[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.82,0:00:05.49,Default,,0000,0000,0000,,In this session we're going to\Nbe having a look at simple equations
Dialogue: 0,0:00:05.49,0:00:09.38,Default,,0000,0000,0000,,in one variable and\Nthe equations will be linear.
Dialogue: 0,0:00:09.38,0:00:11.71,Default,,0000,0000,0000,,That means that there are no x-squared\Nterms
Dialogue: 0,0:00:11.71,0:00:18.37,Default,,0000,0000,0000,,and no x-cube terms, just\Nx's and numbers. So let's have a
Dialogue: 0,0:00:18.37,0:00:25.29,Default,,0000,0000,0000,,look at the first one. \N3x plus 15 equals x plus 25.
Dialogue: 0,0:00:25.29,0:00:29.90,Default,,0000,0000,0000,,An important thing to remember\Nabout any equation is this equal
Dialogue: 0,0:00:29.90,0:00:34.97,Default,,0000,0000,0000,,sign represents a balance. What\Nthat equal sign says is that
Dialogue: 0,0:00:34.97,0:00:40.96,Default,,0000,0000,0000,,what's on the left hand side is\Nexactly the same as what's on
Dialogue: 0,0:00:40.96,0:00:43.26,Default,,0000,0000,0000,,the right hand side.
Dialogue: 0,0:00:43.31,0:00:48.68,Default,,0000,0000,0000,,If we do anything to one side of\Nthe equation, we have to do it
Dialogue: 0,0:00:48.68,0:00:52.62,Default,,0000,0000,0000,,to the other side. If we don't,\Nthe balance is disturbed.
Dialogue: 0,0:00:53.37,0:00:55.11,Default,,0000,0000,0000,,If we can keep that in mind,
Dialogue: 0,0:00:55.87,0:00:59.74,Default,,0000,0000,0000,,then whatever operations we\Nperform on either side of the
Dialogue: 0,0:00:59.74,0:01:04.77,Default,,0000,0000,0000,,equation, so long as it's done\Nin exactly the same way on other side,
Dialogue: 0,0:01:04.77,0:01:06.71,Default,,0000,0000,0000,,we should be alright.
Dialogue: 0,0:01:07.52,0:01:12.25,Default,,0000,0000,0000,,Our first step in solving any\Nequation is to attempt to gather
Dialogue: 0,0:01:12.25,0:01:16.19,Default,,0000,0000,0000,,all the x terms together and all\Nthese free floating numbers together.
Dialogue: 0,0:01:16.19,0:01:22.10,Default,,0000,0000,0000,,To begin with we've\Ngot 3x on the left and and x on the right.
Dialogue: 0,0:01:22.10,0:01:28.40,Default,,0000,0000,0000,,If we take an x away from both sides,\Nwe take one x off the left,
Dialogue: 0,0:01:28.40,0:01:34.31,Default,,0000,0000,0000,,and one x off the right.\NThat will give us 2x plus 15 equals 25.
Dialogue: 0,0:01:34.31,0:01:38.25,Default,,0000,0000,0000,,We need to get the numbers\Ntogether. These free numbers.
Dialogue: 0,0:01:38.25,0:01:42.96,Default,,0000,0000,0000,,We can see that if we take 15 away\Nfrom the left, and 15 away from the right
Dialogue: 0,0:01:42.96,0:01:47.58,Default,,0000,0000,0000,,then we will have no numbers remaining \Non the left, just a 2x
Dialogue: 0,0:01:47.58,0:01:51.87,Default,,0000,0000,0000,,and taking 15 away on the right,\Nthat gives us 10.
Dialogue: 0,0:01:51.87,0:01:53.85,Default,,0000,0000,0000,,So, x must be equal to 5.
Dialogue: 0,0:01:54.63,0:01:57.48,Default,,0000,0000,0000,,That was relatively\Nstraightforward.
Dialogue: 0,0:01:57.48,0:02:01.96,Default,,0000,0000,0000,,Lots of plus signs, no minus signs, \Nwhich we know can be complicated,
Dialogue: 0,0:02:01.96,0:02:05.62,Default,,0000,0000,0000,,also no brackets. So let's\Nintroduce both of those.
Dialogue: 0,0:02:06.65,0:02:14.19,Default,,0000,0000,0000,,We will take 2x plus 3\Nis equal to
Dialogue: 0,0:02:14.19,0:02:17.21,Default,,0000,0000,0000,,6 minus bracket 2x minus three bracket.
Dialogue: 0,0:02:18.41,0:02:23.63,Default,,0000,0000,0000,,Now this right hand\Nside needs a little bit of dealing with.
Dialogue: 0,0:02:23.63,0:02:28.85,Default,,0000,0000,0000,,It needs getting\Ninto shape so we have to remove
Dialogue: 0,0:02:28.85,0:02:34.94,Default,,0000,0000,0000,,this bracket. \N2x plus 3 equals 6,
Dialogue: 0,0:02:34.94,0:02:39.72,Default,,0000,0000,0000,,now we take away 2x, thats minus 2x\Nand then we're taking away minus 3.
Dialogue: 0,0:02:39.72,0:02:44.51,Default,,0000,0000,0000,,When we take away minus 3,\Nthat makes it plus 3.
Dialogue: 0,0:02:45.08,0:02:49.72,Default,,0000,0000,0000,,Before we go any further, we\Nneed to tidy this right hand
Dialogue: 0,0:02:49.72,0:02:55.92,Default,,0000,0000,0000,,side up a little bit more \N2x plus 3 is equal to 9 minus 2x
Dialogue: 0,0:02:55.92,0:03:00.95,Default,,0000,0000,0000,,and now we're in the same\Nposition at this line as we were
Dialogue: 0,0:03:00.95,0:03:05.59,Default,,0000,0000,0000,,when we started there. So we\Nneed to get the xs together,
Dialogue: 0,0:03:05.59,0:03:11.40,Default,,0000,0000,0000,,which we can best do by adding\N2x to each side.
Dialogue: 0,0:03:11.40,0:03:15.27,Default,,0000,0000,0000,,On the right, minus 2x plus 2x. \NThis side gives us no x.
Dialogue: 0,0:03:15.29,0:03:22.24,Default,,0000,0000,0000,,And on the left it's going to\Ngive us 4X.
Dialogue: 0,0:03:22.24,0:03:29.18,Default,,0000,0000,0000,,4X plus 3 is equal to 9\Nbecause minus 2X plus 2X gives no x.
Dialogue: 0,0:03:29.18,0:03:35.20,Default,,0000,0000,0000,,Now I can take 3\Naway from each side, 4X equals 6
Dialogue: 0,0:03:35.20,0:03:42.14,Default,,0000,0000,0000,,and so x is 6 divided by 4 \Nand we want to write that in its
Dialogue: 0,0:03:42.14,0:03:46.77,Default,,0000,0000,0000,,lowest terms which is 3 over 2.\NPerfectly acceptable answer.
Dialogue: 0,0:03:46.78,0:03:51.45,Default,,0000,0000,0000,,Or one and a half. So either of these\Nare acceptable answers.
Dialogue: 0,0:03:51.45,0:03:56.12,Default,,0000,0000,0000,,This one (6 over 4) isn't because it's not in its\Nlowest terms, it must be reduced
Dialogue: 0,0:03:56.12,0:04:01.17,Default,,0000,0000,0000,,to its lowest terms. So either\Nof those two are OK as answers.
Dialogue: 0,0:04:01.75,0:04:06.04,Default,,0000,0000,0000,,With all of these equations, we\Nshould strictly check back by
Dialogue: 0,0:04:06.04,0:04:10.72,Default,,0000,0000,0000,,taking our answer and putting it\Ninto the first line of the
Dialogue: 0,0:04:10.72,0:04:15.79,Default,,0000,0000,0000,,equation and seeing if we get\Nthe right answer.
Dialogue: 0,0:04:15.79,0:04:22.42,Default,,0000,0000,0000,,So with this five, \N3 fives are 15, and 15 is 30. \N5 plus 25 is also 30
Dialogue: 0,0:04:22.42,0:04:26.71,Default,,0000,0000,0000,,The balance that we talked about at\Nthe beginning is maintained.
Dialogue: 0,0:04:26.71,0:04:31.78,Default,,0000,0000,0000,,If you take this number, 3 over 2,\Nor one and a half.
Dialogue: 0,0:04:31.80,0:04:35.41,Default,,0000,0000,0000,,And substitute it back into here \Nwe should find again that
Dialogue: 0,0:04:35.41,0:04:39.34,Default,,0000,0000,0000,,it gives us the right answer,\Nthat both sides give the same value.
Dialogue: 0,0:04:39.34,0:04:42.95,Default,,0000,0000,0000,,They balance. \NSolet's just do that.
Dialogue: 0,0:04:42.95,0:04:47.54,Default,,0000,0000,0000,,Let us take one and a half. \NSo 2 times one and a half is 3,
Dialogue: 0,0:04:48.13,0:04:53.52,Default,,0000,0000,0000,,plus three is 6, so we've got\Nsix on the left side.
Dialogue: 0,0:04:53.52,0:04:57.37,Default,,0000,0000,0000,,Here we have 6, takeaway, \Nnow let's deal with this bracket,
Dialogue: 0,0:04:57.37,0:05:01.60,Default,,0000,0000,0000,,just this bracket on its own, nothing else.\N2x takeaway, three, that's nothing,
Dialogue: 0,0:05:01.60,0:05:06.22,Default,,0000,0000,0000,,so we are just left with 6.\NWe calculated this side to be 6.
Dialogue: 0,0:05:06.22,0:05:11.23,Default,,0000,0000,0000,,Six equals six,\Nso again, we've got that
Dialogue: 0,0:05:11.23,0:05:15.08,Default,,0000,0000,0000,,balance, so we know that we've\Ngot the right answer.
Dialogue: 0,0:05:16.25,0:05:21.44,Default,,0000,0000,0000,,Let's carry on and have a look\Nat one or two questions with
Dialogue: 0,0:05:21.44,0:05:25.03,Default,,0000,0000,0000,,more brackets and this\Ntime numbers outside those
Dialogue: 0,0:05:25.03,0:05:30.22,Default,,0000,0000,0000,,brackets as well. So in a sense,\Nwhat we're doing is we are
Dialogue: 0,0:05:30.22,0:05:33.81,Default,,0000,0000,0000,,increasing the complexity of the\Nequation, but the simple
Dialogue: 0,0:05:33.81,0:05:38.99,Default,,0000,0000,0000,,principles that we've got so far\Nare going to help us out because
Dialogue: 0,0:05:38.99,0:05:42.98,Default,,0000,0000,0000,,no matter how complicated it\Ngets and this does look
Dialogue: 0,0:05:42.98,0:05:46.57,Default,,0000,0000,0000,,complicated, the same ideas work\Nall the time.
Dialogue: 0,0:05:46.60,0:05:52.32,Default,,0000,0000,0000,,We begin by multiplying out the\Nbrackets and taking care,
Dialogue: 0,0:05:52.32,0:05:58.56,Default,,0000,0000,0000,,in particular, with any minus signs\Nthat come up so
Dialogue: 0,0:05:58.56,0:06:05.32,Default,,0000,0000,0000,,8 times by x is 8x and eight times by\Nminus three is minus 24.
Dialogue: 0,0:06:05.32,0:06:09.48,Default,,0000,0000,0000,,We've multiplied everything\Ninside that bracket by what's
Dialogue: 0,0:06:09.48,0:06:16.24,Default,,0000,0000,0000,,outside, so we've 8 times by x\Nand we've 8 times by minus 3
Dialogue: 0,0:06:16.24,0:06:20.73,Default,,0000,0000,0000,,Now we remove this\Nbracket were taking away 6,
Dialogue: 0,0:06:20.73,0:06:24.28,Default,,0000,0000,0000,,that's minus six and we're\Ntaking away minus 2x,
Dialogue: 0,0:06:24.92,0:06:30.82,Default,,0000,0000,0000,,a minus minus gives us a\Nplus. So that's plus two X
Dialogue: 0,0:06:30.82,0:06:38.53,Default,,0000,0000,0000,,equals, two times x here,\Nis 2x, 2 times by 2 is 4.
Dialogue: 0,0:06:38.53,0:06:44.95,Default,,0000,0000,0000,,and now we have to multiply by\Nminus five, so we've got
Dialogue: 0,0:06:44.95,0:06:52.44,Default,,0000,0000,0000,,minus five times by 5. That's minus 25.\NAnd minus five times by minus x.
Dialogue: 0,0:06:52.44,0:06:54.58,Default,,0000,0000,0000,,So that's plus 5x.
Dialogue: 0,0:06:55.30,0:07:01.01,Default,,0000,0000,0000,,Each side needs tidying up. We\Nneed to look at this side and
Dialogue: 0,0:07:01.01,0:07:06.71,Default,,0000,0000,0000,,gather x terms and numbers together.\Nso with 8x plus 2x that's 10x.
Dialogue: 0,0:07:07.37,0:07:13.72,Default,,0000,0000,0000,,Take away 24, takeaway six, \Nthat's taking away 30 altogether.
Dialogue: 0,0:07:13.72,0:07:16.60,Default,,0000,0000,0000,,2x and 5x gives us 7x.
Dialogue: 0,0:07:17.37,0:07:22.69,Default,,0000,0000,0000,,Add on 4 takeaway 25. Now\Nthat's the equivalent of taking
Dialogue: 0,0:07:22.69,0:07:29.54,Default,,0000,0000,0000,,away 21. Let's get the xs together.\NWe can take 7x away from each side
Dialogue: 0,0:07:29.54,0:07:35.33,Default,,0000,0000,0000,,so we would have 3x minus 30 there\Nequals just
Dialogue: 0,0:07:35.33,0:07:38.00,Default,,0000,0000,0000,,minus 21 because we've taken that\N7x away.
Dialogue: 0,0:07:38.00,0:07:44.31,Default,,0000,0000,0000,,Add the 30 to each side,\Nbecause minus 30 + 30 gives us
Dialogue: 0,0:07:44.31,0:07:50.82,Default,,0000,0000,0000,,nothing. And add it on over\Nhere. So we get three x equals,
Dialogue: 0,0:07:50.82,0:07:57.30,Default,,0000,0000,0000,,30 added to minus 21, just\Nthe same as 30 takeaway 21,
Dialogue: 0,0:07:57.30,0:07:59.15,Default,,0000,0000,0000,,the answer is 9.
Dialogue: 0,0:07:59.17,0:08:03.30,Default,,0000,0000,0000,,And three times by something\Nto give us 9 means the
Dialogue: 0,0:08:03.30,0:08:08.17,Default,,0000,0000,0000,,something, the x, must be 3.\NAgain we ought to be able
Dialogue: 0,0:08:08.17,0:08:12.67,Default,,0000,0000,0000,,to take that 3, put it back\Ninto this line and see that
Dialogue: 0,0:08:12.67,0:08:17.54,Default,,0000,0000,0000,,in fact we've got the correct\Nanswer. So let's just try.
Dialogue: 0,0:08:17.54,0:08:22.42,Default,,0000,0000,0000,,3 minus 3 is zero.\N8 times by 0 is still 0,
Dialogue: 0,0:08:22.42,0:08:23.54,Default,,0000,0000,0000,,so we can forget about that term.
Dialogue: 0,0:08:25.09,0:08:29.64,Default,,0000,0000,0000,,2 times by 3three is 6, so six\Ntakeaway six is nothing, so
Dialogue: 0,0:08:29.64,0:08:33.14,Default,,0000,0000,0000,,we've actually got nothing\Nthere, zero, and nothing there, zero.
Dialogue: 0,0:08:33.14,0:08:37.34,Default,,0000,0000,0000,,so there's nothing on the left side of the equation.
Dialogue: 0,0:08:37.34,0:08:42.59,Default,,0000,0000,0000,,The right hand side should come out to\N0 as well. Let's see that it does.
Dialogue: 0,0:08:42.59,0:08:48.19,Default,,0000,0000,0000,,3 plus 2 is 5 and 2 fives are 10.\N5 takeaway 3 is 2,
Dialogue: 0,0:08:48.19,0:08:53.09,Default,,0000,0000,0000,,and 5 twos are 10. \NSo we've got 10 takeaway 10,
Dialogue: 0,0:08:53.09,0:08:55.19,Default,,0000,0000,0000,,this gives us nothing again, \Nso we've got nothing on the right.
Dialogue: 0,0:08:55.19,0:09:00.33,Default,,0000,0000,0000,,The equation balances with this value,
Dialogue: 0,0:09:00.33,0:09:03.08,Default,,0000,0000,0000,,so this is our one and only solution.
Dialogue: 0,0:09:03.08,0:09:10.30,Default,,0000,0000,0000,,We take a final\Nexample of this kind with
Dialogue: 0,0:09:10.30,0:09:15.82,Default,,0000,0000,0000,,x plus 1 times by\N2x plus 1.
Dialogue: 0,0:09:16.37,0:09:23.95,Default,,0000,0000,0000,,Equals x plus 3\Ntimes by 2x plus 3
Dialogue: 0,0:09:23.95,0:09:26.80,Default,,0000,0000,0000,,- 14.
Dialogue: 0,0:09:27.59,0:09:31.87,Default,,0000,0000,0000,,Now, I did say that these were\Nlinear equations that there
Dialogue: 0,0:09:31.87,0:09:34.59,Default,,0000,0000,0000,,would be no x-squared terms in them.
Dialogue: 0,0:09:34.59,0:09:38.32,Default,,0000,0000,0000,,But when we start\Nto multiply out these
Dialogue: 0,0:09:38.32,0:09:41.61,Default,,0000,0000,0000,,brackets, we are going to\Nget some x-squared terms.
Dialogue: 0,0:09:42.99,0:09:49.15,Default,,0000,0000,0000,,Let's have a look, to show you what\Nactually does happen.
Dialogue: 0,0:09:49.15,0:09:52.23,Default,,0000,0000,0000,,We do x times by two x. \NThat's two x-squared.
Dialogue: 0,0:09:52.23,0:09:57.88,Default,,0000,0000,0000,,x times by one, that is plus x.
Dialogue: 0,0:09:58.53,0:10:01.78,Default,,0000,0000,0000,,One times two x, that is\Nplus 2x.
Dialogue: 0,0:10:01.78,0:10:05.72,Default,,0000,0000,0000,,1 times by 1
Dialogue: 0,0:10:05.72,0:10:08.81,Default,,0000,0000,0000,,+ 1. \NEquals...
Dialogue: 0,0:10:10.11,0:10:13.93,Default,,0000,0000,0000,,x times by two x,\Nthat's two x-squared.
Dialogue: 0,0:10:15.79,0:10:19.50,Default,,0000,0000,0000,,x times by three is 3x..
Dialogue: 0,0:10:20.19,0:10:23.64,Default,,0000,0000,0000,,3 times by 2x is 6x.
Dialogue: 0,0:10:24.49,0:10:30.00,Default,,0000,0000,0000,,3 times by 3 is 9 \Nand finally takeaway 14.
Dialogue: 0,0:10:30.00,0:10:35.01,Default,,0000,0000,0000,,Now we need to tidy up\Nboth sides here.
Dialogue: 0,0:10:35.87,0:10:42.71,Default,,0000,0000,0000,,Two x-squared plus 3x\Nplus one equals
Dialogue: 0,0:10:42.71,0:10:48.18,Default,,0000,0000,0000,,2 x-squared plus 9x, \N(taking these two terms together)
Dialogue: 0,0:10:48.82,0:10:52.45,Default,,0000,0000,0000,,Plus 9, minus 14.
Dialogue: 0,0:10:54.51,0:10:58.71,Default,,0000,0000,0000,,And this needs a little bit more\Ntidying, but before we do that,
Dialogue: 0,0:10:58.71,0:11:03.23,Default,,0000,0000,0000,,let's just have a look at what\Nwe've got.
Dialogue: 0,0:11:03.23,0:11:06.78,Default,,0000,0000,0000,,We've got 2 x-squared there, and \N2 x-squared there. They are both
Dialogue: 0,0:11:06.78,0:11:10.66,Default,,0000,0000,0000,,positive, so I can take two x-sqaured\Naway from both sides.
Dialogue: 0,0:11:10.66,0:11:14.21,Default,,0000,0000,0000,,That means that the two \Nx-squared vanishes from both
Dialogue: 0,0:11:14.21,0:11:18.09,Default,,0000,0000,0000,,sides, so let's do that. Take\Ntwo x-squared from each side,
Dialogue: 0,0:11:18.09,0:11:22.93,Default,,0000,0000,0000,,and that leaves us 3x plus one,\Nis equal to 9x, and now we can
Dialogue: 0,0:11:22.93,0:11:24.87,Default,,0000,0000,0000,,do this bit 9 takeaway 14,
Dialogue: 0,0:11:24.94,0:11:29.42,Default,,0000,0000,0000,,Well, that's going to be\Ntakeaway five. I still have the
Dialogue: 0,0:11:29.42,0:11:34.71,Default,,0000,0000,0000,,five to subtract. Now we're back\Nto where we used to be getting
Dialogue: 0,0:11:34.71,0:11:38.37,Default,,0000,0000,0000,,the xs together, getting the\Nnumbers together. We can
Dialogue: 0,0:11:38.37,0:11:43.66,Default,,0000,0000,0000,,subtract 3x from both sides, so\Nthat gives me one equals 6x minus 5
Dialogue: 0,0:11:43.66,0:11:49.36,Default,,0000,0000,0000,,We can add the five\Nto both sides so that
Dialogue: 0,0:11:49.36,0:11:52.62,Default,,0000,0000,0000,,6 equals 6X, and so 1 is equal to x.
Dialogue: 0,0:11:53.81,0:11:58.12,Default,,0000,0000,0000,,And again, we can check that\Nthis works. We can substitute it back in.
Dialogue: 0,0:11:58.12,0:12:03.86,Default,,0000,0000,0000,,1 plus 1 is 2,\Ntwo ones are two, and one is 3.
Dialogue: 0,0:12:03.86,0:12:08.53,Default,,0000,0000,0000,,So effectively we've got 3 times\Nby two at this side, which gives 6.
Dialogue: 0,0:12:08.53,0:12:16.04,Default,,0000,0000,0000,,Here 1 + 3 is 4\Ntwo times by one is 2, + 3 is 5
Dialogue: 0,0:12:16.04,0:12:21.39,Default,,0000,0000,0000,,so we've got four times by 5.\NSo altogether there there's 20.
Dialogue: 0,0:12:22.10,0:12:26.94,Default,,0000,0000,0000,,Takeaway 14 is again 6, so\Nagain, this equation is balanced
Dialogue: 0,0:12:26.94,0:12:31.34,Default,,0000,0000,0000,,exactly when we take x to be\Nequal to 1.
Dialogue: 0,0:12:32.41,0:12:36.70,Default,,0000,0000,0000,,Now, those equations that we've\Njust looked at have really been
Dialogue: 0,0:12:36.70,0:12:39.82,Default,,0000,0000,0000,,about whole numbers. The\Ncoefficients have been whole
Dialogue: 0,0:12:39.82,0:12:43.33,Default,,0000,0000,0000,,numbers. Everything's been in\Nterms of integers. What happens
Dialogue: 0,0:12:43.33,0:12:47.23,Default,,0000,0000,0000,,when we start to get some\Nfractions in there? Some
Dialogue: 0,0:12:47.23,0:12:51.91,Default,,0000,0000,0000,,rational numbers? How do we deal\Nwith that? So again, we're going
Dialogue: 0,0:12:51.91,0:12:56.59,Default,,0000,0000,0000,,to add in more complexity, but\Nagain, the rules are the same.
Dialogue: 0,0:12:56.59,0:13:02.44,Default,,0000,0000,0000,,What we do to one side of the\Nequation we must do to the other
Dialogue: 0,0:13:02.44,0:13:04.78,Default,,0000,0000,0000,,side in order to preserve that
Dialogue: 0,0:13:04.78,0:13:10.93,Default,,0000,0000,0000,,particular balance. So let's\Nintroduce some fractions
Dialogue: 0,0:13:10.93,0:13:14.07,Default,,0000,0000,0000,,along with some brackets.
Dialogue: 0,0:13:14.07,0:13:22.05,Default,,0000,0000,0000,,4 bracket X plus 2, all over 5\Nis equal to
Dialogue: 0,0:13:22.05,0:13:25.47,Default,,0000,0000,0000,,7 plus 5x over 13.
Dialogue: 0,0:13:26.18,0:13:28.95,Default,,0000,0000,0000,,We've got some fractions here.
Dialogue: 0,0:13:29.79,0:13:35.40,Default,,0000,0000,0000,,Numbers in the denominator 5 and\N30. We want rid of those we want
Dialogue: 0,0:13:35.40,0:13:40.22,Default,,0000,0000,0000,,to be able to work with whole\Nnumbers with integers, so we
Dialogue: 0,0:13:40.22,0:13:45.83,Default,,0000,0000,0000,,have to find a way of getting\Nrid of them. That means we
Dialogue: 0,0:13:45.83,0:13:50.24,Default,,0000,0000,0000,,have to multiply everything\Nbecause what we do to one side,
Dialogue: 0,0:13:50.24,0:13:55.05,Default,,0000,0000,0000,,we must do to the other.
Dialogue: 0,0:13:55.05,0:13:59.86,Default,,0000,0000,0000,,The common denominator for five and\N13 is 65, that's five times by 13.
Dialogue: 0,0:13:59.86,0:14:05.08,Default,,0000,0000,0000,,So let's do that. Let's\Nmultiply everything by 65 and
Dialogue: 0,0:14:05.08,0:14:10.78,Default,,0000,0000,0000,,I'll write it down in full so\Nthat we can see it happening.
Dialogue: 0,0:14:10.78,0:14:17.29,Default,,0000,0000,0000,,we have 65 times 4, brackets x+2 over 5.\NThen we have 65 times by 7.
Dialogue: 0,0:14:17.29,0:14:22.17,Default,,0000,0000,0000,,Remember, I said we have to\Nmultiply everything, so it's not
Dialogue: 0,0:14:22.17,0:14:26.65,Default,,0000,0000,0000,,just these fraction bits, it's\Nany spare numbers that there are
Dialogue: 0,0:14:26.65,0:14:27.87,Default,,0000,0000,0000,,around as well.
Dialogue: 0,0:14:28.39,0:14:34.55,Default,,0000,0000,0000,,Plus 65 times by\N5x over 30.
Dialogue: 0,0:14:35.26,0:14:41.07,Default,,0000,0000,0000,,Now let's look at each term and\Nmake it simpler. Tidy it up.
Dialogue: 0,0:14:41.07,0:14:46.88,Default,,0000,0000,0000,,For a start, five will divide\Ninto 65, so 5 into five goes one
Dialogue: 0,0:14:46.88,0:14:49.37,Default,,0000,0000,0000,,and five into 65 goes 13 times.
Dialogue: 0,0:14:50.11,0:14:57.95,Default,,0000,0000,0000,,And then four times by 13? \NWell, that's 52, so we have
Dialogue: 0,0:14:57.95,0:15:01.87,Default,,0000,0000,0000,,52 times by x +2 equals.
Dialogue: 0,0:15:02.79,0:15:07.42,Default,,0000,0000,0000,,That's in a nice familiar form\Nwhere used to that sort of
Dialogue: 0,0:15:07.42,0:15:11.28,Default,,0000,0000,0000,,former. We've arrived at it by\Nchoosing to multiply everything
Dialogue: 0,0:15:11.28,0:15:14.76,Default,,0000,0000,0000,,by this common denominator.\NLet's tidy this side up.
Dialogue: 0,0:15:15.49,0:15:21.25,Default,,0000,0000,0000,,7 times by 65. Well, that's\Npretty tall order. Let's try it.
Dialogue: 0,0:15:21.25,0:15:26.05,Default,,0000,0000,0000,,Seven 5s are 35. Seven 6s are 42 and\Nthree is 45.
Dialogue: 0,0:15:26.76,0:15:33.96,Default,,0000,0000,0000,,Here with 65 times 5 x over 13.\N13 goes into 13 once,
Dialogue: 0,0:15:33.96,0:15:41.15,Default,,0000,0000,0000,,and 13 goes into 65 five times.\NSo we five times by 5x is 25x.
Dialogue: 0,0:15:41.15,0:15:46.33,Default,,0000,0000,0000,,You may say 'what happened to\Nthese ones?'. Well if I divide by
Dialogue: 0,0:15:46.33,0:15:49.96,Default,,0000,0000,0000,,one it stays unchanged so I\Ndon't have to write them down.
Dialogue: 0,0:15:50.97,0:15:56.88,Default,,0000,0000,0000,,And now we left with an\Nequation that were used to
Dialogue: 0,0:15:56.88,0:16:01.44,Default,,0000,0000,0000,,handling. We've met these kind\Nbefore, so let's multiply out
Dialogue: 0,0:16:01.44,0:16:05.98,Default,,0000,0000,0000,,the brackets, get the xs\Ntogether and solve the equation.
Dialogue: 0,0:16:05.98,0:16:12.36,Default,,0000,0000,0000,,So we multiply out the brackets\N52 times 2 is 104
Dialogue: 0,0:16:12.36,0:16:19.18,Default,,0000,0000,0000,,equals 455 plus 25x. \NTake the 25x away from each side,
Dialogue: 0,0:16:19.18,0:16:21.46,Default,,0000,0000,0000,,gives me 27x there.
Dialogue: 0,0:16:21.47,0:16:27.51,Default,,0000,0000,0000,,And no x is there.\NTake the 104 away
Dialogue: 0,0:16:27.51,0:16:32.21,Default,,0000,0000,0000,,from each side,\Nwhich gives me 351.
Dialogue: 0,0:16:33.41,0:16:38.70,Default,,0000,0000,0000,,And now lots of big numbers,\Nreally, that shouldn't be a
Dialogue: 0,0:16:38.70,0:16:43.51,Default,,0000,0000,0000,,problem to us. 351 divided by 27 will\Ncertainly go once
Dialogue: 0,0:16:44.19,0:16:48.78,Default,,0000,0000,0000,,(there's one 27 in the 35 and eight\Nover and 27s into 8
Dialogue: 0,0:16:48.78,0:16:53.37,Default,,0000,0000,0000,,go three times,\Nso answer is x equals 13 and we
Dialogue: 0,0:16:53.37,0:16:57.31,Default,,0000,0000,0000,,should go back and check it\Nand make sure that it's right.
Dialogue: 0,0:16:57.31,0:16:58.62,Default,,0000,0000,0000,,So let's do that.
Dialogue: 0,0:16:59.46,0:17:02.82,Default,,0000,0000,0000,,13 + 2? Well, that's 15
Dialogue: 0,0:17:02.82,0:17:09.82,Default,,0000,0000,0000,,15 divided 5 is 3 and\Nnow times by 4, so that's 12.
Dialogue: 0,0:17:09.82,0:17:15.31,Default,,0000,0000,0000,,Hang on to that number 12 \Nat that side.
Dialogue: 0,0:17:15.31,0:17:20.40,Default,,0000,0000,0000,,5 times 13 divided by 13. \NSo the answer is just five
Dialogue: 0,0:17:20.40,0:17:25.89,Default,,0000,0000,0000,,plus the 7 is again 12. \NThe same as this side.
Dialogue: 0,0:17:25.89,0:17:29.42,Default,,0000,0000,0000,,So the answer is correct. \NIt balances.
Dialogue: 0,0:17:30.19,0:17:35.15,Default,,0000,0000,0000,,Let's practice that one again\Nand have a look at another example.
Dialogue: 0,0:17:35.15,0:17:39.09,Default,,0000,0000,0000,,We take X plus 5 over 6
Dialogue: 0,0:17:39.09,0:17:45.94,Default,,0000,0000,0000,,minus x plus one over 9
Dialogue: 0,0:17:46.44,0:17:50.26,Default,,0000,0000,0000,,is equal to x plus three over 4.
Dialogue: 0,0:17:50.26,0:17:53.84,Default,,0000,0000,0000,,We haven't got any brackets.
Dialogue: 0,0:17:53.84,0:17:58.08,Default,,0000,0000,0000,,Does that make any\Ndifference? The thing you have
Dialogue: 0,0:17:58.08,0:18:00.08,Default,,0000,0000,0000,,to remember is that this line.
Dialogue: 0,0:18:00.93,0:18:04.70,Default,,0000,0000,0000,,Not only acts as a division\Nsign, but it acts as a bracket.
Dialogue: 0,0:18:05.23,0:18:11.66,Default,,0000,0000,0000,,It means that all of x plus 5\Nis divided by 6.
Dialogue: 0,0:18:12.41,0:18:18.08,Default,,0000,0000,0000,,So it might be as well if we\Nkept that in mind and put
Dialogue: 0,0:18:18.08,0:18:22.54,Default,,0000,0000,0000,,brackets around these terms so\Nthat we're clear that
Dialogue: 0,0:18:22.54,0:18:27.40,Default,,0000,0000,0000,,we've written down that these\Nare to be kept together and are
Dialogue: 0,0:18:27.40,0:18:32.66,Default,,0000,0000,0000,,all divided by 6. These two are\Nto be kept together and all
Dialogue: 0,0:18:32.66,0:18:37.12,Default,,0000,0000,0000,,divided by 9. And similarly here\Nthey are all divided by 4.
Dialogue: 0,0:18:37.97,0:18:42.80,Default,,0000,0000,0000,,Next step we need a common\Ndenominator. We need a number
Dialogue: 0,0:18:42.80,0:18:47.63,Default,,0000,0000,0000,,into which all of these will\Ndivide exactly. Now we could
Dialogue: 0,0:18:47.63,0:18:50.70,Default,,0000,0000,0000,,multiply them altogether and\Nwe'd be certain.
Dialogue: 0,0:18:51.25,0:18:57.58,Default,,0000,0000,0000,,But the arithmetic would be\Nhorrendous. 6 times 9 times 4 is very big.
Dialogue: 0,0:18:57.58,0:19:03.42,Default,,0000,0000,0000,,Can we find a smaller\Nnumber into which six, nine, and
Dialogue: 0,0:19:03.42,0:19:05.37,Default,,0000,0000,0000,,four will all divide?
Dialogue: 0,0:19:06.01,0:19:12.41,Default,,0000,0000,0000,,Well, a candidate for that is 36.\N36 will divide by 6l
Dialogue: 0,0:19:12.41,0:19:18.80,Default,,0000,0000,0000,,36 will divide by 9 and 36 will divide\Nby 4, so lets multiply
Dialogue: 0,0:19:18.80,0:19:25.73,Default,,0000,0000,0000,,throughout by that number 36. \NWe have 36, because we've put the
Dialogue: 0,0:19:25.73,0:19:30.53,Default,,0000,0000,0000,,brackets in we are quite clear\Nthat we're multiplying
Dialogue: 0,0:19:30.53,0:19:37.46,Default,,0000,0000,0000,,everything over that 6 by the 36.\NMinus 36 times x plus one.
Dialogue: 0,0:19:37.48,0:19:44.35,Default,,0000,0000,0000,,All over 9 equals 36 times by x,\Nplus three all over 4, so we
Dialogue: 0,0:19:44.35,0:19:49.85,Default,,0000,0000,0000,,made it quite clear by using the\Nbrackets what this 36 is
Dialogue: 0,0:19:49.85,0:19:57.29,Default,,0000,0000,0000,,multiplying . 6 into six goes\Nonce and six into 36 goes 6 times.
Dialogue: 0,0:19:57.29,0:20:01.23,Default,,0000,0000,0000,,9 into nine goes once and 9 into 36\Ngoes 4.
Dialogue: 0,0:20:01.23,0:20:08.43,Default,,0000,0000,0000,,4 into 4 goes\Nonce and four into 36 goes 9.
Dialogue: 0,0:20:09.16,0:20:14.71,Default,,0000,0000,0000,,So now I have this bracket to\Nmultiply by 6 this bracket to
Dialogue: 0,0:20:14.71,0:20:20.26,Default,,0000,0000,0000,,multiply by 4 and this bracket\Nto multiply by 9. I don't have
Dialogue: 0,0:20:20.26,0:20:25.39,Default,,0000,0000,0000,,to worry about the ones because\NI'm dividing by them so they
Dialogue: 0,0:20:25.39,0:20:29.66,Default,,0000,0000,0000,,leave everything unchanged. So\Nlet's multiply out six times by
Dialogue: 0,0:20:29.66,0:20:35.63,Default,,0000,0000,0000,,6X plus 30 (6 times by 5).\NThis is a minus four I'm
Dialogue: 0,0:20:35.63,0:20:38.62,Default,,0000,0000,0000,,multiplied by, so I need to be a
Dialogue: 0,0:20:38.62,0:20:45.33,Default,,0000,0000,0000,,bit careful. Minus four times x\Nis minus 4x,
Dialogue: 0,0:20:45.33,0:20:48.24,Default,,0000,0000,0000,,Minus 4 times 1 is minus 4
Dialogue: 0,0:20:48.87,0:20:52.10,Default,,0000,0000,0000,,Equals 9X\Nand 9 threes
Dialogue: 0,0:20:52.10,0:20:59.31,Default,,0000,0000,0000,,are 27. So now\Nwe need to simplify this side
Dialogue: 0,0:20:59.31,0:21:06.39,Default,,0000,0000,0000,,6x takeaway 4x. That's just two\NX30 takeaway, four is 26, and
Dialogue: 0,0:21:06.39,0:21:09.93,Default,,0000,0000,0000,,that's equal to 9X plus 27.
Dialogue: 0,0:21:11.15,0:21:17.49,Default,,0000,0000,0000,,Let me take 2X away from each\Nside, so I have 26 equals 7X
Dialogue: 0,0:21:17.49,0:21:23.38,Default,,0000,0000,0000,,plus 27 and now I'll take the\NSeven away from each side and
Dialogue: 0,0:21:23.38,0:21:30.18,Default,,0000,0000,0000,,I'll have minus one is equal to\N7X and so now I need to divide
Dialogue: 0,0:21:30.18,0:21:36.52,Default,,0000,0000,0000,,both sides by 7 and so I get\Nminus 7th for my answer. Don't
Dialogue: 0,0:21:36.52,0:21:41.05,Default,,0000,0000,0000,,worry that this is a fraction,\Nsometimes they workout like
Dialogue: 0,0:21:41.05,0:21:44.13,Default,,0000,0000,0000,,that. Don't worry, that is the\Nnegative number. Sometimes they
Dialogue: 0,0:21:44.13,0:21:47.62,Default,,0000,0000,0000,,workout like that. Let's have a\Nlook at another one 'cause this
Dialogue: 0,0:21:47.62,0:21:51.70,Default,,0000,0000,0000,,is a process that you're going\Nto have to be able to do quite
Dialogue: 0,0:21:51.70,0:21:58.24,Default,,0000,0000,0000,,complicated questions. So we'll\Ntake 4 - 5 X.
Dialogue: 0,0:21:58.76,0:22:00.01,Default,,0000,0000,0000,,All over 6.
Dialogue: 0,0:22:00.85,0:22:04.66,Default,,0000,0000,0000,,Minus 1 - 2 X all over
Dialogue: 0,0:22:04.66,0:22:10.07,Default,,0000,0000,0000,,3. Equals\N13 over 42.
Dialogue: 0,0:22:11.17,0:22:16.45,Default,,0000,0000,0000,,What are we going to do? First\Nof all, let's remind ourselves
Dialogue: 0,0:22:16.45,0:22:17.77,Default,,0000,0000,0000,,that this line.
Dialogue: 0,0:22:19.15,0:22:21.04,Default,,0000,0000,0000,,Not only means divide.
Dialogue: 0,0:22:21.76,0:22:28.80,Default,,0000,0000,0000,,Divide 4 - 5 X by 6 but it means\Ndivide all of 4 - 5 X by 6. So
Dialogue: 0,0:22:28.80,0:22:33.02,Default,,0000,0000,0000,,let's put it in a bracket to\Nremind ourselves and let's do
Dialogue: 0,0:22:33.02,0:22:34.08,Default,,0000,0000,0000,,the same there.
Dialogue: 0,0:22:34.81,0:22:40.15,Default,,0000,0000,0000,,Now we need a common\Ndenominator, six and three and
Dialogue: 0,0:22:40.15,0:22:47.63,Default,,0000,0000,0000,,40. Two, well, six goes into 42\Nand three goes into 42 as well.
Dialogue: 0,0:22:47.63,0:22:52.43,Default,,0000,0000,0000,,So let's choose 42 as our\Ndenominator, an multiply
Dialogue: 0,0:22:52.43,0:22:59.37,Default,,0000,0000,0000,,everything by 42. So will have\N42 * 4 - 5 X or
Dialogue: 0,0:22:59.37,0:23:05.78,Default,,0000,0000,0000,,over 6 - 42 * 1 -\N2 X all over 3.
Dialogue: 0,0:23:06.35,0:23:11.89,Default,,0000,0000,0000,,Equals 42 *\N13 over 42.
Dialogue: 0,0:23:14.10,0:23:19.44,Default,,0000,0000,0000,,Six goes into six once and six\Ngoes into 42 Seven times.
Dialogue: 0,0:23:20.40,0:23:26.67,Default,,0000,0000,0000,,Three goes into three once and\Nfree goes into 4214 times.
Dialogue: 0,0:23:27.26,0:23:30.91,Default,,0000,0000,0000,,42 goes into itself once and
Dialogue: 0,0:23:30.91,0:23:35.54,Default,,0000,0000,0000,,again once. So now I need to\Nmultiply out these brackets.
Dialogue: 0,0:23:36.22,0:23:37.86,Default,,0000,0000,0000,,And simplify this side.
Dialogue: 0,0:23:38.70,0:23:45.25,Default,,0000,0000,0000,,So 7 times by 4 gives us\N28 Seven times. My minus five
Dialogue: 0,0:23:45.25,0:23:52.31,Default,,0000,0000,0000,,gives us minus 35 X. Now here\Nwe have a minus sign and the
Dialogue: 0,0:23:52.31,0:23:59.36,Default,,0000,0000,0000,,14, so it's minus 14 times by\N1 - 14 and then it's minus
Dialogue: 0,0:23:59.36,0:24:05.92,Default,,0000,0000,0000,,14 times by minus 2X. So it's\Nplus 28X. Remember we've got to
Dialogue: 0,0:24:05.92,0:24:10.45,Default,,0000,0000,0000,,take extra care when we've got\Nthose minus signs.
Dialogue: 0,0:24:10.51,0:24:17.40,Default,,0000,0000,0000,,One times by 13 is just\N13. Now we need to tidy
Dialogue: 0,0:24:17.40,0:24:24.29,Default,,0000,0000,0000,,this side up 28 takeaway 14\Nis just 1435 - 35 X
Dialogue: 0,0:24:24.29,0:24:30.60,Default,,0000,0000,0000,,plus 28X. Or what's the\Ndifference there? It is 7 so
Dialogue: 0,0:24:30.60,0:24:37.49,Default,,0000,0000,0000,,it's minus Seven X equals 30.\NTake the 14 away from each
Dialogue: 0,0:24:37.49,0:24:40.93,Default,,0000,0000,0000,,side. We've minus Seven X equals
Dialogue: 0,0:24:40.93,0:24:46.66,Default,,0000,0000,0000,,minus one. And divide both\Nsides by minus 7 - 1 divided
Dialogue: 0,0:24:46.66,0:24:51.65,Default,,0000,0000,0000,,by minus Seven is just a 7th\Nagain fractional answer, but
Dialogue: 0,0:24:51.65,0:24:53.02,Default,,0000,0000,0000,,not to worry.
Dialogue: 0,0:24:54.40,0:25:01.14,Default,,0000,0000,0000,,When we looked at these, now we\Nwant to have a look at a type
Dialogue: 0,0:25:01.14,0:25:04.73,Default,,0000,0000,0000,,of equation which occasionally\Ncauses problems. This particular
Dialogue: 0,0:25:04.73,0:25:07.87,Default,,0000,0000,0000,,equation or kind of equation\Nlooks relatively
Dialogue: 0,0:25:07.87,0:25:12.32,Default,,0000,0000,0000,,straightforward. Translate\Nnumbers get rather more
Dialogue: 0,0:25:12.32,0:25:16.51,Default,,0000,0000,0000,,difficult. It can cause\Ndifficulties, so we got three
Dialogue: 0,0:25:16.51,0:25:19.10,Default,,0000,0000,0000,,over 5 equals 6 over X
Dialogue: 0,0:25:19.10,0:25:23.14,Default,,0000,0000,0000,,straightforward. But what do we\Ndo? Let's think about it. First
Dialogue: 0,0:25:23.14,0:25:27.03,Default,,0000,0000,0000,,of all, in terms of fractions,\Nthis is a fraction 3/5, and it
Dialogue: 0,0:25:27.03,0:25:28.52,Default,,0000,0000,0000,,is equal to another fraction
Dialogue: 0,0:25:28.52,0:25:34.14,Default,,0000,0000,0000,,which is 6. Well family\Nobviously 3/5 is the same
Dialogue: 0,0:25:34.14,0:25:40.54,Default,,0000,0000,0000,,fraction as 6/10 and so\Ntherefore X has got to be equal
Dialogue: 0,0:25:40.54,0:25:44.75,Default,,0000,0000,0000,,to 10. That's not going to\Nhappen with every question. It's
Dialogue: 0,0:25:44.75,0:25:46.77,Default,,0000,0000,0000,,not going to be as easy as that.
Dialogue: 0,0:25:47.42,0:25:51.64,Default,,0000,0000,0000,,We're going to have to juggle\Nwith the numbers, so how do we
Dialogue: 0,0:25:51.64,0:25:55.54,Default,,0000,0000,0000,,do that? Well, again, we need a\Ncommon denominator. We need a
Dialogue: 0,0:25:55.54,0:25:59.77,Default,,0000,0000,0000,,number that will be divisible by\N5 and a number that will be
Dialogue: 0,0:25:59.77,0:26:01.72,Default,,0000,0000,0000,,divisible by X. And the obvious
Dialogue: 0,0:26:01.72,0:26:04.88,Default,,0000,0000,0000,,choice is 5X. So let us
Dialogue: 0,0:26:04.88,0:26:12.25,Default,,0000,0000,0000,,multiply. Both sides by 5X.\NSo we 5X times by
Dialogue: 0,0:26:12.25,0:26:19.28,Default,,0000,0000,0000,,3/5 equals 5X times by\N6 over X and now
Dialogue: 0,0:26:19.28,0:26:26.31,Default,,0000,0000,0000,,five goes into five once\Nand five goes into five
Dialogue: 0,0:26:26.31,0:26:33.03,Default,,0000,0000,0000,,once there. X goes into X\Nonce an X goes into X. Once
Dialogue: 0,0:26:33.03,0:26:37.83,Default,,0000,0000,0000,,there, let's remember that we're\Nmultiplying by these, so we have
Dialogue: 0,0:26:37.83,0:26:43.52,Default,,0000,0000,0000,,one times X times three. That's\N3X and divided by one, so it's
Dialogue: 0,0:26:43.52,0:26:50.51,Default,,0000,0000,0000,,still three X equals 5 {\i1} 1 {\i0} 6,\Nwhich is just 30 and divided by
Dialogue: 0,0:26:50.51,0:26:52.69,Default,,0000,0000,0000,,one, so it's still 30.
Dialogue: 0,0:26:53.27,0:26:59.27,Default,,0000,0000,0000,,3X is equal to 30, so X must be\Nequal to 10, which is what we
Dialogue: 0,0:26:59.27,0:27:03.98,Default,,0000,0000,0000,,had before. Is there another way\Nof looking at this equation?
Dialogue: 0,0:27:03.98,0:27:05.33,Default,,0000,0000,0000,,Well, yes there is.
Dialogue: 0,0:27:05.84,0:27:13.23,Default,,0000,0000,0000,,If two fractions are equal that\Nway up there also equal the
Dialogue: 0,0:27:13.23,0:27:15.08,Default,,0000,0000,0000,,other way up.
Dialogue: 0,0:27:16.89,0:27:21.39,Default,,0000,0000,0000,,This makes it easier still\Nbecause all that we need to do
Dialogue: 0,0:27:21.39,0:27:25.89,Default,,0000,0000,0000,,now is multiply by the common\Ndenominator and we can see what
Dialogue: 0,0:27:25.89,0:27:29.64,Default,,0000,0000,0000,,that common denominator is. It's\Nquite clearly 6 because six
Dialogue: 0,0:27:29.64,0:27:32.64,Default,,0000,0000,0000,,divides into six and three\Ndivides into 6.
Dialogue: 0,0:27:32.65,0:27:39.75,Default,,0000,0000,0000,,So we had five over three is\Nequal to X over 6, and we're
Dialogue: 0,0:27:39.75,0:27:45.83,Default,,0000,0000,0000,,going to multiply by this common\Ndenominator of six. So six goes
Dialogue: 0,0:27:45.83,0:27:51.92,Default,,0000,0000,0000,,into six once on each occasion,\Nthree goes into three once and
Dialogue: 0,0:27:51.92,0:27:59.01,Default,,0000,0000,0000,,into six twice, so again X is\Nequal to 10 two times by 5,
Dialogue: 0,0:27:59.01,0:28:01.04,Default,,0000,0000,0000,,one times by X.
Dialogue: 0,0:28:01.61,0:28:03.51,Default,,0000,0000,0000,,Whichever way you use.
Dialogue: 0,0:28:04.54,0:28:08.77,Default,,0000,0000,0000,,Doesn't matter. They should come\Nout the same. There's no reason
Dialogue: 0,0:28:08.77,0:28:12.27,Default,,0000,0000,0000,,why they shouldn't, but you do\Nhave to be careful. The number
Dialogue: 0,0:28:12.27,0:28:15.78,Default,,0000,0000,0000,,work can be a bit tricky\Nsometimes. Let's have a look at
Dialogue: 0,0:28:15.78,0:28:16.94,Default,,0000,0000,0000,,just a couple more.
Dialogue: 0,0:28:17.48,0:28:23.67,Default,,0000,0000,0000,,Five over 3X\Nis equal to
Dialogue: 0,0:28:23.67,0:28:26.76,Default,,0000,0000,0000,,25 over 27.
Dialogue: 0,0:28:28.06,0:28:34.90,Default,,0000,0000,0000,,OK. What I think I'm going to\Ndo with these is flip them over
Dialogue: 0,0:28:34.90,0:28:41.04,Default,,0000,0000,0000,,3X over 5 is equal to 27 over\N25. I can see straight away.
Dialogue: 0,0:28:41.04,0:28:46.29,Default,,0000,0000,0000,,I've got a common denominator\Nhere of 25. Five goes into 25
Dialogue: 0,0:28:46.29,0:28:52.86,Default,,0000,0000,0000,,exactly and so does 25. So if I\Ndo that multiplication 25 * 3 X
Dialogue: 0,0:28:52.86,0:28:56.80,Default,,0000,0000,0000,,over 5 is equal to 25 * 27 over
Dialogue: 0,0:28:56.80,0:29:03.42,Default,,0000,0000,0000,,25. 25 goes into itself\Nonce on each occasion.
Dialogue: 0,0:29:04.75,0:29:10.14,Default,,0000,0000,0000,,Five goes into itself once and\Nfive goes into 25 five times. So
Dialogue: 0,0:29:10.14,0:29:17.20,Default,,0000,0000,0000,,I have 15X5 times by three. X is\Nequal to 27, and so X is 27 over
Dialogue: 0,0:29:17.20,0:29:22.18,Default,,0000,0000,0000,,15. Dividing both sides by 15\Nand there is here a common
Dialogue: 0,0:29:22.18,0:29:27.58,Default,,0000,0000,0000,,factor between top and bottom of\Nthree, which gives me 9 over 5,
Dialogue: 0,0:29:27.58,0:29:31.72,Default,,0000,0000,0000,,so that's an acceptable answer\Nbecause it's in its lowest
Dialogue: 0,0:29:31.72,0:29:34.63,Default,,0000,0000,0000,,forms. Or I could write it as
Dialogue: 0,0:29:34.63,0:29:38.93,Default,,0000,0000,0000,,one. And four fifths, which is\Nalso an acceptable answer.
Dialogue: 0,0:29:40.06,0:29:45.72,Default,,0000,0000,0000,,Now, some of you may not like\Nwhat I did there when I flipped
Dialogue: 0,0:29:45.72,0:29:51.37,Default,,0000,0000,0000,,it over, and we might want to\Nthink, well, how would I do it
Dialogue: 0,0:29:51.37,0:29:56.62,Default,,0000,0000,0000,,if I had to start from there. So\Nlet's tackle that in another
Dialogue: 0,0:29:56.62,0:30:02.68,Default,,0000,0000,0000,,way. So again we five over 3X is\Nequal to 25 over 27 this time.
Dialogue: 0,0:30:03.22,0:30:08.16,Default,,0000,0000,0000,,We're not going to flip it over.\NLet's look for a common
Dialogue: 0,0:30:08.16,0:30:12.28,Default,,0000,0000,0000,,denominator here between these\Ntwo, so we want something that
Dialogue: 0,0:30:12.28,0:30:16.82,Default,,0000,0000,0000,,3X will divide into exactly, and\Nsomething that 27 will divide
Dialogue: 0,0:30:16.82,0:30:22.17,Default,,0000,0000,0000,,into exactly, well. Three will\Ndivide into 27, so the 27, so to
Dialogue: 0,0:30:22.17,0:30:28.35,Default,,0000,0000,0000,,speak ought to be a part of our\Nanswer. What we need is an X,
Dialogue: 0,0:30:28.35,0:30:34.53,Default,,0000,0000,0000,,because if we had 27 X, 3X would\Ndivide into it 9 times and the
Dialogue: 0,0:30:34.53,0:30:37.00,Default,,0000,0000,0000,,27 would just divide into it.
Dialogue: 0,0:30:37.09,0:30:43.84,Default,,0000,0000,0000,,X times, so that's going to\Nbe our common denominator. The
Dialogue: 0,0:30:43.84,0:30:51.21,Default,,0000,0000,0000,,thing that we are going to\Nmultiply both sides by 27 X.
Dialogue: 0,0:30:51.40,0:30:58.42,Default,,0000,0000,0000,,So we can look at this and we\Ncan see that X divides into X
Dialogue: 0,0:30:58.42,0:31:04.97,Default,,0000,0000,0000,,one St X device into X. Once\Nthere we can also see that three
Dialogue: 0,0:31:04.97,0:31:10.12,Default,,0000,0000,0000,,divides into three and three\Ndivides into 27 nine times and
Dialogue: 0,0:31:10.12,0:31:17.14,Default,,0000,0000,0000,,over here 27 goes into 27 once\Neach time. So I have 9 * 1
Dialogue: 0,0:31:17.14,0:31:19.95,Default,,0000,0000,0000,,* 5 and 95 S 45.
Dialogue: 0,0:31:20.75,0:31:27.36,Default,,0000,0000,0000,,Divided by 1 * 1, which is one,\Nso it's still 45 equals 1 times
Dialogue: 0,0:31:27.36,0:31:35.08,Default,,0000,0000,0000,,by 25. And times by the X\Nthere 25 X. Now I need to divide
Dialogue: 0,0:31:35.08,0:31:40.82,Default,,0000,0000,0000,,both sides by 25 so I have 45\Nover 25 equals X.
Dialogue: 0,0:31:41.53,0:31:47.01,Default,,0000,0000,0000,,This is not in its lowest\Nterms. I can divide top on
Dialogue: 0,0:31:47.01,0:31:52.96,Default,,0000,0000,0000,,bottom by 5, giving me 9 over\N5 again, which again I can
Dialogue: 0,0:31:52.96,0:31:55.70,Default,,0000,0000,0000,,write as one and four fifths.
Dialogue: 0,0:31:57.84,0:32:00.84,Default,,0000,0000,0000,,Let's take one final example.
Dialogue: 0,0:32:01.44,0:32:06.13,Default,,0000,0000,0000,,This time, let's look at some\Nfractions, but this time,
Dialogue: 0,0:32:06.13,0:32:11.29,Default,,0000,0000,0000,,mysteriously, the X is already\Non the top. That's really good
Dialogue: 0,0:32:11.29,0:32:17.39,Default,,0000,0000,0000,,for us. All we need to do is\Nlook at what's our common
Dialogue: 0,0:32:17.39,0:32:18.79,Default,,0000,0000,0000,,denominator, well, 7.
Dialogue: 0,0:32:19.96,0:32:26.62,Default,,0000,0000,0000,,And 49 what number divides\Nexactly by both of these and it
Dialogue: 0,0:32:26.62,0:32:34.39,Default,,0000,0000,0000,,will be 49. So we just need\Nto do 49 times by 19 X
Dialogue: 0,0:32:34.39,0:32:41.60,Default,,0000,0000,0000,,over 7 equals 49 * 57 over\N4949 goes into 49 once each
Dialogue: 0,0:32:41.60,0:32:46.78,Default,,0000,0000,0000,,time. And Seven goes into\N49 Seven times.
Dialogue: 0,0:32:47.38,0:32:53.23,Default,,0000,0000,0000,,And so I have 7 times by 19 X\Nequals 57 and you might say,
Dialogue: 0,0:32:53.23,0:32:56.74,Default,,0000,0000,0000,,well, hang on a minute,\Nshouldn't you have multiplied
Dialogue: 0,0:32:56.74,0:33:01.81,Default,,0000,0000,0000,,out that first? Well, I didn't\Nwant to. Why didn't I want to?
Dialogue: 0,0:33:01.81,0:33:05.32,Default,,0000,0000,0000,,Well, sometimes when you play\Ndarts, your arithmetic improves
Dialogue: 0,0:33:05.32,0:33:10.78,Default,,0000,0000,0000,,and triple 19 on a dartboard is\N57. So 19 divides into 57 three
Dialogue: 0,0:33:10.78,0:33:15.46,Default,,0000,0000,0000,,times, and I don't want to lose\Nthat relationship. So I'm going
Dialogue: 0,0:33:15.46,0:33:18.58,Default,,0000,0000,0000,,to divide each side by 19 so 7X.
Dialogue: 0,0:33:18.65,0:33:24.30,Default,,0000,0000,0000,,Is equal to three, which means X\Nmust be 3 over 7.
Dialogue: 0,0:33:24.98,0:33:28.47,Default,,0000,0000,0000,,Playing darts does help with\Narithmetic. We finished there
Dialogue: 0,0:33:28.47,0:33:31.96,Default,,0000,0000,0000,,with simple linear equations.\NThe important thing in dealing
Dialogue: 0,0:33:31.96,0:33:36.62,Default,,0000,0000,0000,,with these kinds of equations\Nand any kind of equation is to
Dialogue: 0,0:33:36.62,0:33:41.66,Default,,0000,0000,0000,,remember that the equal sign is\Na balance. What it tells you is
Dialogue: 0,0:33:41.66,0:33:46.32,Default,,0000,0000,0000,,that what's on the left hand\Nside is exactly equal to what's
Dialogue: 0,0:33:46.32,0:33:51.36,Default,,0000,0000,0000,,on the right hand side. So\Nwhatever you do to one side, you
Dialogue: 0,0:33:51.36,0:33:55.63,Default,,0000,0000,0000,,have to do to the other side,\Nand you must follow.
Dialogue: 0,0:33:55.65,0:33:58.67,Default,,0000,0000,0000,,The rules of arithmetic\Nwhen you do it.