1 00:00:02,046 --> 00:00:03,986 >> Welcome to the Cypress College Math Review 2 00:00:03,986 --> 00:00:07,226 On multiplying and dividing fractions. 3 00:00:07,826 --> 00:00:12,306 Jumping into objective number one, we need to do some multiplying of fractions. 4 00:00:12,536 --> 00:00:17,206 Our first example tells us to use the diagram to answer the following: 5 00:00:17,926 --> 00:00:20,476 Using the diagram, we want to address Part A first. 6 00:00:20,586 --> 00:00:21,856 What fraction is shaded? 7 00:00:22,286 --> 00:00:31,366 Looks like I've got 1, 2, 3, 4, 5, 6, 7, 8 of these sections that are shaded. 8 00:00:31,606 --> 00:00:35,516 Of course, that means that we've got the remaining sections that are not shaded. 9 00:00:35,676 --> 00:00:40,036 I'm going to label those as 9, 10, 11, and 12. 10 00:00:40,426 --> 00:00:47,366 Meaning, all together we've got a total of 12 sections in this diagram. 11 00:00:47,696 --> 00:00:49,666 In order to address what fraction is shaded then, 12 00:00:49,996 --> 00:00:52,786 we simply count up the shaded sections, which we have done already. 13 00:00:53,266 --> 00:01:00,076 We had 8 shaded sections and a total of 12 sections. 14 00:01:00,706 --> 00:01:03,286 Therefore, the fraction that is shaded would be 8 over 12. 15 00:01:03,796 --> 00:01:08,576 Of course, mathematically we can reduce this to become a 2 over 3. 16 00:01:10,226 --> 00:01:12,626 That takes us over to Part B of the question then, 17 00:01:13,526 --> 00:01:16,356 where they are asking us: what is 1/2 of 2/3? 18 00:01:17,476 --> 00:01:23,276 Now, the key part here is to recognize that 2/3 was represented by the shaded region. 19 00:01:24,336 --> 00:01:29,806 So this question is asking us to take 1/2 of the shaded region, but we still have to account 20 00:01:29,806 --> 00:01:31,156 for the total number of sections. 21 00:01:32,126 --> 00:01:38,536 As we look at the shaded region, 1/2 of that would be represented by the four sections 22 00:01:38,536 --> 00:01:42,006 that we can take -- whether it's top or bottom or however we organize it doesn't matter. 23 00:01:42,006 --> 00:01:43,786 But we had 8 sections that are shaded. 24 00:01:43,996 --> 00:01:46,686 Half of those then would mean we've got four shaded sections. 25 00:01:47,436 --> 00:01:51,196 So of the shaded sections, we've got 4 shaded sections, 26 00:01:51,196 --> 00:01:53,606 representing 1/2 of the shaded sections. 27 00:01:54,126 --> 00:01:56,976 When the question asks, what is 1/2 of 2/3, though, 28 00:01:56,976 --> 00:02:00,286 we have to remember it's out of the total diagram. 29 00:02:00,766 --> 00:02:04,836 Therefore, we end up with 12 total sections. 30 00:02:05,156 --> 00:02:08,476 One-half of our 2/3 then would be 4 over 12. 31 00:02:09,366 --> 00:02:12,936 And of course, we can reduce that to simply be a 1/3. 32 00:02:15,096 --> 00:02:18,256 That takes us to our next example with multiplying fractions. 33 00:02:18,696 --> 00:02:22,776 First, we want to note that when we multiply by a proper fraction, the answer is less 34 00:02:22,776 --> 00:02:26,386 than the original number to keep context on what we've done. 35 00:02:26,666 --> 00:02:30,316 When we multiply fractions, our first step is to cross cancel 36 00:02:30,776 --> 00:02:33,256 and our second step will be to multiply across. 37 00:02:34,326 --> 00:02:36,406 Let's go ahead and jump into the example then. 38 00:02:36,606 --> 00:02:42,196 In our example, they are asking us to multiply 3 over 15 times 20 over 21. 39 00:02:42,196 --> 00:02:45,876 I'm just going to rewrite the problem in order to create some spacing. 40 00:02:46,326 --> 00:02:50,776 So I've got 3 over 15 being multiplied by my 20 over 21. 41 00:02:51,136 --> 00:02:53,966 And again, our first step is to cross cancel. 42 00:02:54,716 --> 00:02:57,546 So what I'm looking to do is cancel with common factors. 43 00:02:58,286 --> 00:03:04,796 The first thing I see is that I've got a common factor of 5 between 15 and 20. 44 00:03:05,936 --> 00:03:10,276 Since I have a common factor of 5, I'm going to cancel out the 15 and 20 45 00:03:10,846 --> 00:03:13,566 and reduce those by the factor of 5. 46 00:03:14,206 --> 00:03:17,626 Reducing 15 by the factor of 5 means I'm left with a 3. 47 00:03:18,196 --> 00:03:20,956 Reducing 20 by a factor of 5 means I am left with 4. 48 00:03:21,556 --> 00:03:27,636 Now, looking at 3 and 21, I can cross cancel with another factor of 3. 49 00:03:29,566 --> 00:03:36,936 When I cancel 3 and 21, 3 reduces to just 1 since 3 divided by 3 is 1, and 21, of course, 50 00:03:36,936 --> 00:03:39,676 reduces to 7 since 21 divided by 3 is 7. 51 00:03:40,316 --> 00:03:44,776 It appears that I can't cancel anymore with the remaining factors, 52 00:03:44,806 --> 00:03:46,776 so I'm going to gather my terms together. 53 00:03:47,266 --> 00:03:51,906 I now have 1 over 3 times 4 over 7. 54 00:03:52,436 --> 00:03:57,126 It doesn't appear I can cancel out anymore since I have no shared factors, which means, 55 00:03:57,126 --> 00:03:59,006 like you see in step two, I multiple across. 56 00:03:59,376 --> 00:04:03,716 Multiplying across, I'm left with 1 times 4, which is just a 4, 57 00:04:04,006 --> 00:04:06,566 and 3 times the 7, giving me a 21. 58 00:04:07,286 --> 00:04:09,946 Which means our answer is 4 over 21. 59 00:04:09,946 --> 00:04:12,126 And of course we cannot reduce that fraction. 60 00:04:13,716 --> 00:04:16,706 Let's go ahead and jump into our next example now, where we are being asked 61 00:04:16,706 --> 00:04:21,126 to multiply four fractions, which means organization is becoming more important. 62 00:04:21,806 --> 00:04:28,216 The example asks us to multiply 2 over 9 times 3 over 10 times 4 over 11 times 5 over 12. 63 00:04:28,996 --> 00:04:32,546 I'm going to rewrite the problem again, just to create some spacing. 64 00:04:34,156 --> 00:04:39,136 Because, remember, our first step will be to cross cancel. 65 00:04:39,606 --> 00:04:42,506 Anywhere we've got some common factors. 66 00:04:42,906 --> 00:04:49,346 Now, the issue here is to remember that we don't have to cross cancel only fractions 67 00:04:49,346 --> 00:04:50,656 that are right next to each other. 68 00:04:50,826 --> 00:04:55,596 Meaning, our options are not only to look for cross cancelling amongst consecutive fractions. 69 00:04:55,816 --> 00:04:59,266 We can cross cancel anywhere available from numerator to denominator. 70 00:04:59,706 --> 00:05:03,746 To demonstrate that, I'm going to cross cancel 2 and 12. 71 00:05:04,686 --> 00:05:09,296 Again, just to be clear, we could cross cancel 2 and 10, since they have a common factor of 2. 72 00:05:10,506 --> 00:05:13,826 Instead, just to show that this is an option, I'm going to cross cancel 2 and 12. 73 00:05:14,856 --> 00:05:18,486 Sharing the common factor of 2, 2 reduces to be just 1, 74 00:05:18,486 --> 00:05:21,516 and of course 12 divided by 2 leaves us with 6. 75 00:05:22,396 --> 00:05:27,816 Looking for other options, I'm going to cross cancel 3 and 9 here with the common factor 76 00:05:27,816 --> 00:05:31,276 of 3 -- 9 reduces to 3, 3 reduces to 1. 77 00:05:31,686 --> 00:05:38,596 Continuing our search, I'm going to cross cancel 5 and 10, since they have a common factor 78 00:05:38,596 --> 00:05:42,596 of 5 -- 10 reduces to be 2, 5 reduces to be 1. 79 00:05:43,036 --> 00:05:46,876 Now at this point we've got a lot of numbers being manipulated here. 80 00:05:46,956 --> 00:05:50,946 So I'm going to regather my fractions to see where I'm at to ensure I don't make a mistake. 81 00:05:51,606 --> 00:05:56,816 After all this cross cancelling, I now have 1 over 3 times 1 82 00:05:56,816 --> 00:06:01,986 over 2 times 4 over 11 times 1 over 6. 83 00:06:04,516 --> 00:06:07,606 We have an option to cross cancel using the numerator of 4. 84 00:06:08,436 --> 00:06:13,256 You can either cross cancel 4 with 2 or we can cross cancel 4 with 6. 85 00:06:13,626 --> 00:06:16,456 I'm going to cross cancel 4 with 2. 86 00:06:16,986 --> 00:06:20,476 That leaves me with 1 and 2 in the numerator. 87 00:06:21,366 --> 00:06:27,076 Recognize that we can also cancel 2 with 6 since they have a common factor of 2. 88 00:06:27,586 --> 00:06:32,456 Two, of course, reduces to 1 and 6 reduces to 3. 89 00:06:34,386 --> 00:06:38,406 Gathering my fractions again, now that we've done more cross cancelling, it appears I have 1 90 00:06:38,406 --> 00:06:45,506 over 3 times 1 over 1 times 1 over 11 times 1 over 3. 91 00:06:47,216 --> 00:06:50,326 Which means at this point, double check to make sure we can't cross cancel. 92 00:06:50,326 --> 00:06:54,996 We have one's in the numerator all the way across the board, so we can move on to step two 93 00:06:54,996 --> 00:06:59,526 to multiply straight across: 1 times 1 times 1 times 1 is of course just 1. 94 00:06:59,906 --> 00:07:05,376 And in the denominator, we have 3 times 1 times 11 times 3 to get us our 99. 95 00:07:05,886 --> 00:07:10,786 Since we cannot reduce that fraction, our answer is the 1 over 99. 96 00:07:13,800 --> 00:07:15,800 Pause the video and try these problems 97 00:07:19,060 --> 00:07:22,260 >> That takes us to our objective two, which is dividing fractions. 98 00:07:23,096 --> 00:07:26,226 First, we want to discuss reciprocals for a fraction. 99 00:07:27,016 --> 00:07:29,976 We need to interchange the numerator and the denominator 100 00:07:30,316 --> 00:07:31,756 when we are trying to find the reciprocal. 101 00:07:32,126 --> 00:07:36,876 For example, the reciprocal of 3 over 10 is 10 over 3. 102 00:07:40,046 --> 00:07:41,586 Jumping into the first example, then, 103 00:07:42,046 --> 00:07:45,366 they ask us to use the following diagram to answer the following. 104 00:07:45,996 --> 00:07:49,356 As we did previously, Part A asks, what fraction is shaded? 105 00:07:49,896 --> 00:07:54,176 Looking at our diagram like we did previously, it looks like I've got one section, 106 00:07:54,176 --> 00:07:59,376 two sections that are shaded, while this third and fourth section are not shaded. 107 00:08:00,156 --> 00:08:03,316 Which means, to address what fraction is shaded, 108 00:08:03,646 --> 00:08:08,376 we've got two shaded sections out of four total sections. 109 00:08:08,886 --> 00:08:12,866 Which, of course, we can reduce to be 1/2. 110 00:08:12,866 --> 00:08:14,666 That takes us over to Part B of the question. 111 00:08:14,756 --> 00:08:18,666 What is 1/2 of 1/2? 112 00:08:18,666 --> 00:08:25,056 Remember, 1/2 represented the shaded section, and they are asking us to take 1/2 of 1/2. 113 00:08:25,576 --> 00:08:29,926 So instead of considering both shaded sections, we are now only taking half 114 00:08:29,926 --> 00:08:32,866 of the two or one of those sections. 115 00:08:33,386 --> 00:08:39,716 Which means I have 1 shaded section, but of course, this is out of the total 4 sections. 116 00:08:40,266 --> 00:08:43,346 Therefore, 1/2 of 1/2 is 1/4. 117 00:08:44,196 --> 00:08:49,816 This brings us to an important note that dividing by 2 is the same as multiplying by 1/2. 118 00:08:49,886 --> 00:08:52,706 What we're trying to accomplish here is to make the connection 119 00:08:52,706 --> 00:08:55,386 between multiplication and division of fractions. 120 00:08:57,666 --> 00:09:01,216 Now, let's go ahead and jump into our last example of dividing fractions. 121 00:09:02,046 --> 00:09:07,026 Note first to multiply the first fraction by the reciprocal of the second fraction. 122 00:09:07,776 --> 00:09:14,516 That is, we need to take the reciprocal of the divisor in order to divide these fractions. 123 00:09:14,986 --> 00:09:19,866 Our example asks us to divide 2 over 15 divided by 1 over 5. 124 00:09:20,546 --> 00:09:25,696 Now, as previously stated, we want to take the first fraction here, 2 over 15, 125 00:09:25,916 --> 00:09:32,496 but we want to multiply it by the reciprocal of 1 over 5, which would be 5 over 1. 126 00:09:33,266 --> 00:09:37,496 Now that we have converted this into multiplication, I can go into the steps 127 00:09:37,496 --> 00:09:40,526 of multiplication where step number one is to cross cancel. 128 00:09:41,576 --> 00:09:49,746 Cross cancelling 15 and 5 with a common factor of 5 means that 15 reduces to 3, 5 reduces to 1. 129 00:09:50,316 --> 00:09:54,356 We can't cross cancel 2 and 1 at all. 130 00:09:55,746 --> 00:10:03,526 Therefore, I can multiply straight across: 2 times 1 gives me 2, 3 times 1 gives me 3. 131 00:10:04,116 --> 00:10:07,276 We can't reduce 2/3. 132 00:10:07,896 --> 00:10:10,106 Therefore, my answer is 2/3. 133 00:10:12,520 --> 00:10:16,770 Pause the video and try these problems 134 00:10:20,040 --> 00:10:23,360 Thank you again for joining us with the Cypress College Math Review.