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 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?

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.

Solution:

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

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

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.

The ritz carlton hotel used a customer opinion questionare to obtain performing data about its dining and entertainment services. Customers were asked to rate six factors: Welcome, Service, food, menu, appeal, atmosphere and overall experience. Data were recorded for each factor 1 for fair 2 for average and 3 for good and 4 for excellent a) The customers response provided data for six variables. Are the variables quantative or qualitative? b) What measurement of scale if used?

Solution:

1. A quantitative variable is a phenomenon measured in amounts, that is, numerical units. For example, a numerical score is a quantitative variable.
2. The measurement is based on a scale of 1 to 4.

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.

Many academic institutions offer a sabbatical policy. Every seven years a professor is given a year free of teaching. A professor earning 70,000 per year who works for a total of 42 years what is the present value of the amount they will earn while on sabbatical, if the interest rate is 6% EAR?

Solution:

1. In his seventh year, the professor receives $70,000. Since the question does not state when he receives this amount, let’s assume it is received at the end of the seventh year as a lump sum. The present value of this is $46,554 (i.e. $70,000/(1.06^7))
2. Similarly, the professor receives $70,000 at the end of year 14, and the present value of this is $30,961.07
3. Similarly, the professor receives $70,000 at the end of year 21, and the present value of this is $20,590.88
4. Similarly, the professor receives $70,000 at the end of year 28, and the present value of this is $13,694.11
5. Similarly, the professor receives $70,000 at the end of year 35, and the present value of this is $9,107.37
6. Similarly, the professor receives $70,000 at the end of year 42, and the present value of this is $6,056.92

Now we can add up each of the six present values, to arrive at the value of all the amounts received during the sabbaticals: $126,964.34 (if you add up the numbers above you get $126,964.35, but that’s due to rounding error).

If the professor receives the money at the middle of each year, you can re-calculate the above (e.g. by discounting the seventh year by 6.5 years rather than by 7 years).  You might also want to recalculate this assuming that the salary is paid monthly, and see how it compares to the above result.

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

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 12^th count, take 1200 * 1.5¹¹ = 1200 *86.5 = 103,797