# Category: College

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:

- Notice that 444 * 111 = 4*111*111
- The square root of 4 is 2
- The square root of 111*111 is 111
- So the square root of 4*111*111 = 2*111
- The answer is therefore 222

Friday, October 1st, 2010

### Solution:

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

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.

Friday, October 1st, 2010

### Solution:

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

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).

Tuesday, September 7th, 2010

### Solution:

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

MUx(64,25) =

MUx(49,36) =

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:

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