominator 12, to go from 4
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to 12 you have to multiply by[br]3, so multiply the numerator
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by 3 as well, you get 3.
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So let's add these.
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So 6/12 plus 4/12, plus 3/12 is[br]going to be equal to-- our
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denominator's going to be 12--[br]it's going to be 6 plus 4,
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plus 3, which is equal to 6 plus[br]4 is 10, plus 3 is 13.
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So it's going to be[br]equal to 13/12.
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And this is as an improper[br]fraction.
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Or we could say that this is the[br]same thing, this is equal
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to 12/12 plus 1/12, or we could[br]say the same thing as
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12/12 is just 1, right?
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12 divided by 12 is 1.
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So this is 1 and 1/12.
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So when they pool their money,[br]they get 1 and 1/12 of the
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price of the ice cream that[br]they want to buy.
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So they say what fraction of[br]the cost of ice cream was
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added as tax?
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This is the exact amount that[br]they needed to pay.
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So clearly, 1 is the non-taxed[br]price of the ice cream, so
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this 1/12 was the amount[br]added as tax.
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So the answer to the question[br]is 1/12 of the price
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was added as tax.
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