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Numbers, such as -1, 0, 1, 2, and 3, that have no fractional part. Integers include the counting numbers (1, 2, 3, ...), their negative counterparts (-1, -2, -3, ...), and 0. FAQ: Why did we decide to multiply 10, 9, 8 and -6?
A: The key word here is LEAST. Least means lowest not closest to 0 (think farthest to the left on a number line.) Since we have access to negative integers, we are looking for the most negative number. Since we are multiplying 4 integers together, in order for the product to be negative, we need to have 1 or 3 negative numbers.
We want the absolute values of the integers to be as high as possible so we can get the largest negative number. If we list all the integers out on a number line we have:
{-6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
The 4 largest integers (after absolute value) are 7, 8, 9 ,10 but none of these are negative and as we learned above we need either one or three of the integers to be negative. The next 4 possible integers could be 10, 9, 8, -6. This satisfies the criteria of one negative integer and also gives us the largest possible product because we have chosen the numbers with the largest absolute values. A fraction can only be undefined if the denominator equals zero.
First of all, remember that zero itself is neither positive nor negative, so neither P nor Q nor S could equal zero by itself. The sum [positive] + [positive] can never be zero, but the sum [positive] + [negative] could be zero: (Q + S) and (P + S) could be zero, so fractions (A) and (B) could be undefined. The differences [positive] - [negative] or [negative] - [positive] can never be zero, but [positive] - [positive] could be zero: (P - Q) could be zero, which means fraction (E) could be undefined. Fractions (A), (B), and (E) are the only ones that could be undefined.
FAQ: For answer choice E, how can we assume that P and Q might be equal?
Good question! The question states that P and Q are both positive numbers which means both are greater than 0; however, there are no other limitations on what P and Q could be. P and Q could be equal since the question does not explicitly state that they are not equal. Another way to think about fractions with denominators that are multiples of ten.
So, one-twentieth. I could write that as one half times one-tenth. Well one half, I can write that as a decimal. That's 0.5 and one-tenth, of course, is ten to the negative one. Well, if i multiply these, multiplying by ten to the negative one, as we found two videos ago, this slides the decimal place over one place to the left and I get 0.05.
Again, one-fortieth. Well, this is 0.25 times ten to the negative one. Again slide the decimal place over one place to the left. I get 0.025 and that's the decimal for one-fortieth.
1/40= 1/4 1/10= 0.25 0.1= 0.025
1/20 = 1/2 1/10 = 0.5 0.1 = 0.05 - The easiest way to approach this is to plug in numbers for different cases.
(p^2 + PQ)= odd
We can simply use 1 for our odd number and 2 for our even number. And notice we're gonna have four different cases, four different possibilities here. They both could be even, both P and Q could be even. Both P and Q could be odd numbers. Or it may be that P is even and Q is odd. Or it may be that P is odd and Q is even. So, we have to test all four of those cases separately.
case 1: P and Q are both even
4 + 4 = 8 does not work
case 2 , case 3
and case 4: P= odd, Q=even
1 + 4 = 5 works
That works and that's the only case among the four that works. Therefore we know, in this problem it must be true that P is an odd number and Q is an even number. We are able to figure that out from the four-case method. We might also solve this problem using logic
Notice if P is even, it automatically makes both terms even. And then we get even plus even, which equals even and that doesn't work. There's no way we can get an odd number from even plus even. So therefore, P has to be odd. Now if P is odd, the first term is definitely odd. If the sum is odd, the second term must be even.
If P is odd, the only way P times Q can be even, is if Q is an even number. Therefore, P is odd and Q is even. So, that's the way to solve it with logic. Let's start with the four-case method.
Even= 2 Odd=1
Notice that the cases that work are the cases where Q is odd. If Q is odd, then the expression is odd, and if Q is even, then the expression doesn't work. So we, the conclusion that we can draw definitely is, Q is odd, and we really can't draw anything about P.
It's definitely true that Q is odd, and we have, we can draw no conclusion about P. Now again using logic, think about this. 4P must be even, because 4 is an even number. So, the only way that the sum will be odd is if Q is an odd number. So, Q must be odd.
4P will be even regardless of whether P is even or odd, so whether P is even or odd does not matter at all to the result. When we set two ratios equal, that's called a proportion. To solve this, we are going to set up a proportion. We will use rates of the form:
rate= thermometers/minutes
For the first case, we know we have 135 thermometers and 18 minutes. For the second case, we will use X for the number of thermometers, and change 1 hour to 60 minutes.
135/18= X/60
To solve this, let's resist the urge to reach automatically for a calculator. Do NOT cross-multiply first. With proportions, always cancel before you multiply. First of all, notice that 135 is divisible by 9 (because 1 + 3 + 5 = 9), and of course 18 is divisible by 9. Let's cancel a factor of 9 in the fraction on the left.
15/2=X/60 so X= (60 * 15)/2= 450
In one hour, this machine will make 450 thermometers. ;
