将变量转化为数字，第一部分

在这节课中，我将向你们介绍将变量转化为数字的策略，这可能是SAT数学部分最重要的策略。所以听好了，如果你想重温这一集，直到你真的，真的明白发生了什么，我强烈推荐你去看。现在，将变量转化为数字将使许多数学问题变得更简单。不是每次考试一两个，而是每个部分有几个，所以你可以经常这样做，另外，这个策略很好，因为它适用于每个人。如果你是数学巨星，它会让最难的问题变得中等难度，如果你真的在数学上挣扎，它会让中等难度的问题变得很容易，所以它对所有人都有帮助，这个策略有三个基本步骤。首先，你要给变量赋值所以不管变量是什么，你都要让它成为一个数字。下一步，你要用你选择的那些数字来解决你的问题，最后，简化与你提出的解决方案相匹配的答案选项。让我们用大量的例子来更详细地了解这三个步骤。
所以，我将向你们展示几个例子，我们将把变量转化为数字，如果你能使用“真正的数学”来解决这些问题，那就太好了，然而，我希望你练习遵循这些步骤，因为你需要习惯它们，有时在解决更简单的问题时，为了流利地解决更难的问题。所以，即使你能正确地做到这一点，我也希望你能理解这个策略，并能够应用它。
让我们从这个问题开始;w和x的平均值或算术平均值是6 y和z的平均值是10。w x y x的平均值是多少?第一步是把变量转化为数字我想你们通常要避免使用数字1和2避免重复使用数字。举个例子，w和x，我需要取w和x的两个值但是如果可能的话，我应该避免1和2我不应该重复它们它们的平均值需要等于6。我可以用很多不同的数字但是我们试试5和7因为我知道5和7的平均值是6 y和z的平均值是10。同样，我可以选择很多数字，但我还是简单地说9和11，因为这些数字的平均值也是10。我已经完成了这一步，也就是把它们转化成数字，现在可以进行第二步了;用我选的数来解方程。问题是，w x y z的平均值是多少w x y z的平均值是多少我把它们都加起来，然后除以4因为这就是平均值计算器可能会有用但答案是8。 So that's pretty easy. I'm ready for step three which is just finding the match. I got eight, c is eight. The answer is c.
让我们看另一个例子。如果a, b, c和d是连续的偶数，并且'a'小于'b'小于'c'小于'd'那么b + d比a + c大多少。所以，我开始将这些变量转化为数字，我必须遵守问题中的任何限制。我被告知a b c和d是连续的偶数，这意味着偶数的顺序是2 4 6 8 10等等，我被告知a是最小的，它最大到d。那么，我将继续分配2,4,6和8，我必须保持它们的顺序，因为它说，2a是最小的，它从这里开始上升。我想我已经完成了第一步，也就是分配数字。准备进入第二步，也就是回答问题。问题是b + d比a + c大多少?我们来算一下，b + d是b是多少?B是4 d是8，结果是12这比a + c大多少，a是多少?已知a = 2 c = 6，结果是8。 So the question is, this is how much greater than this and the answer is that's it's four greater so that's the answer. Now that we've gotten the answer to the question we're ready for the final step which is go find the answer choice that matches. We said it's four greater and that's b.
让我们看另一个例子;在一个正整数序列的第一项之后，每一项与它前面的一项之比为3:1。第五届和第二届的比例是多少?”我想指出的一点是，这可能会让你们想起代数二的某些章节这可能会派上用场但很多人看到这个会说，我不记得公式了，我做不出来。我还是放弃吧。”但是有这么多的SAT题目，如果你们读一下，想一下，就不会有问题，这里也是一样。我们不打算用一个疯狂的，花哨的公式，只是想一下，这是关于这个问题我想说的一点。我想说的另一点是，如果没有变量，那么问题中就隐藏了变量。题目讨论了第一项和其他项，所以隐变量是第一项我们可以把第一项设为任何我们想要的变量。为了简单起见，我们先从数字1开始。 The first term let's make it one and then the pattern is that the ratio of each term to the term immediately preceding it, is three to one. Ratio means there's a multiplication so the second term it's going to be multiplied by three. The third term it's going to be multiplied by three again so that's nine, the fourth term is going to be multiplied by three again so that's 27. The fifth term is going to be multiplied by three again so that's 81, so I think we've gotten enough progress here on picking our numbers and we're ready to actually answer the question. What's the ratio of the fifth term to the second term?
所以我们要处理这一项和这一项。81比3的比率是多少?这就是答案，但你可以看到，它没有被约简，所以它与我们最后第三步的答案选项不匹配。所以如果我们把这两个都除以3，就变成了27比1，当然，最后一步是找到匹配项，我们的匹配项是d，下一个例子。“如果w的10%等于y, w的40%等于z, z的百分之几是y?”好的，这个有点难，我们看看这里。选择某些数字比其他数字更有意义，例如，我注意到我们在这里取了10%的w，在这里取了40%的w，很多时候当你处理百分比时，使用像100这样的数字是很好的。取100的百分之一很容易。如果我让你取11.54的10%那就没那么有趣了除非你真的喜欢。好了，我们设w是100这相对来说比较容易处理。 So 10 percent of w, 10 percent of 100 is y, what's 10 percent of 100? Well, hopefully you know it's ten but if not, your calculator could help you with that and since that's 10 and it's y we'll write it there so we don't lose track and then 40 percent of w, 40 percent of that same 100 equals z, what's 40 percent of 100? It's 40. Now I should point out here that since w appears twice, it has to be the same number both times. You don't get to make up different numbers as you work through the problem. So we finished assigning numbers and we're ready to answer the question, the second step. What percent of z is y? What percent of... let's see, z is 40 and y is 10, so the question is, what percent of 40 is 10 or in other words 10 is what percent of 40? Now there are many ways to solve this but it's basically asking as a percentage, 10 is what compared to 40? Well, it's one fourth and you can ask your calculator if you need to but maybe you remember that one fourth is 25 percent. So we've got our answer and then step three is just to look for the answer right there, b, 25 percent.
另一个例子，'如果m被2除，余数为零。如果4m + 1除以2余数是多少?现在，这有点不同。以前，当我们选择数字时，我们必须遵循的规则是很明显的。这个比较稳定。我们事先不知道m是多少，我们必须一直计算，直到找到m的值，当我们把它除以2，余数是0。例如，如果我们要尝试数字3，它不会工作，让我演示给你看。当数字m除以2时，余数为零。好吧，让我们试试这个。3除2得1，减去2得1。 So, three divided by two is one with a remainder of one and that violates the requirement. Our requirement is the remainder zero, this is not for that requirement. So that value of three for m is not going to work. Now, you could play around a little bit but one value that would work is four and I'll just demonstrate for you, so you feel comfortable with that. When four is divided by two, it goes in two times, two times two is four, subtract it and that's your remainder of zero. So sure enough in choosing a value of four for 'm' we've met the rules stated in the equation or rather stated in the question and now we're ready to actually answer the question being asked. What is the remainder if 4m + 1 is divided by two? So let's see 4m +1, what is that? Well we decided that m was four, so four m plus one is four times four, that's 16 plus one that's 17. So the question is, what's the remainder when 17 is divided by two? Let's do that. Not too complicated, divided by two. Eight times two is 16, subtract and it's eight remainder one. So the question is what's the remainder and the answer is one. Now we go look for our answer here and there it is, b.
最后一个例子。如果y等于4乘以x的立方，当x翻倍时y的值会怎样?这和我们之前见过的问题有点不同因为它们是前后的。x有一个初始值，这使得y根据方程是一个确定的值，然后x翻倍，结果，y又变了。所以有原来的情况和新的情况。我们来做一下前后对比。我们试一下x = 3。在这之前，我们有x = 3 y = 4x，也就是3³。3的立方是27 27乘以4正好是108，所以这是y的初始值，现在我们在后面的情况下重复它x已经翻倍了。原来是3，现在变成了2倍大，也就是6，我们再来看数学。 Four times the new value of x which is six also cubed and we go through all this Math it's pretty complicated, we get 864. Now we're ready to answer the question, what happens to the value of y when x is doubled? So y went from being 108 to 864 and the answer is, it got multiplied by eight. It would be really easy to verify that by dividing this by this that would give you a result of eight, it's eight times as big. So it's step three which is basically nothing at all. Let's just say, oh, hey, look! That's the answer choice E, y has now been multiplied by eight. So let's tie together everything we've covered with these examples here.
概括一下，把变量转化为数字有三个基本步骤，可能需要一段时间来适应，但这绝对值得你投入时间来熟练运用这个策略。首先，你要给变量赋值。当你这样做的时候，尽量避免1和2，尽量不重复不同变量的数字，也尽量选择符合规则的数字，所以如果问题说两个数字相乘等于另一个数字，确保这个陈述是正确的。如果你被告知这个数字必须是连续的偶数，确保它是连续的，以此类推。下一步，一旦你得到了这些数字，用这些数字来解题，最后，找到与你确定的答案相匹配的答案选项，这就是把变量转化为数字。