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I would like to see your solution. If each column (or each row) must add up to 15 and there are 3 columns, that's 45 people. And you only have 36.
Just thinking this through since it doesn't sound right. Seems like it should be 50%-50%.
We'll label the 3 prizes as G1 and G2 (goats) and C (car). There are 6 different arrangements:
# Door1 Door2 Door3
1 C G1 G2
2 C G2 G1
3 G1 C G2
4 G1 G2 C
5 G2 C G1
6 G2 G1 C
Pick a door, any door. Let's say Door 2. In 2 of the cases (3 and 5), you don't want to switch. In the other 4 case (1, 2, 4, and 6), you do want to switch. So using this logic, you want to switch 2/3 of the time so it makes sense to switch.
Let's look at it from Monty Hall's point of view. He knows where the car is. Let's say the arrangement is #5 from above. Now there 4 different possibilities:
1/3 of the time the contestant will pick door 1, Monty will show what's in Door3, and the contestant would want to switch.
1/6 of the time the contestant will pick Door 2 AND Monty will show what's in Door 1. The contestent should keep the original pick.
1/6 of the time the contestent will pick Door 2 AND Monty will show what's in Door 3. The contestent should keep the original pick.
1/3 of the time, the contestant will pick Door 3, Monty will show what's behind Door 1 and the contestent would want to switch.
So, again, 2/3 of the time the contestent should switch.
I guess I convinced myself that it's better to switch. I would prefer that someone find a hole in my logic!
When I've seen magic squares, they usually tell you the "magic number" and you have to make each row/column/diagonal add up to that number. It looks like in this case, you can make the magic number whatever you want. Let's pick -36 for the first one. Now that we have the magic number, we can figure out A in the diagram below: -16 + -13 + A + -7 = -36; So A is 0. Now that you filled in A, you have enough information to figure out B, then C, etc.
Here it is solved for magic number of -36
You can use the same procedure to solve the other two puzzles. In each case, you only have one missing number in the diagonal going from top-right to bottom left. Replace the ? in that diagonal (2nd row, 3rd column) with whatever number you want (I picked 0 just to make it easy), find out the sum of the diagonal, and then continue on filling in the remaining spaces.
The answer is 4! (pronounced 4 factorial) which is 4 * 3 * 2 * 1. So the answer is 24.
To help you understand, think about this. How many ways can you arrange 2 numbers? Two, right? 12 and 21.
Now add in a 3rd number. One way of doing this is to take the 2 ways to arrange 2 numbers and add in the 3rd:
12 - Add a 3 in to get 123, 132, and 312
21 - Add a 3 in to get 213, 231, and 321.
So for every solution for 2 numbers, there's 3 ways to add in a 3rd number. So that's 2 * 3 solution.
To add in a 4th number, take each of the 3 digit numbers and work from there:
123 - Add the 4 in to get the following four possibilities: 1234, 1243, 1423, 4123.
So for each of the ways you could make a 3 digit number, there are 4 possibilities when you add in a 4th digit. There are 6 (3*2*1) ways to make a 3 digit number, multiply that by 4 when you add in a fourth digit and you get 24 (4*3*2*1)
Add the first 2 equations together:
I must have made the same mistake that you did because I got the same answer the first time. Think about this. What if you used 1 liter of 9% and 3 liters of 12%? You add 9 + 12 + 12 + 12 AND then divide by 4. I forget the "divide by 4" as you probably did. We're essentially calculating the average concentration.
Let X be the amount of 9% solution. 4 - X is the amount of 12% solution.
Let's say your 2 digit number is XY which is 10X + Y. You have 2 equations to work with:
X+Y=5 or X=5-Y
and
10X + Y = 10Y + X - 27 or 9X = 9Y - 27 or X = Y - 3
Substitute 5-Y for X in the 2nd equation:
5 - Y = Y - 3
8 = 2Y
Y = 4
Therefore X=1; your original is 14 and the reverse is 41. Just as luca-deltodesco said.
You're not doing any thing wrong. The correct solution isn't listed. It should be z-x+1.
You can also use similar triangles to solve this as you suggested. (I assume that your figure is a rectangle)
First, figure out the lenght of PR as Dross pointed out. Triangle PQR is similar PVQ because two of the angles are equal - both are right triangles and angles PVQ and RPQ are obviously equal since they are the same angle. Because they are similar:
You know the values for PQ (18) and for RQ (12), so you can solve for PV. Then you can subtract the length of PV from PR to get VR.
Next, the triangle RVU is similar to PQR. They are both right angles and angle RPQ = URV (alternate interior angles). So:
You know PQ, PR and VR so you can solve for UR.
so just to reiterate, can you just ignore the part that says -3 ≤ x² ?
Well, ignore is a pretty strong word. In your equation, it has no effect on the answer.
Sorry, but I was a little off there in my first post. Your step from
Where I (we?) went wrong was I was looking for the -3 to be the lower bound and 5 to be the upper bound (actually the square roots). The answer to the above equation is really equal to the intersection of the two equations:
All values of x satisfy the first equation. For the second, the following set of x's satisfy the equation:
So the answer is the set of x's which meet both criteria which in this case is simply:
A better example to show how the intersection works is to subtract 5 rather than 1, like I did in my first post.
Now you must take the intersection of these two sets of answers. The first equation limits x to values between -3 and 3. The second limits x to values greater than 1 or less than -1. The intersection gives us the following 2 intervals:
Again, sorry about the first post. Although I don't think any of it was wrong, it just made the waters muddier. This one probably didn't help much either!
I don't remember the rules regarding absolute values, inequalities, and square roots, but your mistake was going from
There's some rule involving square roots and inequalities. For example:
I know . The is the easy part. How do I prove mathematically ? Anyone?I'm thinking this through as I go, so bear with me. Consider a more interesting but very similar problem:
. First off, consider only those values of x where (without absolute value) is a positive number.Now, consider those values where
(without the absolute value) results in a negative number. This is the points between . When you take the absolute value of a negative number, you're essentially multiplying by -1.So your answer must satisfy all of these limitations:
This is all the points between -3 and -1 plus the points beween 1 and 3.
I didn't figure out your answer and I probably just made things worse but maybe my comments will job someone else's memory. Bottom line is that the problem is not as simple as it first appears. You need to manipulate the equations based on the value of x. I need to stop for now. Maybe I'll come back later this evening and try to clear up this mess.
Sorry about the duplicate answer. At least we came up with the same answer!
Picture two triangles on the graph. The first triangle will have corner points of (6,2), (6,0) and (z,0). The second will have (10,4), (10,0), and (z,0). These are right triangles. You want to find z so the the hypotenuses (h) of these 2 triangles are equal. Use the Pythagorean theorem.
For the first triangle:
Set those two equations to be equal to each other and you have:
So the point is (9.5, 0)
The number of different ways to choose 2 balls from the 12 total is (12 choose 2) or:
Let's do part b first. The number of ways to choose 2 balls the same color is the sum of the ways to choose 2 red balls plus the number of ways to choose 2 white plus the number of ways to choose 2 green. The number of ways to choose 2 red is (4 choose 2):
The number of ways to choose 2 white is (5 choose 2) which is equal to 10.
The number of ways to choose 2 green is (3 choose 2) which is equal to 3.
The sum of these possibilities is 19. So the probability of choosing 2 balls of the same color is 19 out of 66 or 19/66= 28.78%
So now part a is real easy. If they're not the same color, they must be different colors. We accounted for 19 of the 66 ways, so the other 47 ways must give you balls of different colors. So the probability is 47 out of 66 or 71.21%.
We can verify this number. The 3 possible ways to pick different color balls are: Red & white, red & green, and white & green. The number of ways to pick a red and white is equal to the number of different ways to pick a red ball multipled by the number of ways to pick a white ball. This is (4 choose 1) times (5 choose 1) which is 4 * 5 = 20 . Likewise the number of ways to pick a red & green is (4 choose 1) * (3 choose 1) = 3 * 4 - 12. White and Green: 5 * 3 = 15. Add these 3 together: 20 + 12 + 15 = 47.
Just playing around here to try it out. Feel free to delete this in the efforts of tidiness.
x^2 + y^2 = z^2 \mbox { sure looks nicer than x**2 + y**2 = z**2}Is there any easy way to see what your reply will look like before you submit it?
Yes, that's it exactly. Simplifying it:
1.35x = 670
x =~ $496.30
So far the student has 196 points. To get an A, he needs to get 90% of the possible 350 points (350 * .9). That's 315. For a B, he needs to get 80% of 350 or 280. The best he can do is get a 100 on the final exam to give him a total of 296. Not enough for an A. He needs to score at least an 84.
From Pi Man with help from his daughter MM (MathMeggie).
got to include all of the sixty's:
6, 16, 26, 36, 46, 56, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 76, 86, 96. That's 20 (there's two in 66).
dividend / divisor = quotient with a remainder
Just do your division as if both numbers (dividend and divisor) where positive. When you're done, if both numbers were positive, the quotient is positive. If both numbers were negative, the quotient is positive. If one (and only one) of the two is negative, then the quotient is negative.
So -45 / 7 would be -6 with a remainder of 3. Actually, now that you brought it up, it probably should be a remainder of -3 since [quotient * divisor + remainder = dividend].
-6 * 7 + (-3) = 45.
Or if the remainder is always supposed to be positive, the answer could be -7 with a remainder of 4: -7 * 7 + 4 = 45. Bottom line is I thought I knew the answer but I guess I don't. It's been too long since I studied that I guess.
The other is a rectangle with all 4 sides = 4 for an area and perimeter of 16. So the feared number is 17 which is halfway between 16 and 18.
For the 2nd question, if you allow zeroes, there are 64 combinations:
16,0,0,0
15,1,0,0
14,2,0,0 14,1,1,0
13,3,0,0 13,2,1,0 13,1,1,1
12,4,0,0 12,3,1,0 12,2,2,0 12,2,1,1
11,5,0,0 11,4,1,0 11,3,2,0 11,3,1,1 11,2,2,1
10,6,0,0 10,5,1,0 10,4,2,0 10,4,1,1 10,3,3,0 10,3,2,1 10,2,2,2
9,7,0,0 9,6,1,0 9,5,2,0 9,5,1,1 9,4,3,0 9,4,2,1 9,3,3,1 9,3,2,2
8,8,0,0 8,7,1,0 8,6,2,0 8,6,1,1 8,5,3,0 8,5,2,1 8,4,4,0 8,4,3,1 8,4,2,2 8,3,3,2
7,7,2,0 7,7,1,1 7,6,3,0 7,6,2,1 7,5,4,0 7,5,3,1 7,5,2,2 7,4,4,1 7,4,3,2 7,3,3,3
6,6,4,0 6,6,3,1 6,6,2,2 6,5,5,0 6,5,4,1 6,5,3,2 6,4,4,2 6,4,3,3
5,5,5,1 5,5,4,2 5,5,3,3 5,4,4,3
4,4,4,4
On the trianle issue: The sum of the length of any 2 sides must be greater than the 3rd side. So sides of 2, 7 and 9 is not possible because 2 + 7 is not greater than 9. The next set (7,12, 15) is valid because 7 + 12 > 15, 12 + 15 > 7 and 7 + 15 > 12. Really all you need to check is to make sure the sum of the 2 shorter sides is greater than the longest side. So the third set is also valid since 4 + 27 > 30.
I come up with a different solution.
A: Consider all of the different pairings the 2 could be in. There are (15 choose 2)possibilities. That's 105 different pairings. Now count how many pairings are next to each other: (1,2), (2,1), (2,3), (3,2), ... (14,15) and (15, 14). That's 28 pairings out of the possible 105. That's .2666, which is double Luca's answer. And that explains my mistake. There are 15 * 14 possible pairings, not (15 choose 2). So that makes it 28 possibilities out of 210 which is .133333.
B: You still have the 210 different pairings but the number of pairings that meet the requirements is smaller: (1,7), (7,1), (2, 8), (8,2),..., (9,15) & (15,9). There are 18 good pairings out of the 210, or .0857
2. I have no idea what you're asking for here.