Stationery Story Word Problem

[geekily]

"A girl bought some pencils, erasers, and paper clips at the stationery store. The pencils cost 10 cents each, the erasers cost 5 cents each, and the clips cost 2 for 1 cent. If she bought 100 items altogether at a total cost of $1, how many of each item did she buy?"

Last problem on my homework, and I don't know how to set it up. I tried making 2 equations: x + y + z = 100 and .1x + .05y + .005z = 1, but I'm not sure where to go from there, or even if I'm on the right track. I could do it with 2 variables just fine, but 3 kind of throws me off.

Can someone push me in the right direction, please? Thanks so much!

 

[skeeter]

using your equations ...

x + y + z = 100

.1x + .05y + .005z = 1

multiply the second equation by 100 ...

10x + 5y + .5z = 100

now you have two equations that both equal 100 ... set them equal to each other

x + y + z = 10x + 5y + .5z

.5z = 9x + 4y

z = 18x + 8y

since z = 100 - (x + y) ...

100 - x - y = 18x + 8y

100 = 19x + 9y

since x and y are whole numbers, x can be 1, 2, 3, or 4 (5 is too big)

also, 100 - 19x has to be a multiple of 9 ... if this is the case, x can only = 1

so ...

y = 9

z = 90

1 + 9 + 90 = 100

1(.10) + 9(.05) + 90(.005) = $1.00

 

[soroban]

Hello, geekily!

This problem is a bit different from the usual "systems of equations" type.
. . It requires a bit of imagination, too.

A girl bought some pencils, erasers, and paper clips at the stationery store.
The pencils cost 10¢ each, the erasers cost 5¢ each, and the clips cost 2 for 1¢.
If she bought 100 items altogether at a total cost of $1, how many of each item did she buy?

Your equations are correct.
I would express the money in cents, eliminating those ugly decimals.

We have: \(\displaystyle \:\begin{array}{ccc}x\,+\,y\,+\,z& \,=\, & 100 \\ 10x\,+\,5y\,+\,\frac{1}{2}z & = & 100\end{array}\)

We have a fraction ... Multiply the second equation by 2.

. . \(\displaystyle \begin{array}{cccc}x\,+\,y\,+\,z & \,=\, & 100 & \1)\\ 20x\,+\,10z\,+\,z & = & 200 & \2)\end{array}\)

Subtract (1) from (2): \(\displaystyle \:19x\,+\,9y \:=\:100\)


How can we solve for two variables with only one equation?
. . Well, recall that the answers must be positive integers.

Solve for \(\displaystyle y:\;y\:=\:\frac{100\,-\,19x}{9}\)

We see that \(\displaystyle x \,\leq\;5\)
. . and we find that only \(\displaystyle x\,=\,1\) makes \(\displaystyle y\) a whole number.

Hence: \(\displaystyle \,x\,=\,1,\,y\,=\,9,\,z\,=\,90\)

She bought \(\displaystyle 1\) pencil, \(\displaystyle 9\) erasers, and \(\displaystyle 90\) paper clips.


Edit: I was too slow . . . again.

 

[geekily]

Thank you both so much for your answers! I read them both, and they both helped me to better understand the problem and gain more perspective. Thanks for your time! I really appreciate it!