Answere, Week 7, Problem Solving practice test
1. Flower Boxes: 12 pots, 21 small boxes, 15 large boxes
Formulas: You know that you need 48 boxes. Lets let x be the small pots, y be the middle size boxes, and z be the bigger boxes. Since the sum of all 3 is 48, you have: x + y + z = 48
You know that the total you must spend is $129. Since that is from all 3 types of pots, and each x pot costs $1, each y box costs $2, and each z box costs $5, you have this equation: 1x + 2y + 5z = 129
Since you have 3 times as many boxed as you have pots, and x is the number of pots, you have this final equation: 3x = y + z
Your 3 equations, then are x + y + z = 48
x + 2y + z = 129
3x = y + z
Now you have 9 different ways to proceed. You need to solve for one of the variables. I would pick the last one: Subtract z from both sides to get:
3x – z = y
Now plug this into the other 2 equations:
x + (3x-z) + z = 48
x + 2(3x-z) + 5z = 129
Now you have 2 equations and 2 variables. Notice that you no longer have a y variable. I will pick the first equation ans slove for x, then plug the answer into the 2nd equation to solve for z.
x + (3x-z) + z = 48 or 4x – z + z = 48 or 4x = 48, or x = 12. Now you know that you must buy 12 small pots. Plug 12 in for x in the second equation:
x + 2(3x-z) + 5z = 129 or 12 + 2(3(12) – z) + 5z = 129 or 12 + 72 – 2z + 5z = 129 or 84 +3z = 129 or 3z = 45 or z = 15
now plug x = 12 and z = 15 into any of the initial equations to find that y = 21.
2. Music company: You have 2 equations for the 2 companies: Since company A charges 75 up front, you start with an expenditure of $75, and the rate of increase is $28 per month The equation for Comany A is:
y = 75 + 28x (where y is the total expense and x is the month)
Similarly, Company B’s formula is y = 120 + 22.5x
Find when they are both the same, and this ask if this is more or less than 7 months. To find when they are the same, since both equations = y, set them equal to each other using substitution:
75 + 28x = 120 + 22.5x Solve for x, the number of months before the 2 companies will cost the same:
7.5x = 45 or x = 6. Therefore, in 6 months each plan will cost you the same.. ie 75 + 28(6) = 120 + 22.5(6) Since you are gone in another month, you should choose comany B, since it will cost you only $22.5 for month 7, while Company A will charge $28. Answer: Company B
3. To ship 14,000 hats, you can fill 2 crates of 5000. This will leave you with 4000 hats to ship. Divide 300 into 4000. You will get 13 boxes with a remainer of 100 hats. You will need 9 bags to send these.. (12 x 9 = 108.. so you will have 1 partial bag) So, the total cost to ship will be:
10,000(1) + 13(300)(2) + 9(12)(5) or 10,000 + 7800 + 540 or 18, 340 cents or $183.40
4. The 2 equations you can use are: A = P(1 + r/n)^nt, where r is the rate, n is the number of times you will compound per year, and t is the number of years invested. P is the Principal amount. The other equation you will use is the one for compounding continuously, which is: A = Pe^rt, where e = the natural log or 2.72.
Your choices, then, are P(1 + .0675/4)^4(2) or P(2.72)^.0650(2)
Which one has the highest effective yield? The base interest is Pr^t, where r is the base interest rate if you compounded at the end of the year. Notice t=2 and P = P in this example. You are, then, comparing (1 + .0657/4)^4 with (2.72)^.065 Which is bigger? The first is 1.06923. The second is 1.06716. Therefore, the effective yield for bank A is larger. Choose Bank A.