The hot water tap can fill a tub in 25 minutes. The cold water tap takes 20 to fill same tub. How long to fill the tub with both taps open at the same time?

Question

I don't understand how you can solve the problem without knowing anything about the size and shape of the tub

Answer 1

Your "I don't understand" question shows that you are thinking !!  Keep it up!

 

If all the information you had was about one hot water tap, you could not determine the size and the shape of the tub.  You're right about that.

 

But, the problem states that there are two hot water taps.  That's important.  And -- they fill "the same tub."  That's very important, because that means two expressions may set equal (without knowing their exact value).  Problems of the D-I-R-T (Distance-Is-Rate-times-Time) type usually have some variable set equal or some relationship between variables so we may solve the problem.

 

Now, notice the units:    tub per minute         (1/25 tub/minute and 1/20 tub/minute).

  We usually think of the rate as "minutes per tub," so this is what confuses most students.

 

So, for this problem, what is Rate*Time ?  I's not distance; it's Tubs (that is, one tub):

                                    Tub = (tub/minute)*(minutes)         [note: x is the "how long?"]

Hot water tap:     H-part-Tub = (1/25 tub/minutes)*(x minutes)            

     +

Cold water tap"    C-part-Tub = (1?20 tub/minutes)*(x minutes)

-----------------

 both taps            Whole Tub =  ((1/25 + 1/20) tubs/min)*(x minutes)

 

Once you have the units correct, the problem is much easier:

                 1 = (1/25 + 1/20)x

                 1 = (4/100 + 5/100)x

                 1 = (9/100)x

               100/9 = x

 

    x = 11 1/9 minutes         [note: this must be between 20/2 and 25/2]

 

[Also note:  almost nobody measures minutes in ninths, but this is a math problem.]

Answer 2

Isabella, those things don't matter in this case. We instead want to look at how much gets done each minute.

 

1/25 + 1/20 = 1/x

 

Once we find a common denominator, it actually goes away and we just deal with the numerators.

 

4x + 5x = 100

9x = 100

x = 11 minutes