Friday, October 1st, 2010

### Solution:

- Let m = the number of men (an integer)
- Let w = the number of women (an integer)
- Let c = the number of children (an integer)
- Number of bushels = 3m+2w+.5c = 100
- Number of people = m+w+c=100
- 0<m<100
- 0<w<100
- 0<c<100

If we assume that there are 10 children, then we can solve for two equations and two unknowns:

- 3m+2w+5 = 100
- m+w+10=100

We can solve this by subtracting two times the second equation from the first equation:

- 3m+2w+5 = 100
- 2m+2w+20=200 (this is two times the second equation shown above)
- m – 15 = -100 (this is the result of subtracting two times the second equation from the first equation)
- m = -85
- w=100-10- (-85) = 175

The problem with this result is that the values for m and w are outside the allowable range, so clearly the assumption that there are 10 children is wrong. Let’s try increasing the number of children to 80 and see how that impacts the result from m:

- 3m+2w+40 = 100
- 2m+2w+160=200
- m -125 = -100
- m = 25
- w = 100-80-25=-5

Here, the result for m is within the range, but the value for w is outside the range. Let’s try reducing the number of children to 70:

- 3m+2w+35 = 100
- 2m+2w+140=200
- m -105 = -100
- m = 5
- w = 100-70-5= 25

Now we have a solution that makes sense (m=5, w=25, c=70). However, there may be other answers that work, so let’s try increasing the number of students to 72 and see what happens:

- 3m+2w+36 = 100
- 2m+2w+142=200
- m -106 = -100
- m = 6
- w = 100-72-6= 22

This solution also makes sense (m=6, w=22, c=72). Clearly, there can be several correct solutions to this question.

Friday, October 1st, 2010

### Solution:

The answer is forty three hundredths, which is written as 0.43.

Friday, October 1st, 2010

### Solution:

- If you have 26 choices for each letter in a word, then there are 26 such one-letter words, 26
^{2} two-letter words, 26^{3} three-letter words, etc. There would be 26^{8} eight-letter words.
- There would be 26
^{7} eight-letter words that end with the letter N, which is the same as the number of 7 letter words (just add “N” to the end of each seven-letter word.
- There would be 26
^{6} eight-letter words that begin with R and end with N (just add R to the front and N to the end of all six-letter words.
- There would be 2*26
^{7} eight-letter words that begin with an A or B (add A to the beginning of every seven-letter word, then add B to the beginning of every seven-letter word).
- There would be 2*26
^{7} eight-letter words that begin with A or end with B (add A to the beginning of every seven-letter word, then add B to the end of every 7 letter word).

Friday, October 1st, 2010

### Solution:

Each ordered paid represents x and y: (x,y)

To calculate y, you can plug in x.

If y=3x-1, then the given values of x will result in the pairs (0,-1) (2,5) (-1,-4).

If y=3-2x, then the given values of x will result in the pairs (-1,5) (0,3) (1,1).

If y=2x+1, then the given values of x will result in the pairs (0,1) (-1,-1) (1,3).

Friday, July 23rd, 2010

### Solution:

First, let’s draw the trapezoid so that we can visualize the problem. We have not drawn the trapezoid exactly to scale, but this should be accurate enough for our purposes:

Now, we know that MN = 16 because the median equals the average of the top and bottom of the trapezoid

We also know that the distance of Mp = 6, because it just be half the length of BC (the sides of triangle AMp are half the size of the sides of triangle ABC because the triangles are similar and we know that AM is half the length of AB since MN is the median).Similarly,the length of qN = 6.

Now it’s easy to solve for pq:

- Mp + pq + qN = MN = 16
- 6 + pq + 6 = 16
- pq = 4