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Category: College

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

What is 43 over 100 as a decimal.

Friday, October 1st, 2010

Solution:

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

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).

If Utility Function is Expressed as U(x,y) = x^0.5 y^0.5 What is the Marginal Utility at Point (64,25) and (49,36)? Treat Y as a Constant. Would the Answer be MUx(x,y) = y. MUx(64,25) = 5 ? and MUx(49,36) = 6?

Tuesday, September 7th, 2010

Solution:

MUx(x,y) =     Derivative of U(x,y), treating y constant.

MUx(64,25) =

MUx(49,36) =

Use Mathematical Induction to Prove That the Statements are true for Every Positive Integer n. 1*3+2*4+3*5+⋯+n(n+2)=n(n+1)(2n+7)/6

Monday, September 6th, 2010

Solution:

First, let’s make sure we understand what we mean by “mathematical induction”: “Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. It is done by proving that the first statement in the infinite sequence of statements is true, and then proving that if any one statement in the infinite sequence of statements is true, then so is the next one.” (Wikipedia) We have pasted below another more detailed explanation of how to create a proof using mathematical induction (see “Appendix” below).

Let’s now take the equation you provided: 1*3+2*4+3*5+…+n(n+2)=

We can show that this is true for n=1:         1*3 = = = 3

Now let us assume that the statement is true for n = k. If it is, then we will prove that it has to be true for n=k+1:

WTS è =       +

=   

=

=

=

=       QED

In the above proof, WTS means “want to show” and QED means “quod erat demonstratum” (“which was to be demonstrated”).

Appendix

Here’s another more detailed explanation of how to create a proof using mathematical induction:

A Jeweler Needs to mix an Alloy with 16% Gold Content & an Alloy with a 28% Gold Content to Obtain 32oz. of a new Alloy with a 25% Gold Content. How Many oz. of Each of the Original Alloys Must be Used?

Monday, September 6th, 2010

Solution:

  • A = number of ounces of the first alloy
  • B = number of ounces of the second all
  • A+B=32
  • 0.16 A +0.28 B = 32 x 0.25
  • Since A + B = 32, we know that A = 32 – B. You can then re-write the equation above as 0.16 (32-B) + 0.28 B = 32 x 0.25
  • This can be simplified to 5.12- 0.16 B + 0.28 B =8
  • Or, 0.12 B = 2.88
  • Or B = 24
  • So A = 8