Help with Unit/Rate problem

Valmont

I am really stuck on a problem and I hope somebody can either explain the logic to me or give me a helpful hint as per point five in the forum charter. Here we go:

6 people can paint 5 walls in 46 minutes. How many minutes will it take for 10 people to paint 9 walls?

I eventually caved and asked for a hint (this is from khanacademy.org) and it has given me the formula: w = r. t. p
W=walls, r=rate, t=time, and p=people. Fair enough I can get the answer with this but I really don't understand the formula. Why does multiplying those three terms (describing qualitatively different entities) give me how many walls were painted?

Thanks in advance.

locum-motion

Valmont wrote: »

I am really stuck on a problem and I hope somebody can either explain the logic to me or give me a helpful hint as per point five in the forum charter. Here we go:

6 people can paint 5 walls in 46 minutes. How many minutes will it take for 10 people to paint 9 walls?

I eventually caved and asked for a hint (this is from khanacademy.org) and it has given me the formula: w = r. t. p
W=walls, r=rate, t=time, and p=people. Fair enough I can get the answer with this but I really don't understand the formula. Why does multiplying those three terms (describing qualitatively different entities) give me how many walls were painted?

Thanks in advance.

The unit of 'Rate' is 'walls per minute per person".

5 walls = rate x 46 minutes x 6 people

=> 5 walls / rate = 46 minutes x 6 people

=> 1 / rate = 46 minutes x 6 people / 5 walls

=> rate = 5 walls / 46 minutes x 6 people

=> rate = 5 / (46x6) walls per minute per person

=> rate = 5/276 walls per minute per person

=> rate = 0.00362319 0.018116 walls per minute per person.

Then take this rate and put it into the formula with the new data:

9 walls = 0.00362319 0.018116 walls per minute per person x t minutes x 10 people

=> 9 walls = 0.0362319 0.18116 walls per minute x t minutes

=> 9 walls / 0.0362319 0.18116 walls per minute = t minutes

=> 9 / 0.0362319 0.18116 minutes = t minutes

=> 248.4 49.68 minutes = t minutes


Edit: Correcting my mistake as pointed out by Yakuza

Yakuza

In terms of units, you will have [latex]people \times \frac{walls}{minute} \times minutes [/latex], or the compound unit PeopleWalls

In the case of your question:


[latex] 5 PeopleWalls = 6 people \times x \frac{walls}{minute} \times 46 minutes [/latex]

If you solve for x, then you will have the time it takes 1 person to paint 1 wall

You're now looking to solve for y the equation 9 = 10 * x * y where x the value you get from above.

Edit : Locum-motion beat me to the punch. I should have refreshed before posting

I agree with LM's answer for the first part (the rate), but I think there's a boo-boo with the second part (0.00362319 is 1/276 not 5/276) as my answer is 5 times smaller than theirs!

MathsManiac

Valmont wrote: »

I am really stuck on a problem and I hope somebody can either explain the logic to me or give me a helpful hint as per point five in the forum charter. Here we go:

6 people can paint 5 walls in 46 minutes. How many minutes will it take for 10 people to paint 9 walls?

I eventually caved and asked for a hint (this is from khanacademy.org) and it has given me the formula: w = r. t. p
W=walls, r=rate, t=time, and p=people. Fair enough I can get the answer with this but I really don't understand the formula. Why does multiplying those three terms (describing qualitatively different entities) give me how many walls were painted?

Thanks in advance.

Alternatively, without using a formula, you can use what is sometimes called the "unitary" method, as it seeks to work down to a single unit and back up again. At each stage, you're either multiplying or dividing, and a little thought should tell you which:

6 people can paint 5 walls in 46 minutes.
1 person can paint 5 walls in 46*6 = 276 minutes.
1 person can paint 1 wall in 276/5 = 55.2 minutes.
1 person can paint 9 walls in (55.2)*9 = 496.8 minutes.
10 people can paint 9 walls in 496.8/10 =49.68 minutes.

(Problems like this always assume a proportionality that might not be true in reality, especially in extreme cases. For example, the same logic used in this question would tell you that 3312 people could paint a wall in one second, which I doubt!)

locum-motion

Yakuza wrote: »

...theirs!

I presume you've used "theirs" as it's non-gender specific.
For future reference, I'm a "him".

locum-motion

MathsManiac wrote: »

...
(Problems like this always assume a proportionality that might not be true in reality, especially in extreme cases. For example, the same logic used in this question would tell you that 3312 people could paint a wall in one second, which I doubt!)

LOL.

Yakuza

locum-motion wrote: »

I presume you've used "theirs" as it's non-gender specific.
For future reference, I'm a "him".

Yes, I did. I referred to "Princess Petulia Pissy Pants" as a she in the Countdown thread; turns out she's a he, so I go with non gender-specific terms when I'm not sure

locum-motion

Yakuza wrote: »

Yes, I did. I referred to "Princess Petulia Pissy Pants" as a she in the Countdown thread; turns out she's a he, so I go with non gender-specific terms when I'm not sure

I would have thought that PPPP's name was a fairly safe bet!

Valmont

Thanks everyone. The unitary method makes perfect sense. Looking at how straightforward it is I definitely need to slow down and not over-complicate things.