evaluate the following and break even analysis

Question

Evaluate the following:

1. rose bought 9 pens and 7 pencils and paid a total of $14.80. Ann bought 2 pens and 10 pencils and paid $7. find the price for a single pen and a single pencil
2. one number is 5 times another, the larger number is 9 more than twice the smaller. find the two numbers
3. the difference between the two numbers is 4. the larger is 8 less than twice the smaller. what are the two numbers?
4. eight discs and five zip discs cost a total of $27.50. two discs and four zip discs cost $16.50. find the unit of the unit for each
5. A 20-ft board is cut into two pieces, one of which is 6ft longer than the other. how long is each piece?

Break-even analysis:

1. find the break-even point if the sales revenue $R is given by R=45 n, where n is the number of items sold, $C represents the cost of producing the items, and is given by C=32n+4160

Answer 1

use elimination, substitution or matrix (linear) algebra

9x +7y = 14.8

2x +10y = 7

x+5y = 3.5

7x +35y = 7(3.5) =24.5

45x+35y=74

(45-7)x = 74-24.5

38x = 49.5

x=49.5/38= 9.9/7.8= about $1.30 for a pen, closer to $1.30 and a 1/4 of a cent

y = (3.5-1.3)/5 =2.2/5 =about $0.44 for a pencil, or (3.5-1.3025)/5=2.1975/5= $.4395

check the answers:

2(1.3)+ 10(.44)= 2.6+4.4= 7 fits Ann's cost perfectly

9(1.3) + 7(.44)= 11.7+3.08 = 14.78, close to Rose's 14.80, just 2 cents off

2(1.3025) +10(.4395)= 2.605+4.395= 7

9(1.3025) +7(.4395)= 11.7225+3.0765 = 14.799= $14.79 and 9/10 of a cent

44 cents for a pencil, $1.30 for a pen

(assuming no arithmetic mistakes, when it doesn't come out evenly, it's a clue there may be a mistake, somewhere)

L=5s

L=9+2s

5s=9+2s

3s = 9

s = 9/3 = 3= the smaller number

L=3(5) = 9+2(3) = 15 = the Larger number

L-s = 4

L=4+s

L= 2s-8

4+s=2s-8

s = 4+8=12 = smaller number

L =4+12 = 16 = Larger number

8x+5y = 27.5

2x+4y = 16.5

8x+16y = 4(16.5) = 66

8x+ 5y =27.5

11y = 66-27.5=38.5

y = 38.5/11 = $3.50 = price of zip disc

x = 16.5-4(3.5))/2 = (16.5-14)/2 =2.5/2 = $1.25 = price of a disc

8(1.25) +5(3.5) = 10+17.5=27.5

2(1.25)+ 4(3.5) = 2.5+14 =16.5

20 foot board is cut in 2 pieces, one 6 feet more than the other

L+s = 20

L=s+6

20-s = s+6

2s =20-6=14

s = 14/2 =7 feet for the smaller piece

L=7+6 = 13 feet for the Larger piece

45n= 32n +4160

13n = 4160

n = 4160/13 =320 = breakeven output where profits = zero