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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?

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 … read more

What is 43 over 100 as a decimal.

Solution:

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

1.079 fl qt = ? ml. My book says that 1L = 1.057 fl qt. but how do i get the L to ml.

Solution:

  1. 1L = 1.057 fl qt
  2. 1L = 1000 ml
  3. 1.079 fl qt = 1.079 x 1.057 L = 1.140503 L
  4. 1.140503 L = 1140.503 ml
… read more

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?

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.… read more

Son accepted to college. Tuition will not increase for 4 years he is attending. First tuition pymt of 10,000 due in 6 months. Same payment due every 6 months unitl a total of 8 pymts made.The college offers a bank account with fixed APR of 4%(semiannually) for the next 4 yrs. You can with draw money every 6 months. How much money must you deposit today if you intend on not making any further deposits and will make all tuition pymts from this acct, leaving the acct empty when the last pymt is made?

Solution:

The APR is 4%, so the semi-annual interest rate is 2%. If you deposit $10,000 in the bank after every six months, you will have $85,829.69 in the bank at the end of 4 years:

  1. After six months you will have $10,000
  2. After one year you will have 10,000×1.02+10,000 = $20,200
  3. After 1.5 years you will have $20,200×1.02+10,000 = $30,604
  4. After 2 years you will have $30,604×1.02+10,000 = $41,216.08
  5. After 2.5 years you will have $41,216.08×1.02+10,000 = $52,040.40
  6. After 3 years you will have $52,040.40×1.02+10,000 = $63,081.21
  7. After 3.5 years you will have $63,081.21×1.02+10,000 = $74,342.83
  8. After 4 years you will have $74,342.83×1.02+10,000 = $85,829.69

The present value of that $85,829.69, discounting at 2% is $73,254.81 (i.e. $85,829.69/(1.02^8)), so if you put this amount in the bank now, and make the payments every six months, there will be nothing left in the account after four years.… read more

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?

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
… read more

A colony of bats is counted every two months. The first four counts are 1200, 1800, 2700, and 4050. If this growth rate continues, (4 points) 1. What is the recurrence relation of the bat population? (2 points) 2. How many bats are there at the 12th count? Show all work. (Hint: solve the recurrence relation above)

Solution:

  1. Recurrence relation
    • Note that 1800/1200 = 1.5.
    • Next, take 1.5* 1800 =2700, and that is the next population.
    • Now 1.5*2700 = 4050.
  2. To calculate the number of bats at the 12th count, take 1200 * 1.511 = 1200 *86.5 = 103,797
… read more

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

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
… read more

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?

Solution:

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

MUx(64,25) =

MUx(49,36) =

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afterChange: function() { var dropdown = document.getElementById('languageDropDown'); var selOpt = … read more

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

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, … read more

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?

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

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languageChangeHandler: function(defaultev) … read more

Solve the Triangle. Round Angle Measures to the Nearest Minute & Side Measures to the Nearest Tenth. a=11.4, b=13.7, c=12.2

Solution:

These are the formulas used to solve triangles:

1. The sum of the internal angles equals 180o …A + B + C = 180o

2. The ‘sine rule’ …

3. The ‘cosine rule’ …

a² = b² + c² – 2bc cosA

or

b² = a² + c² – 2ac cosB

or

c² = b² + a² – 2ba cosC

In the problem that you gave us, we know the length of the three sides of the triangle:

a = 11.4

b = 13.7

c = 12.2

When no angles are known, the cosine rule is the only option, so 11.42 =13.72 +12.22 – 2*13.7*12.2*cosA….Therefore cosA = 0.617955.

You can now use a calculator or a table to find the value of A. You should get A = 0.9047 radians = 51.833 degrees = 51 degrees and 50 minutes.

Use the sine rule to find one of the remaining angles.

=

So sin(B) =  = 0.944835

You can … read more

Cos 2x=0.32

Solution:

First let’s apply the reverse cosine function to both sides of the equation:

cos-1(cos(2x)) = cos-1 (0.32) = cos-1 ()

Now if you take the inverse cosine the  cosine of 2x, you get 2x…so:

2x = cos-1 ()

2x 71.34°

x 35.67°  (which is approximately the same as 0.622 radians)

This looks like it might be the final answer, but actually it’s only one of the many correct answers. One way to see this is by graphing y=cos(2x) and seeing where it intersects the line y=0.32. If you do this, you will see that the intersections occur at multiple points:

As you can see, the intersections occur at x 0.622 radians, 2.519 radians, 3.764 radians, 5.660 radians, etc. The general form of the solution set is as follows:

… read more

2x-5y=9 and 3x+4y=25. Solve by the method of elimination.

Solution:

1.    3x + 4y=25
2.    2x – 5y=9

We could rewrite these as follows by multiplying the first equation by 5 and the second by 4:

1.    15x + 20y = 125
2.    8x  – 20y = 36

If we add the above two equations, we can eliminate y:

23x = 161

So x =3D 7, and plugging this into the first equation we can figure out
y:  21 + 4y =3D 25,

so y = 1… read more

12 out of 1000 women at age forty who participate in routine screening have breast cancer. 880 out of 1000 women with breast cancer will get positive mammographies. 90 out of 1000 women without breast cancer will also let positive mammographies. If 100 women in this age group undergo a routine screening, about what fraction of women with positive mammographies will actually have breast cancer?

Solution:

1.    Out of 100 women, we expect 1.2 to actually have breast cancer (12 out of 1000)
2.    However, only 88% (880 out of 1000) of these will be detected, so 1.056 (i.e. 88% of 1.2) will be detected
3.    In addition, of the 98.8 women who don’t have breast cancer 9% (90 out of 1000) will trigger a false positive” (9% of 98.8 = 8.892)
4.    So out of 100 women, 9.948 will have positive mammography readings (1.056+9.892)
5.    Of these 9.948, only 1.056 will actually have breast cancer
6.    10.6% = (1.056/1.056+9.892) This ratio is essentially an application of Bayes Theorem

Please click on http://www.thegodofreason.com/bayesintro.pdf to download a PDF that explains how Bayes Theorem works. Page 3 gives an example that is very similar to your problem:

… read more

I had 48 marbles in my jar. I lost 1/8 and gave 1/6 to my best friend. How many marbles were left in my jar?

Solution:

  1. Start with 48
  2. Lost 6  (1/8 of 48 =s 6)
  3. Gave 8 to a friend   (1/6 of 48 is 8)
  4. 48 – 6 – 8
  5. = 48 – 14
  6. = 34
… read more

Solve equation. Round to nearest hundredth if neccessary. 11xsquared+3=5(4xsquared-3)

Solution:

  1. The equation you gave us:
  2. Can be re-written =s: or or
  3. The Solution is:
  4. This can also be written in an approximate decimal form: +1.41 and -1.41
… read more

I need help with solving systems by graphing. this is one of the problems y=5x-2 y=x+6

Solution:

First write down the two equations and then pick several values of x and use those values to calculate the corresponding values of y. You can use the  x,y pairs to then plot each line. You will see that the two lines intersect at 2,8 =i.e. x=2 and y=8).

… read more

Find the domain of the function. f(x)= 2x^2+5x+3/2x^2-5x-3

Solution:

First, let’s plot  f(x) =

Here’s what the graph looks like:

Now, the definition of “domain of a function” as follows: The domain of a given function is the set of “input” values for which the function is defined.” For instance, the domain of cosine would be all real numbers, while the domain of the square root would be only numbers greater than or equal to 0 (ignoring complex numbers in both bases).

As you can see from the above graph, the function is valid for all “real” values of x, except for 0. The set of “real numbers”, as you may know, can be thought of as points on an infinitely long number line. The shorthand way of saying “all real numbers except for x” is as follows:

… read more

The total area of a cylinder is 40. If height = 8, find the radius. (The formula for total area used in class is: Total area = 2πrh + 2πr²)

Solution:

The total area is  40π=2πrh+2πr^2
=2πr*8 + 2πr²
=16πr + 2πr²
=16r + 2r²

If we rearrange the terms and divide the terms by 2 (in order to simplify the equation), we get:

r2 + 8r – 20 = 0

(r+10)  (r-2) = 0

r = 2  or  -10

Therefore, r = 2 (assuming that -10 is not an option!).… read more

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