Simple Linear Algebra Problem: Calvin and Wendy's Screw Purchase Dilemma

[schmolch]

Hi All!

I hope i posted this in the correct forum. Technically its not homework, just a problem from a puzzle-book which i enjoyed very much until i got to this problem.

Homework Statement

Calvin Carpenter needed screws of three different sizes. He bought the same number of screws of each size at $.03, $.04, and $.05 per screw respectively.
Wendy Woodall spent the same amount of money as Calvin, but divided her money equally among screws of the three sizes. In all, Wendy purchased six more screws than Calvin did. How much money did they each spend?

Homework Equations

The Attempt at a Solution

Calvin gets the same number of screws of each size:

3x+4x+5x=k (cents, x=number of screws, k=total price in cents)

Wendy gets screws for the same price each size:

3a+4b+5c=k
3a=4b=5c

I could mess around some more, but i just don't get it.
Im trying to understand this since more than a week and every day i feel dumber. If anyone could explain this to me i would be very gratefull.

thanks in advance,
schmolch

 

[lanedance]

i think its in your definition of how wendy spends, but were close

first for Calvin as you wrote
3x+4x+5x= 12x = k

now wendy spends her money evenly but divides it equally, as there are three types of screws, each one gets k/3 spent on it

so if she buy a,b,c of respectively

then
3a = 4b = 5c = k/3

so now knowing k, give a,b,c & x

however there is one unused fact, wendy got 6 more screws than Calvin...

 

[Architeuthis Dux]

Hi schmolch,

It seems like you are on the right track with setting up equations for Calvin and Wendy's purchases. However, the key to solving this problem lies in the fact that Wendy purchased six more screws than Calvin did. This means that her total number of screws purchased must be equal to Calvin's total number of screws plus six.

So, let's set up an equation for Wendy's total number of screws:

3a + 4b + 5c = x + 6

Where x is the total number of screws Calvin purchased.

Now, we can also set up an equation for the total amount of money spent by Wendy:

3a + 4b + 5c = k

Where k is the total amount of money spent by both Calvin and Wendy.

Since we know that Wendy spent the same amount of money as Calvin, we can set these two equations equal to each other:

x + 6 = k

Now, we can substitute x + 6 for k in our equation for Wendy's total number of screws:

3a + 4b + 5c = x + 6

3a + 4b + 5c = x + x + 6

3a + 4b + 5c = 2x + 6

Finally, we can substitute our original equation for Calvin's total number of screws (3x + 4x + 5x = k) into this equation:

3a + 4b + 5c = 2(3x+4x+5x) + 6

3a + 4b + 5c = 24x + 6

Now, we have two equations with three variables (a, b, and c) and we can solve for them using basic algebraic methods such as substitution or elimination. Once we have the values for a, b, and c, we can plug them back into our equation for k (total amount of money spent) to find the answer.

I hope this helps you understand the problem better! Don't worry, sometimes it takes time and practice to fully grasp mathematical concepts. Keep at it and don't get discouraged. Good luck!