# One hundred bushels of corn are to be divided among 100 persons. men get 3 bushels each, women 2 and children 1/2. how many men, women and children are present?

Friday, October 1st, 2010

### Solution:

1. Let m = the number of men (an integer)
2. Let w = the number of women (an integer)
3. Let c = the number of children (an integer)
4. Number of bushels = 3m+2w+.5c = 100
5. Number of people = m+w+c=100
6. 0<m<100
7. 0<w<100
8. 0<c<100

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

1. 3m+2w+5 = 100
2. m+w+10=100

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

1. 3m+2w+5 = 100
2. 2m+2w+20=200 (this is two times the second equation shown above)
3. m          – 15 = -100 (this is the result of subtracting two times the second equation from the first equation)
4. m = -85
5. 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:

1. 3m+2w+40 = 100
2. 2m+2w+160=200
3. m          -125 = -100
4. m                 =  25
5. 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:

1. 3m+2w+35 = 100
2. 2m+2w+140=200
3. m          -105 = -100
4. m                    =    5
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:

1. 3m+2w+36 = 100
2. 2m+2w+142=200
3. m          -106 = -100
4. m                    =    6
5. 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.

# What is the square root of 444*111?

Friday, October 1st, 2010

### Solution:

1. Notice that 444 * 111 = 4*111*111
2. The square root of 4 is 2
3. The square root of 111*111 is 111
4. So the square root of 4*111*111 = 2*111
5. The answer is therefore 222

# X has 5 base-ten blocks. She models a number that is less than 999. The number she models has a zero in the tens place and a zero in the ones place. What number is she modeling?

Friday, October 1st, 2010

### Solution:

Base ten blocks are a mathematical manipulative used to learn basic mathematical concepts including addition, subtraction, number sense, place value and counting. You can manipulate the blocks in different ways to express numbers and patterns. Generally, the 3-dimensional blocks are made of a solid material such as plastic or wood and come in four sizes to indicate their individual place value: Units (one’s place), Longs (ten’s place), Flats (hundred’s place) and Big Blocks (thousand’s place). There are also computer programs available that simulate base ten blocks.

Since the number that is being modeled has a zero in the tens place and a zero in the ones place, there are no Longs or Units. Since the number is less than 999, there are no Big Blocks. That leaves only Flats, and there must be five Flats. Since each Flat represents one hundred, the number must be 500.

# Find the minimum number of students needed to guarantee that 4 of them were born: hint use pigeonhole principle. 1. on the same day of the week; 2. in the same month. 3. Which counting principle applies to the questions above?

Friday, October 1st, 2010

### Solution:

1. There are seven days in a week
2. If there are 7 students, they could all have been born on different days of the week
3. If there are 8 students, we could guarantee that 2 of them were born on the same day
4. If there are 15 students, we could guarantee that 3 of them were born on the same day
5. F there are 22 students, we could guarantee that 4 of them were born on the same day

# Suppose a Word is a String of 8 letters of the Alphabet with Repeated Letters Allowed: (5 points, 1 point each) Show all work. Do not Answer with just a Number. 1. How many words are there? 2. How many words end with the letter N? 3. How many words begin with R and end with N? 4. How many words start with A or B? 5. How many words begin with A or end with B

Friday, October 1st, 2010

### Solution:

1. If you have 26 choices for each letter in a word, then there are 26 such one-letter words, 262 two-letter words, 263 three-letter words, etc. There would be 268 eight-letter words.
2. There would be 267 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.
3. There would be 266 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.
4. There would be 2*267 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).
5. There would be 2*267 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).

# Complete These Ordered Pairs: (0, ) (2, ) (-1, ) y=3x-1 (-1, ) (0, ) (1, ) 2x+y=3 (0, ) (-1, ) (1, ) y=2x+1

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).

# What is the cardinality of the following set. P(S) stands for power set. Find p (p ((S) for S = {a, b}

Friday, July 23rd, 2010

### Solution:

1.    The cardinality of a set is a measure of the “number of elements of the set”. For example, the set A = {2, 4, 6} contains = elements, and therefore A has a cardinality of 3.
2.    Given a set S, the power set of S, written , P(S), is the set of all subsets of S, including the empty set and S itself.
3.    If S={a, b}, then P(S) ={{},{a},{b},{a,b}} and the Cardinality of P(S) is 4.

# You are developing new bath soap, and you hire a public opinion survey group to do some market research for you. The group claims that in its survey of 450 consumers, the following were named as important factors in purchasing bath soap: Odor 425 Lathering Case 397 Natural Ingredients 340 Odor and lathering Case 284 Odor and Natural Ingredients 315 Lathering ease and natural ingredients 219 All three factors 147 should you have confidence in these results? Why or why not?

Thursday, July 22nd, 2010

### Solution:

First, take a look at this link to understand the Principle of Inclusion and Exclusion:

http://en.wikipedia.org/wiki/Inclusion%E2%80%93exclusion_principle

Next, make a list of all the subsets:  # A cheese shop carries a large stock of 34 kinds of cheese. By the end of the day, 48 cheese sales have been made, and the items sold must be restocked. How many different restocking orders are Possible?

Monday, July 19th, 2010

### Solution:

1. 48 cheese sales have been made
2. The first sale could have been one of 34 different types of cheese
3. The second sale could also have been one of 34 different types of cheese etc.
4. The 48th sale could also have been one of 34 different types of cheese
5. Each sale is independent of every other sale, so each event increases the number of possible restocking orders by a factor of 34
6. Therefore, the number of possible restocking orders = 3448