math help please

[TechWife]

I am in need of assistance as I am not a math person. I am making care packages that are all identical and contain multiple items. Each items is sold in bulk, but the bulk packages are of different sizes (24 combs, 36 deodorant, 10 cosmetic bags, 50 shampoo, 12 pencils, etc.). Is there an equation that I can use to calculate the number of each package I need to buy so that I can make an even number of care packages without having left over items? Or, if that isn't possible, how can I calculate the least number of items left over?

In case I am not making sense - this is the "word problem" version.

TechWife is making care packages. Each care package will contain 1 cosmetic bag, 1 comb, 1 deodorant, 1 shampoo, 1 pen and 1 notebook. She can save money on each item by buying in bulk. One box of combs contains 24 combs, 1 box of deodorant contains 36 deodorants, 1 box of shampoo contains 50 shampoos, 1 box of pens contains 18 pens and 1 box of notebooks contains 12 notebooks. How many boxes of each item does she need to buy in order to make sure she doesn't have any items left over when she is finished making her care packages?

 

[Tanaqui]

Okay, let's ignore the shampoos right now.

Your other items are all sold in units divisible by 6. For every 2 notebooks you have 3 pens, 4 combs, 6 deodorants. In order to make this match up neatly you'll want to purchase like this:

2 packages of deodorant

3 packages of combs

4 packages of pens

6 notebooks

This gets you into units of 12. EDIT: Specifically, going back to your original numbers, you'll have 72 of each of these items. If you doubled these numbers, you'd have 144 of each item. Then if you bought three boxes of shampoo you'd have 150 shampoos. Those numbers are pretty close, you're left with 6 shampoos to donate to the local food pantry or women's shelter.

 

Edited August 9, 2020 by Tanaqui

[Tanaqui]

I'm assuming these are the real numbers, btw. If not, um, I'll try to write something out with variables instead of numbers but I'm a little brain fogged right now. I seem to have a cold. (I think it's a regular cold, not a scary covid cold. I don't know how I got sick, I don't go anywhere or do anything or see anyone.)

 

Edit: Yeah, brain fog struck. I was so happy with my math I forgot those weren't the original numbers at all. Hold on, lemme redo this. My first step is right, but the part where I end up with 48 of each is still wrong.

Edited August 9, 2020 by Tanaqui

[TrulySusan]

The least common multiple of the numbers you listed is 1800.

Now, just divide 1800 by the number of items to see how many boxes you need.

So, packs of 50 combs would be 1800/50=36 boxes and so on.

Im guessing that’s way more bags than you want to make 😂 

[Bootsie]

Are you concerned simply about having the least amount of items left over?  Or , are you concerned about making the most boxes for the least amount of money?  I would think having, for example, 3 pencils left over is minor compared to having 3 shampoos left over. 

[happi duck]

Except for deodorant 48 might work decently.

2 boxes of combs

1 box shampoo (2 left)

3 boxes pens (6 left)

4 boxes of notebooks

2 boxes deodorant (24 left) or 1 box plus 12 not bulk

Cosmetic bags if 10 pack then 5 boxes (2 left)

1 hour ago, TechWife said:

I am in need of assistance as I am not a math person. I am making care packages that are all identical and contain multiple items. Each items is sold in bulk, but the bulk packages are of different sizes (24 combs, 36 deodorant, 10 cosmetic bags, 50 shampoo, 12 pencils, etc.). Is there an equation that I can use to calculate the number of each package I need to buy so that I can make an even number of care packages without having left over items? Or, if that isn't possible, how can I calculate the least number of items left over?

In case I am not making sense - this is the "word problem" version.

TechWife is making care packages. Each care package will contain 1 cosmetic bag, 1 comb, 1 deodorant, 1 shampoo, 1 pen and 1 notebook. She can save money on each item by buying in bulk. One box of combs contains 24 combs, 1 box of deodorant contains 36 deodorants, 1 box of shampoo contains 50 shampoos, 1 box of pens contains 18 pens and 1 box of notebooks contains 12 notebooks. How many boxes of each item does she need to buy in order to make sure she doesn't have any items left over when she is finished making her care packages?

 

 

[TechWife]

Wow - the big numbers are what I was afraid of. I actually have more items for the care packages, but these examples are real numbers.

They are care packages for caregivers (usually parents) who are staying with children  arriving through the ED. Each bag will have: 

shampoo, conditioner, soap, lotion

wide tooth comb

toothbrush & toothpaste 

deodorant

lip balm

gripper socks

sleep eye shield

small notebook & pen

coloring book for adults & colored pencils w/pencil sharpener 

Each item is inexpensive but the bulk quantities are messing with my brain. For example - there is a bulk package that has 50 bottles of shampoo, 50 bottles of conditioner and 100 bars of soap. I could add a package that has 30 bottles of shampoo and 30 bottles of conditioner but then still have 20 bars of soap left over. Then the lotion comes in a box of 24. The pencil sharpeners are in packs of four, the eye shields in a package of 30. See my problem? 

i think I’m going to get one package of the toiletry combo then add packages of the others one by one until I reach my budget limit because there is no way to make it even up and stay in budget. So, one package of everything, then add one by one to get them as even as possible until I hit my budget. I’ll save the extra and do this again in six months. 

 

[happi duck]

I noticed that the oral surgeon and periodontist both had personalized lip balm to give patients.  If that's true in your area perhaps they might donate some.  Seems worth an email.

Dentists might donate brushes and toothpaste.

If you want small, a hotel might donate shampoo etc.

A hotel or other business may have promo pens to donate.

 

[Ali in OR]

Just want to say this is really cool, and the two times we were rushed to the children's hospital, it was the socks that I wanted most! It was summer and I probably just had flip flops--I just remember my feet being freezing in the hospital and really wished I had socks.

[Tanaqui]

Quote

Wow - the big numbers are what I was afraid of. I actually have more items for the care packages, but these examples are real numbers.

 

It's the shampoo that's screwing this up, that big number of shampoos that doesn't really share enough useful prime factors with everything else. You can make 24 or 36 bags and just resign yourself to having some extra stuff.

[Ottakee]

Those are nice care packages.  I know I would have loved one the many times we ended up inpatient at the hospital.