Building a system of equations

[Suprin]

Homework Statement

It's a verbal problem. It goes like this:

"Peter needs to buy some screws: large, medium, and small. He goes to the hardware store and orders 100 screws for a total of $100. For each large screw he was charged $5.00, $1.00 for each medium one, and $0.05 for each small one.

The point is to express this in a system of linear equations and then figure out how much of each one did Peter order. I can figure out the last part if I can figure out the first part.

Homework Equations

I can use L for the amount of large screws, M for the amount of medium screws, S for the amount of small screws or just XYZ it (for clarity's sake, I wrote both on my papers but I'll stick to LMS here).

The first equation I wrote down goes like this:

$5.00 L + $1.00 M + $0.05 S = $100
or
5L + M + 1/20 S = 100

The Attempt at a Solution

The equation above is my attempt, although upon further reasoning I think it might be wrong. I was thinking maybe it could be something like:

$5.00 L + $1.00 M + $0.05 S = K, where K can stand for any amount of money spent depending on the total purchase.

In the end, I really need some guidance to figure this out. Hopefully someone can give me a hint or a clue as to what it is that I'm not seeing and if my original equation is correct so far. :p

 

[Ray Vickson]

Homework Statement

It's a verbal problem. It goes like this:

"Peter needs to buy some screws: large, medium, and small. He goes to the hardware store and orders 100 screws for a total of $100. For each large screw he was charged $5.00, $1.00 for each medium one, and $0.05 for each small one.

The point is to express this in a system of linear equations and then figure out how much of each one did Peter order. I can figure out the last part if I can figure out the first part.

Homework Equations

I can use L for the large screws, M for the mediums, S for small or just XYZ it (for clarity's sake, I wrote both on my papers but I'll stick to LMS here).

The first equation I wrote down goes like this:

$5.00 L + $1.00 M + $0.05 S = $100
or
5L + M + 1/20 S = 100

The Attempt at a Solution

The equation above is my attempt, although upon further reasoning I think it might be wrong. I was thinking maybe it could be something like:

$5.00 L + $1.00 M + $0.05 S = K, where K can stand for any amount of money spent depending on the total purchase.

In the end, I really need some guidance to figure this out. Hopefully someone can give me a hint or a clue as to what it is that I'm not seeing and if my original equation is correct so far. :p

Besides the equation 5L + M + (1/20)S = 100 (spend $100) you also need to buy a total of 100 screws.

You will have 2 equations in 3 unknowns, so you can, say, solve for L and M in terms of S. Then try integer values of S until you hit integer values of both L and M.

 

[Suprin]

That's the part I'm seriously stuck at; the whole "buys 100 screws" ordeal. I'm pretty sure I know how the solution for the system will come out; more than likely a free variable since there will be a good number of possible solutions.

 

[Ray Vickson]

That's the part I'm seriously stuck at; the whole "buys 100 screws" ordeal. I'm pretty sure I know how the solution for the system will come out; more than likely a free variable since there will be a good number of possible solutions.

What do you mean by the symbols L, M and S?

 

[Suprin]

Large, Medium, and Small. The respective amount of each size of screw.

 

[Ray Vickson]

Large, Medium, and Small. The respective amount of each size of screw.

Yes, I know they referred to large, medium and small, but to what aspects? You have now clarified this: L = number of large to purchase, etc. Always do this from the start; that is: step 1 should always be to define your variables (including units, if applicable). This will help you keep thing straight and will certainly help the poor sap that has to mark the papers; it might even boost your marks.

 

[Suprin]

I did in the very first post. Or are you referring that I should've added the words "number of" or "amount of" as well?

 

[mafagafo]

since there will be a good number of possible solutions.

For the system. I don't believe this problem allows (logically) more than one solution.

 

[Ray Vickson]

I did in the very first post. Or are you referring that I should've added the words "number of" or "amount of" as well?

No, if you read what you wrote you did not explicitly say they were the quantities, although, I assumed that is what you meant. You then seemed to get bogged down, and so I was not sure after all whether you really meant what you said.

Anyway, if you solve for two of the variables in terms of the third, you will, of course, have infinitely many possible solutions---one for each setting of the third variable. However, you need more: all three of your variables must be whole numbers (integers ≥ 0) and that cuts down the possibilities tremendously.

 

[Suprin]

To mafagafo:
That's what a friend just helped me realize on the phone. Still, I *have* to express this as a system of equations. It's the very first instruction on the paper. Solving it should be simple enough.To Ray:
It's all even more detailed on the papers. I'm bogged down because I can't figure out how to write the missing equation(s) for the system. That's where I'm stuck. That's where I need a hand.

 

[Ray Vickson]

To mafagafo:
That's what a friend just helped me realize on the phone. Still, I *have* to express this as a system of equations. It's the very first instruction on the paper. Solving it should be simple enough.


To Ray:
It's all even more detailed on the papers. I'm bogged down because I can't figure out how to write the missing equation(s) for the system. That's where I'm stuck. That's where I need a hand.

Sorry: no more hints.

 

[Mark44]

To mafagafo:
That's what a friend just helped me realize on the phone. Still, I *have* to express this as a system of equations. It's the very first instruction on the paper. Solving it should be simple enough.


To Ray:
It's all even more detailed on the papers. I'm bogged down because I can't figure out how to write the missing equation(s) for the system. That's where I'm stuck. That's where I need a hand.

There is no missing equation. Your two equations represent the system that you are to solve. Use the hint that Ray has given you, that any solution has to have integer values for the three variables.

 

[Suprin]

There is no missing equation. Your two equations represent the system that you are to solve. Use the hint that Ray has given you, that any solution has to have integer values for the three variables.

I don't have 2 equations. I only wrote one.

 

[Mark44]

I saw that you had two equations, but didn't realize that one of them was just the simplified form of the first.

How many screws are ordered?

 

[Dick]

I don't have 2 equations. I only wrote one.

Then you should write the other one. What equation corresponds to the statement that the number of screws equals 100?

 

[Suprin]

I saw that you had two equations, but didn't realize that one of them was just the simplified form of the first.

How many screws are ordered?

100 screws total for a total of $100. My best guess is that the second equation would be:

L + M + S = 100 screws

My guess is that the Small screws purchased are a multiple of 20. Problem is that the professor does want us to show the procedure. I wasn't able to get a hold of her via e-mail or phone to see if she'd accept us literally writing "we guessed and were able to prove it mathematically" as an answer. So, assuming I'm right, these are the 2 equations I have:

$5 L + $1 M + $0.05 S = 100
L + M + S = 100 screws

Where L is the amount of Large screws, M is Medium screws, S for Small screws.


I tried giving S a value that would turn the 0.05 into a whole number (began with 20). Then solved for L in the (L + M + S = 100) equation and just kept solving back and forth, but no idea if that's even the right procedure.

I know that if the problem had another sentence giving a bit more of info (ie: the amount of Large screws are double the amount of Medium screws) then it'd be easier to solve, but that's all there is to the problem.

 

[Ray Vickson]

100 screws total for a total of $100. My best guess is that the second equation would be:

L + M + S = 100 screws

My guess is that the Small screws purchased are a multiple of 20. Problem is that the professor does want us to show the procedure. I wasn't able to get a hold of her via e-mail or phone to see if she'd accept us literally writing "we guessed and were able to prove it mathematically" as an answer. So, assuming I'm right, these are the 2 equations I have:

$5 L + $1 M + $0.05 S = 100
L + M + S = 100 screws

Where L is the amount of Large screws, M is Medium screws, S for Small screws.


I tried giving S a value that would turn the 0.05 into a whole number (began with 20). Then solved for L in the (L + M + S = 100) equation and just kept solving back and forth, but no idea if that's even the right procedure.

I know that if the problem had another sentence giving a bit more of info (ie: the amount of Large screws are double the amount of Medium screws) then it'd be easier to solve, but that's all there is to the problem.

You don't need to guess: that is the other equation. It is obvious that it must hold, since it was given right away as part of the problem statement---you buy 100 screws. In fact, that is exactly why I asked you specifically what you meant by L, M and S, because then the answer would be staring you in the face, or so I thought.

I would suggest that you use the two equations to solve (symbolically) for L and M as functions of S, using exact rational arithmetic (no decimals). The answer ought to be very revealing.

 

[Suprin]

You don't need to guess: that is the other equation. It is obvious that it must hold, since it was given right away as part of the problem statement---you buy 100 screws. In fact, that is exactly why I asked you specifically what you meant by L, M and S, because then the answer would be staring you in the face, or so I thought.

I would suggest that you use the two equations to solve (symbolically) for L and M as functions of S, using exact rational arithmetic (no decimals). The answer ought to be very revealing.

Sorry. It wasn't obvious to me since I didn't know it was fine to have one equation ending in dollars and the other in another unit (in this case screws). I had in fact written it on a piece of paper (somewhere), but the previously stated reasoning was what stalled me.

What Dick and Mark said was what finally confirmed what I suspected.

 

[Mark44]

For your two equations:

$5 L + $1 M + $0.05 S = 100
L + M + S = 100 screws

you aren't consistent with the units in the first equation; e.g. $5L and the other terms on the left side, and a bare 100 on the right side. It's better to omit the units entirely, but if you feel you need to be more explanatory, you can add something in parentheses to indicate what the units are, like so:
5L + 1M + .05S = 100 (in dollars)