1 00:00:00,000 --> 00:00:00,620 2 00:00:00,620 --> 00:00:04,850 We're told that for the past few months, Old Maple Farms 3 00:00:04,849 --> 00:00:08,689 has grown about 1,000 more apples than their chief rival 4 00:00:08,689 --> 00:00:11,000 in the region, River Orchards. 5 00:00:11,000 --> 00:00:14,179 Due to cold weather this year, the harvests at both farms 6 00:00:14,179 --> 00:00:15,919 were down by about a third. 7 00:00:15,919 --> 00:00:19,050 However, both farms made up for some of the shortfall by 8 00:00:19,050 --> 00:00:23,980 purchasing equal quantities of apples from farms in 9 00:00:23,980 --> 00:00:26,539 neighboring states. 10 00:00:26,539 --> 00:00:28,949 What can you say about the number of apples 11 00:00:28,949 --> 00:00:30,129 available at each farm? 12 00:00:30,129 --> 00:00:32,799 Does one farm have more than the other, or do they have the 13 00:00:32,799 --> 00:00:33,669 same amount? 14 00:00:33,670 --> 00:00:35,200 How do I know? 15 00:00:35,200 --> 00:00:37,410 So let's define some variables here. 16 00:00:37,409 --> 00:00:57,219 Let's let M be equal to number of apples at Maple Farms. And 17 00:00:57,219 --> 00:00:58,310 then who's the other guy? 18 00:00:58,310 --> 00:00:59,260 River Orchards. 19 00:00:59,259 --> 00:01:03,750 So let's let R be equal to the number of 20 00:01:03,750 --> 00:01:12,030 apples at River Orchards. 21 00:01:12,030 --> 00:01:14,689 So this first sentence, they say-- let me do this in a 22 00:01:14,689 --> 00:01:17,539 different color-- they say for the past few years, Old Maple 23 00:01:17,540 --> 00:01:22,560 Farms has grown about 1,000 more apples than their chief 24 00:01:22,560 --> 00:01:25,650 rival in the region, River Orchards. 25 00:01:25,650 --> 00:01:31,500 So we could say, hey, Maple is approximately Old River, or M 26 00:01:31,500 --> 00:01:34,680 is approximately River plus 1,000. 27 00:01:34,680 --> 00:01:37,190 Or since we don't know the exact amount-- it says it's 28 00:01:37,189 --> 00:01:39,939 about 1,000 more, so we don't know it's exactly 1,000 more-- 29 00:01:39,939 --> 00:01:44,079 we can just say that in a normal year, Old Maple Farms, 30 00:01:44,079 --> 00:01:49,879 which we denote by M, has a larger amount of apples than 31 00:01:49,879 --> 00:01:51,170 River Orchard. 32 00:01:51,170 --> 00:01:53,750 So in a normal year, M is greater than R, right? 33 00:01:53,750 --> 00:01:56,620 It has about 1,000 more apples than Old Maple Farms. 34 00:01:56,620 --> 00:01:59,660 Now, they say due to cold weather this year-- so let's 35 00:01:59,659 --> 00:02:02,569 talk about this year now-- the harvests at both farms were 36 00:02:02,569 --> 00:02:03,994 down about a third. 37 00:02:03,995 --> 00:02:07,930 38 00:02:07,930 --> 00:02:09,620 So this isn't a normal year. 39 00:02:09,620 --> 00:02:11,659 Let's talk about what's going to happen this year. 40 00:02:11,659 --> 00:02:13,699 In this year, each of these characters are going to be 41 00:02:13,699 --> 00:02:15,280 down by 1/3. 42 00:02:15,280 --> 00:02:18,710 Now if I go down by 1/3, that's the same thing as being 43 00:02:18,710 --> 00:02:20,510 2/3 of what I was before. 44 00:02:20,509 --> 00:02:21,379 Let me do an example. 45 00:02:21,379 --> 00:02:26,699 If I'm at x, and I take away 1/3x, I'm left with 2/3x. 46 00:02:26,699 --> 00:02:29,399 47 00:02:29,400 --> 00:02:33,800 So going down by 1/3 is the same thing as multiplying the 48 00:02:33,800 --> 00:02:35,910 quantity by 2/3. 49 00:02:35,909 --> 00:02:38,759 So if we multiply each of these quantities by 2/3, we 50 00:02:38,759 --> 00:02:40,879 can still hold this inequality, because we're 51 00:02:40,879 --> 00:02:44,009 doing the same thing to both sides of this inequality, and 52 00:02:44,009 --> 00:02:46,340 we're multiplying by a positive number. 53 00:02:46,340 --> 00:02:48,680 If we were multiplying by a negative number, we would have 54 00:02:48,680 --> 00:02:50,460 to swap the inequality. 55 00:02:50,460 --> 00:02:54,400 So we can multiply both sides of this by 2/3. 56 00:02:54,400 --> 00:02:59,525 So 2/3 of M is still going to be greater than 2/3 of R. 57 00:02:59,525 --> 00:03:02,620 And you could even draw that in a number line if you like. 58 00:03:02,620 --> 00:03:04,280 Let's do this in a number line. 59 00:03:04,280 --> 00:03:06,449 This all might be a little intuitive for you, and if it 60 00:03:06,449 --> 00:03:08,979 is, I apologize, but if it's not, it never hurts. 61 00:03:08,979 --> 00:03:10,439 So that's 0 on our number line. 62 00:03:10,439 --> 00:03:15,490 So in a normal year, M is has 1,000 more than R. 63 00:03:15,490 --> 00:03:18,330 So in a normal year, M might be over here and 64 00:03:18,330 --> 00:03:20,310 maybe R is over here. 65 00:03:20,310 --> 00:03:23,949 I don't know, let's say R is over there. 66 00:03:23,949 --> 00:03:29,049 Now, if we take 2/3 of M, that's going to stick us some 67 00:03:29,050 --> 00:03:32,340 place around, oh, I don't know, 2/3 68 00:03:32,340 --> 00:03:33,840 is right about there. 69 00:03:33,840 --> 00:03:40,250 So this is M-- let me write this-- this is 2/3 M. 70 00:03:40,250 --> 00:03:41,909 And what's 2/3 of R going to be? 71 00:03:41,909 --> 00:03:45,250 Well, if you take 2/3 of this, you get to right about there, 72 00:03:45,250 --> 00:03:47,969 that is 2/3R. 73 00:03:47,969 --> 00:03:56,360 So you can see, 2/3R is still less than 2/3M, or 2/3M is 74 00:03:56,360 --> 00:03:59,220 greater than 2/3R. 75 00:03:59,219 --> 00:04:04,889 Now, they say both farms made up for some of the shortfall 76 00:04:04,889 --> 00:04:08,899 by purchasing equal quantities of apples from farms in 77 00:04:08,900 --> 00:04:10,200 neighboring states. 78 00:04:10,199 --> 00:04:21,579 So let's let a be equal to the quantity 79 00:04:21,579 --> 00:04:26,235 of apples both purchased. 80 00:04:26,235 --> 00:04:29,240 81 00:04:29,240 --> 00:04:30,430 So they're telling us that they both 82 00:04:30,430 --> 00:04:31,870 purchased the same amount. 83 00:04:31,870 --> 00:04:34,840 So we could add a to both sides of this equation and it 84 00:04:34,839 --> 00:04:36,409 will not change the inequality. 85 00:04:36,410 --> 00:04:39,670 As long as you add or subtract the same value to both sides, 86 00:04:39,670 --> 00:04:41,640 it will not change the inequality. 87 00:04:41,639 --> 00:04:46,000 So if you add a to both sides, you have a plus 2/3M is a 88 00:04:46,000 --> 00:04:49,149 greater than 2/3R plus a. 89 00:04:49,149 --> 00:04:52,979 This is the amount that Old Maple Farms has after 90 00:04:52,980 --> 00:04:57,080 purchasing the apples, and this is the amount that River 91 00:04:57,079 --> 00:04:58,379 Orchards has. 92 00:04:58,379 --> 00:05:01,439 So after everything is said and done, Old Maple Farms 93 00:05:01,439 --> 00:05:04,199 still has more apples, and you can see that here. 94 00:05:04,199 --> 00:05:07,689 Maple Farms, a normal year, this year they only had 2/3 of 95 00:05:07,689 --> 00:05:10,180 the production, but then they purchased a apples. 96 00:05:10,180 --> 00:05:14,509 So let's say a is about, let's say that a is that many 97 00:05:14,509 --> 00:05:18,039 apples, so they got back to their normal amount. 98 00:05:18,040 --> 00:05:19,810 So let's say they got back to their normal amount. 99 00:05:19,810 --> 00:05:21,610 So that's how many apples they purchased, so 100 00:05:21,610 --> 00:05:23,590 he got back to M. 101 00:05:23,589 --> 00:05:29,359 Now, if R, if River Orchards also purchased a apples, that 102 00:05:29,360 --> 00:05:33,110 same distance, a, if you go along here gets you to right 103 00:05:33,110 --> 00:05:35,370 about over there. 104 00:05:35,370 --> 00:05:38,959 So once again, this is-- let me do it a little bit 105 00:05:38,959 --> 00:05:42,009 different, because I don't like it overlapping, so let me 106 00:05:42,009 --> 00:05:42,740 do it like this. 107 00:05:42,740 --> 00:05:46,259 So let's say this guy, M-- I keep forgetting their names-- 108 00:05:46,259 --> 00:05:50,899 Old Maple Farms purchases a apples, gets them that far. 109 00:05:50,899 --> 00:05:52,359 So that's a apples. 110 00:05:52,360 --> 00:05:55,699 But River Orchards also purchases a apples, so let's 111 00:05:55,699 --> 00:05:56,699 add that same amount. 112 00:05:56,699 --> 00:05:58,699 I'm just going to copy and paste it so it's the exact 113 00:05:58,699 --> 00:05:59,949 same amount. 114 00:05:59,949 --> 00:06:02,310 115 00:06:02,310 --> 00:06:07,449 So River Orchards also purchases a, so it also 116 00:06:07,449 --> 00:06:09,469 purchases that same amount. 117 00:06:09,470 --> 00:06:12,980 So when all is said and done, River Orchards is going to 118 00:06:12,980 --> 00:06:16,390 have this many apples in the year that they had less 119 00:06:16,389 --> 00:06:17,800 production but they went and purchased it. 120 00:06:17,800 --> 00:06:21,420 So this, right here, is-- this value right here 121 00:06:21,420 --> 00:06:26,120 is 2/3R plus a. 122 00:06:26,120 --> 00:06:27,379 That's what River Orchards has. 123 00:06:27,379 --> 00:06:30,829 And then Old Maple Farms has this value right here, which 124 00:06:30,829 --> 00:06:34,289 is 2/3M plus a. 125 00:06:34,290 --> 00:06:36,590 Everything said and done, Old Maple Farms 126 00:06:36,589 --> 00:06:38,659 still has more apples. 127 00:06:38,660 --> 00:06:38,665