Ramdom Question

[Salvadoramerican]

just need some clarifying on some

If 5 adults and 2 teenagers work together, they can do a job in one day. If only 2 aduts work then 6 teenagers must in order to do the job in 1 day. If no adults work and 1 teenager works, how long will it take the teenager to do the job?

If it takes 7 min for Phil DeTubb to fill his tub using the hot water faucet and it only takes 5 min with the cold water faucet, and he leaves the hot faucet on for a minite then turns on the cold faucet so their on together. How long will it take to fill the bathtub?

Let a,b,c be positive integers where a,b,c are not necessary distinct. How many order triples (a,b,c) exists such that a+b+c=11

-this type of question sometimes slows me down
Let AB be the diameter of a circle with center C with CG perpindicular to AB, DE perpindicular to AB, and EF perpindicular to DC. LEt AE=x and BE=y such that the aritmic mean of x and y is 12.5 and the geometric mean is 12. Find CE.
-My real problem is not solving the variables i just use my solver but to actually find the CE.

Any tips on this
How many ordered pairs(x,y) are solutions to the equation[unparseable or potentially dangerous latex formula] where x,y are integers and x is greater than or equal to 0 and y is less than or equal to 0?

Also tips on how to do those problems with the 5 column charts and each column is a,b,c,d and begins with {1,2,3,4: down a row going backwards 5,6,7,8} and ask you in which colum would a number be in. I tend to doi it the long way and wonder if their is a faster way

Wow this is the most i have ever ask but these tend to pile up after a while might as well dump them now

[88bobcat]

[88bobcat]

[007math]

Whoa. That is one complicated looking solution for [unparseable or potentially dangerous latex formula]. What I do:

So let each number be a circle; you have:

o o o o o o o o o o o

That's 11 circles. You are wanting to split it into three parts that add up to 11 (hence 11 circles). The numbers have to be positive so we could "draw" two lines between the circles; in other words, you could draw a line in the following spaces that have a line:

o | o | o | o | o | o | o | o | o | o | o

As we see, we have 10 lines and we need to pick 2: 10 choose 2 = 45.

Just alter this method for when you can account for a number being zero (you can draw a line at the endpoints). I'm not sure how this way would work if the numbers have to be distinct. Maybe someone could show that.

Reading over this, it doesn't seem to make sense. Sorry if this doesn't help...

[SumitG]

Your solution definitely makes sense 007math. Have some self-confidence :)
The only point I'd say you were unclear on was the relationship between the original equation given and how it translates into "circles" and "lines".

As for the problem of them being distinct, you can use PIE, and a method similar to what you've already described. This does take somewhat more work though.

[88bobcat]

[PokémonMaster]

For the bathtub one, because after one minute the tub is 1/7 full, both of the faucets will only take 6/7 of the original time to fill the tub by themselves. Because of that, the tub will take 6 minutes to fill with hot water, and 4 2/7 minutes to fill with cold. To find the time it would take for the faucets to fill the remainder of the tub, take (6^-1+ (30/7)^-1)^-1. Then, to account for the hot water running for 1 minute, add a minute and you should get 3.5 minutes. Was this the answer?

[nsguy1350]

[88bobcat]

[Salvadoramerican]

[007math]

You want a total sum of 11, and since each number must be an integer, I let a circle be a unit. Think of it this way: we pretty much want to cut up 11 into 3 ways. Therefore, I wrote out 11 circles (for each unit) and i cut it up into 3 groups, each group being the unknown variable. And name of this method....ugh....i kind of just figured it out so i have no idea.

[SumitG]

There isn't really a "name" for every method to solving a problem...but here are some key definitions which you would use to explain this particular one.

They call your relation between integers and circles a "bijection" or, "one-to-one correspondence". Essentially, you can map every integer to the same number of circles. This map uniquely sends each integer to a different number of circles - if it sent multiple integers to the same number of circles, it would be called solely injective, whereas if one integer corresponded to multiple numbers of circles you would have a purely surjective map. Maps which are both injective and surjective are bijective. It's a core concept to many areas of math, and the way you've used it here is particularly reminiscent of a "combinatorial proof". Essentially, you find a way to represent something abstract in a more concrete way, then solve the problem in your concrete setting with more applicable intuition, and then biject back to your original abstract setting.