Taps

mrlm1105:

If the hot water tap can fill a bathtub in 60 minutes, and the cold water tap fills it in 30 minutes, how many minutes does it take when they are running together?

First you should think about what would be a reasonable answer. If both taps are running then it is going to take less time than if only one was running. So the answer will have to be less than 30 minutes. Always do reasonableness checks.

You can do this with algebra if you want but i'm going to do it an easier way.
you will get the same answer no matter how much water the bath holds so I am just going to say that it is a 60L bath. That will be an easy number to work with.
The hot water would fill the 60L in 60 minutes. That's 60/60 = 1 L per minutes
The cold water would fill the 60L in 30 minutes. That's 60/30 = 2 L per minutes
Together that will run (1+2) = 3L/ 1min
this can be said the other way around It will take 1 minute to run 3L = 1min / 3L
You have 60L to run
1min/3L * 60L = 20minutes (If you write your fractions upright you will see that the Litres cancel out!)
That passes the reasonable check. Good!

If you want to do it more generally you can replace my 60L with x L. The answer will be the same.

mrlm1105 wrote:If the hot water tap can fill a bathtub in 60 minutes, and the cold water tap fills it in 30 minutes, how many minutes does it take when they are running together?

Melody already provided a good way to solve this........let me show you an alternate way that only involves solving a proportion.

Let's let the bathtub taps run for ANY amount of time we wish........say, for 1 hour.

So, in one hour, the hot water tap has filled 1 whole tub, and the cold water tap - since it can fill one whole tub in 30 minutes - has run twice as long as it takes to fill a whole tub. Thus, it has filled 2 tubs. So, in one hour, working together, both taps have filled 3 tubs.

But we only want ONE tub filled. In other words, we only want 1/3 of 3 tubs filled. Well....by that logic, both taps working together must only have to work for 1 / 3 of an hour - i.e., 20 minutes !!!

If we wanted to "math it up," the following proportion could be set up and solved for "x"............

(60 minutes) / (3 tubs ) = (x minutes) / (1 tub)

The beauty of this procedure is that - as long as we can determine how much work is done by both "things" for a given time period, we can use this proportion method. (I always choose some least common time multiple for the entities involved).

Now....as if I haven't worn this topic thin enough....let me show you one final method for solving this type of problem....this method is EXTREMELY simple and doesn't involve any "thinking" at all - I never like to think too much - I'm lazy !!

First of all, convert everything to minutes ( if need be !!)

Multiply the minutes together..... in this case, we get 60 * 30. = 1800

Them, divide by their sum ...... so,,,,,,, 60 + 30 = 90

Now divide the "multiplied part" by the "added part,"

1800 / 90 = 20

And there you go !!!

Why does this "work??"

As my Algebra 2 teacher in high school used to tell me..... "Magic"