Can someone please translate these math problem to a system of equations?

Question

1. Sunshine Truck Rentals rents a truck at a daily rate of $41.95 plus 29 cents per mile. City Rentals the same size truck for $38.95 plus 31 cents per mile. For what mileage is the mmost cost?

2. Susan wants to have business cards printed. One style will cost $25 plus 2 cents per card. ANother style will cost $10 plus 5 cents per cardd. For how many cards will the cost be the same for both styles?

3. Jermaine was in charge of buying milk for a class picnic for 32 students. Milk is sold in half gallons cartons and gallon cartons at a neighborhood grocery store. The half-gallon carton cost $1.46 and the gallon carton cost $2.39. When he got to the store, there was not much milk left. Jermains bought all 21 cartons they had and paid a total of $41.82. How many carton of each size did he buy?

Answer 1

1. x=41.95+.29y z=38.95+.31y
2. x=$25 +.02n z=$10+.05n
3. 1.46h+2.39f=41.82
these are equations not sure if you want the answers to the questions
but here they are any way

1. well for most cost it increases with mileage so most cost would be the mileage where the truck runs out of petrol
but i think your question is where they both are the same cost and which has the best mileage at that point
so here it is

so total cost for both is as stated x and z
and mileage y

so where x = z the cost is the same
so 41.95+.29y =38.95+.31y
to find y we isolate it on one side of the equation as such

38.95+.31y-.29y-38.95=41.95+.29y-38.95-.29y
.02y=3
.02y/.02=3/.02
y=150 miles

2. ok same again not sure if you want answer but here it is anyway

so we take both equations to be equal at the same cost
so 25+.02y=10+.05y
isolate y from numbers
.05y+10-.02y-10=.02y+25-.02y.10
.03y=15
.03y/.03=15/.03
y=500 cards
and to find the actual cost you just put the answer back into one of the two original equations

3. ok this one is different

so we take the total cost equation which is
1.46h+2.39f=41.82
we also know that h+f=21
so we find the value of h
which is equal to 21-f

then we substitute the value for h into the total cost equation and we get
1.46(21-f)+ 2.39f=41.82
30.66-1.46f+2.39f=41.82
30.66+.93f=41.82
30.66+.93f-30.66=41.82-30.66
.93f=11.16
.93f/.93=11.16/.93
f=12
so f = 12 so therefore h= 21-12=9

Answer 2

1) Call the number of miles m. We need to know when the total costs including the mileage is the same for both companies (assuming one day's rental):

41.95 + 0.29m = 38.95 + 0.31m

Bring terms in m to the left and numbers to the right (either do it the logical way by doing the same to each side of the equation or the quick way - move a term to the other side and change the sign):

0.29m - 0.31m = 38.95 - 41.95

0.2m = 3.0 so m = 3.0/0.2 or (since dividing by 0.2 is the same as multiplying by 5) 3 x 5 = 15 miles

another way of doing it is to multiply the top and bottom of the division by 10: 30/2 = 15

2) Use a similar process. Make c the number of cards.

3) This makes two equations. Let x = gallon cartons and y = half gallons.

2.39x + 1.46y = 41.82 and x + y = 21

You should be able to go from there. In (3) you can say that x = 21 - y, so 2.39(21 - y) + 1.46y = 41.82

from that you can find y and, by substituting that value in the most convenient equation, you can find x.

Some people will, no doubt, work the whole thing out for you. The point of these questions, though, is to learn how to do it yourself. If you are not sure, make up some simple problems in words for yourself so that you understand what has to be done.

Answer 3

1. I think there is an error -- the cost will continue to increase as the miles increase, in both cases.

But anyway, the cost S for the truck rental of m miles from Sunshine Truck Rentals will be given by:

S = 41.95 + 0.29m

while the cost C for the truck rental of m miles from City Truck Rentals will be given by:

C = 38.95 + 0.31m

So those are your two systems of equations for this problem.

To actually determine at which mileage the cost is the SAME (more miles than this and City would be more expensive because they charge more per additional mile than Sunshine), you need to set the two equations equal to one another and solve for mileage m:

You are looking for the value of m (miles travelled) at which C = S (cost is the same from either company), so
38.95 + 0.31m = 41.95 + 0.29m
0.31m - 0.29m = 41.95 - 38.95
0.02m = 3
m = 3 / 0.02
m = 150

So if travelling less than 150 miles it will cost less to rent from City, but if travelling more than 150 miles then it will cost less to rent from Sunshine.


2. Same thing, let A be the cost for n cards of the first style and B be the cost for the same number n of the second style,
A = 25+.02n
B = 10+.05n

The number of cards (n) at which the cost is the same is when the equations for the two costs are the same:
10+.05n = 25+.02n
.05n - .02n = 25 - 10
.03n = 15
n = 15/.03
n = 500

The cost is the same at 500 cards. Above this number company A will be cheaper (because their cost per additional card is less), below this company B will be cheaper.

3. You have several things to describe mathematically -- the total cost (given as $41.82), the total number of cartons (given as 21), the number of Gallon cartons (call this G), and the number of Half-gallon cartons (call this H). The 32 students is a red herring, as we don't have anything to relate this to (for example, how much milk per student was purchased):

G + H = 21
G*2.39 + H*1.46 = $41.82

How do you SOLVE that problem then? Well, you express one variable (G) in terms of the other (H):

If G + H = 21, then
G = 21 - H

Therefore, substitute this knowledge about G into the second equation:

G*2.39 + H*1.46 = 41.82
(21-H)*2.39 + H*1.46 = 41.82

Multiply out to eliminate the parenthesis:

50.19-2.39H + 1.46H = 41.82

Combine like terms:

50.19 -0.93H = 41.82

Add the same value to both sides to get all terms in H on one side and all terms in pure numbers on the other:

50.19 -0.93H -41.82 = 41.82 -41.82
8.37 -0.93H = 0
8.37 -0.93H + 0.93H = 0 + 0.93H
8.37 = 0.93H

Now divide both sides by the multiplier of H to get the value of 1H:

H = 8.37 / 0.93 = 9

9 half-gallon cartons were bought.

And since we know the total number of cartons was 21, the number of Gallon cartons is:

G + H = 21
G = 21 - H
G = 21 - 9
G = 12

12 Gallon cartons were bought.

Double check that 12 Gallon cartons at $2.39 each plus 9 Half-gallon cartons at $1.46 each totals $41.82 to verify your answer.

Answer 4

1. 41.95 + 0.29x
38.95 + 0.31x
2. 25 + .02x = 10 + .05x
15 = .03x
x= 500
3.x+y=21
1.46x + 2.39y = 41.82

Answer 5

1

Sunshine: 41.95 + .29x=

City Rentals: 38.95 + .31x=


2. 25 + .02x = 10 + .05x