Answer: 6 minutes
===========================================================
Step-by-step explanation:
Let's say the tub is 30 gallons. I'm picking this number because it's the LCM of 10 and 15. Picking the LCM means that later division operations result in a whole number.
The hot tap takes 10 minutes to fill the tub. Its rate is 30/10 = 3 gallons per minute.
Rate = (amount done)/(time)
The cold tap works at a rate of 30/15 = 2 gallons per minute.
When both taps are running at the same time, they combine to a total rate of 3+2 = 5 gallons per minute. Therefore, it will take 30/5 = 6 minutes if both taps are opened.
Time = (amount done)/(rate)
-------------------------
Another approach:
The hot water tap takes 10 minutes to do 1 job. That "1 job" refers to "fill the bathtub". The unit rate here is 1/10 of a job per minute.
Meanwhile, the cold tap has a unit rate of 1/15 of a job per minute.
The combined rate is 1/10 + 1/15 = 3/30+2/30 = 5/30 = 1/6 of a job per minute.
x = number of minutes it takes to fill the bathtub if both taps are going at once.
We can say:
(unit rate)*(time) = amount done
(1/6 of a job per min)*(x minutes) = 1 job
(1/6)x = 1
x = 6 minutes is the final answer
-------------------------
Another approach:
The template for problems like this is:
1/A + 1/B = 1/C
where
- A = amount of time for the hot tap water (working alone)
- B = amount of time for the cold tap water (working alone)
- C = amount of time if the two taps work together
We end up with the equation
1/10 + 1/15 = 1/x
This can be rearranged into
(1/10+1/15)x = 1
and also rearrange into
(1/6)x = 1
Follow the steps mentioned earlier to end up with 6 minutes as the final answer