Gregory P.

University of Chicago graduate

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SAT

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Question:

What is the average (arithmetic mean) of all the multiples of ten from 10 to 190 inclusive?

Gregory P.

Answer:

We can find the average of a set of numbers by adding each number up and dividing that by how many numbers we have in the set. So from the problem we have 10, 20, 30, all the way up to 190. How many numbers is that? You might be tempted to say 18, by doing 19 - 1. But imagine it this way: you are counting from 1 to 19. How many numbers is that? That is 19. Make sure to be careful when counting numbers, because it depends on if the smallest value and the largest value are counted or not. In this case both 10 and 190 are counted, and so you have to add 1. So now we have the number we need to divide by to get the average. How do we add up 10 + 20 + ... + 190? Well, you can remember the formula for sum of an arithmetic sequence, and plug in these numbers. The formula is the sum of the first and last term, multiplied by the distance between each term, divided by 2. So we get (10 + 190) * 19 / 2. This gives us 1900. Divide by 19 and you get the average to be 100. The simpler way is to realize that the numbers are even spaced from 10 to 190, and so the average is the same as the midpoint of the numbers. The middle number from 1 to 19 is 10, and so the middle number from 10 to 190 is 100.

ACT

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Question:

What is the greatest common factor of 42, 126, and 210?

Gregory P.

Answer:

The easiest and most sure way of finding GCF is by looking at the prime factorization of each number. This means we break up the number into the product of as many whole numbers as possible. For example, from your multiplication table you know 42 = 6 x 7. You can't break 7 into other numbers. Can you break up 6 further? Yes, 6 = 2 x 3. Can you break 2 or 3 into smaller whole numbers? No, because they are prime. Prime numbers can only be divided by themselves, or 1. So we have 42 = 2 x 3 x 7. Now lets do this with 126. First, since it is an even number, we can cut it in half. So 126 = 2 x 63. We know 63 = 7 x 9, and 9 = 3 x 3. So we get 126 = 2 x 3 x 3 x 7. For 210, lets do the same thing. 210 = 2 x 105. 105 ends in 5, so we can divide it by 5. 105 = 5 x 21. 21 = 3 x 7. And we are done. So 210 = 2 x 3 x 5 x 7. So now we have to find as many numbers as possible that are being shared by these prime factorizations of the 3 numbers. We see there is a 2, a 3, and a 7 shared in all three, but no more. So then we can conclude that 2 x 3 x 7, or 42, is the greatest common factor of 42, 126, and 210, since all these numbers can be obtained by multiplying some number to 42. (42 x 1 = 42!)

Algebra

TutorMe

Question:

How would you solve the equation for x: 2x + 7 = 5x - 5?

Gregory P.

Answer:

Let's group the x's together first. This is the variable we want to figure out. We can do this by subtracting both sides of the equation by 2x. We do this because we have an equation, and whatever we change one side by, we must do so in the other. Then the x's on the left side goes away and we are left with just a lone 7. On the right hand side we had 5x, and we took away 2x, so we are left with 5x - 2x = 3x, and a -5. Now we want all the numbers on the other side so we can figure out what x needs to be to fit the equation. So we do the same thing, and add by 5 to get rid of the number on the right hand side, leaving just 3x. Then 5 has to get added over to the left side, meaning we get 7 + 5 = 12. Thus our final equation looks like this : 12 = 3x. Now 3x means 3 times x. 3 times what number is equal to 12? You should know this from your multiplication table, and it is 4. Another way to look at it is dividing both sides by 3, because then you are left with just x on one side, which means you have 4 = x, and x is what we wanted to know.

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