1.1. At Wal-Mart, in the hardware department, a customer buys five gallons of paint and six brushes and pays $97.52 for them, including 6% sales tax. Another person buys eight gallons of paint and...

1.1. At Wal-Mart, in the hardware department, a customer buys five gallons of paint and six
ushes and pays $97.52 for them, including 6% sales tax. Another person buys eight gallons of
paint and five
ushes and pays $146.28, including the sales tax. Find the price of a gallon of paint
and that of a
ush.
Suppose the paint sells for $x per gallon and the
ushes are $y each. The total for the first
customer is 5x + 6y. Adding 6% sales tax, then it becomes 1.06(5x + 6y). The total is $97.52. Thus,
we get the equation
1.06(5x + 6y) = 97.52
Likewise, the equation for the second customer is
1.06(8x + 5y) = 146.28
To solve the equations, go to WolframAlpha and write the following instruction
1.06*(5*x+6*y)=97.52,1.06*(8*x+5*y)=146.28
The result comes out as x = 16. and y = 2. Thus paint is $16 per gallon and
ushes are $2 each. ♥
Solution:
We have to solve for the following equations:
1.06 (5x + 6y) = 97.52; this equation becomes 5.3x + 6.36y = 97.52 – Equation (1)
1.06 (8x + 5y) = 146.28; this equation becomes 8.48x+5.3y = 146.28 – Equation (2)
Multiplying equation (1) by 8.48, we get: 44.944x + 53.9328y = 826.9696 – (3)
Multiplying equation (2) by 5.3, we get: 44.944x + 28.09y = 775.284 – (4)
Subtracting equation (4) from (3), ie., (3) – (4), we get:
44.944x + 53.9328y – 44.944x - 28.09y = 826.9696 – 775.284
 25.8428y = 51.6856
 y = 51.6856/25.8428
 y = 2
Now, substituting y=2 in any one of the above equations, we get:
5.3x + 6.36 (2) = 97.52 – Equation (1)
 5.3x + 12.72 = 97.52
 5.3x = 84.8
 x = 84.8/5.3
 x = 16
Therefore, the answers are:
Price of a gallon of paint = $16
Price of a
ush = $2
1.2. The cash flows from two projects under different states of the economy are as follows:
State of the economy Probability Project A Project B
Poor 20% $3000 $0
Average 30% $4000 $7000
Good 50% $6000 $15,000
http:
www.wolframalpha.com
Calculate the following:
(A) Expected cash flows for the two projects $4800, $9600 ♥
(B) Standard deviation of the cash flows $1249, $5919.46 ♥
(C) Coefficient of co
elation between the two projects .9901 ♥
Solution:
(A)
Expected cash flows = Sum of [Probability x Cash flows from each state of economy)
Project A:
 (20% x 3000) + (30% x 4000) + (50% x 6000)
 600 + 1200 + 3000
 $4,800
Project B:
 (20% x 0) + (30% x 7000) + (50% x 15000)
 0 + 2100 + 7500
 $9,600
(B)
Standard deviation of cash flows:
Standard deviation = Square root of Sum of Pi x (Ri - E)2
where: Pi = respective probabilities and Ri= respective returns and E = expected return
Project A:
Deviation Square of Probability
Pi x
Square of
Period Return, Ri
(Ri - Mean
4800) deviation Pi deviation
Poor $3,000 ($1,800) $3,240,000 20% $648,000
Average $4,000 ($800) $640,000 30% $192,000
Good $6,000 $1,200 $1,440,000 50% $720,000
Sum of square of deviations x
espective Pi = $1,560,000
Standard deviation = Square root of 1,560,000  $1,249
Project B:
Deviation Square of Probability
Pi x Square
of
Period
Return,
Ri
(Ri - Mean
9600) deviation Pi deviation
Poor $0 ($9,600) $92,160,000 20% $18,432,000
Average $7,000 ($2,600) $6,760,000 30% $2,028,000
Good $15,000 $5,400 $29,160,000 50% $14,580,000
Sum of square of deviations x
espective Pi = $35,040,000
Standard deviation = Square root of 35,040,000  $5,919.46
(c)
Co-efficient of co
elation
Co
elation(r) =* NΣXY - (ΣX)(ΣY) / Sqrt(*NΣX2 - (ΣX)2+*NΣY2 - (ΣY)2])]
where
N = Number of values or elements
X = First Score
Y = Second Score
ΣXY = Sum of the product of first and Second Scores
ΣX = Sum of First Scores
ΣY = Sum of Second Scores
ΣX2 = Sum of square First Scores
ΣY2 = Sum of square Second Scores

Here, let us assume the X value as for Project A x respective probability and Y value as
for Project B x respective probability.

Step 1: Count the number of values
N=3

Step 2: Find ΣX, ΣY, ΣXY, ΣX2 and ΣY2

Project A Pi X
$3,000 20% $600
$4,000 30% $1,200
$6,000 50% $3,000
Project B Pi Y
$0 20% $0
$7,000 30% $2,100
$15,000 50% $7,500
X Y X*Y X*X Y*Y
$600 $0 0 360000 0
$1,200 $2,100 2520000 1440000 4410000
$3,000 $7,500 22500000 9000000 56250000
$4,800 $9,600 $25,020,000 $10,800,000 $60,660,000
ΣX = 4800; ΣY = 9600; ΣXY = 25,020,000; ΣX2 = 10,800,000 and ΣY2 = 60,660,000

Step 3:
Now, we have to substitute the calculated figures in the above formula.
Co
elation(r) =* NΣXY - (ΣX)(ΣY) / Sqrt(*NΣX2 - (ΣX)2+*NΣY2 - (ΣY)2])]
[3*25,020,000 – 4800*9600]/Sqrt([3*10,800,000 – 4800^2] [3*60,660,000 – 9600^2])
Solving for the above equation, we get:
Co-efficient of Co
elation = 0.99
1.3. Paris...