Two taps and a tank
Holding a rate as a rate, rather than averaging the times that produced it.
Scenario
A tank is filled by two taps fed from separate supplies, so neither slows the other when both are open.
Tap A, running alone, fills the empty tank in 6 hours. Tap B, running alone, fills it in 3 hours.
Each tap runs at a constant rate, and the tank starts empty unless a question says otherwise.
Answers in order, explanations withheld, and a report on where the chain broke.
- 17.1Rational expressionsFoundation
What fraction of the tank does tap A fill in one hour?
- A1/6
- B6
- C1/3
- D1/2
Show the answer and reasoningHide the reasoning
1/6
A tap that takes 6 hours to fill the tank fills one sixth of it each hour. Rate and time are reciprocals of one another, which is the move the rest of the set depends on.
- 17.2Rational expressionsStandard
Builds on: The rate from question 1. Adding the times instead of the rates is the error this question exists to separate out.
What fraction of the tank do both taps together fill in one hour?
- A1/2
- B1/9
- C2/9
- D1/4
Show the answer and reasoningHide the reasoning
1/2
Rates add: 1/6 + 1/3 = 1/6 + 2/6 = 1/2. Times do not add, which is what the 1/9 choice assumes by combining 6 and 3 in the denominator.
- 17.3Rational expressionsStandard
Builds on: Question 2. Converting the combined rate back into a time is the second reciprocal, and it is the step at which an averaged answer becomes obviously absurd.
How long do the two taps take to fill the tank together?
- A2 hours
- B4.5 hours
- C9 hours
- D3 hours
Show the answer and reasoningHide the reasoning
2 hours
Filling half the tank each hour means the whole tank takes 2 hours. Averaging the two times gives 4.5 hours, which is longer than tap B manages on its own — a good sign that the method was wrong.
- 17.4Rational expressionsChallenge
Builds on: Questions 1 and 2. The first says how much A managed alone and the second says how fast the remainder is filled; neither figure alone answers this.
Tap A runs alone for 3 hours, and then tap B is opened as well. How much longer until the tank is full?
- A1 hour
- B2 hours
- C30 minutes
- D1.5 hours
Show the answer and reasoningHide the reasoning
1 hour
In 3 hours tap A fills 3 x 1/6 = 1/2 of the tank, leaving 1/2. Both taps together fill 1/2 in an hour, so one more hour finishes it — 4 hours in total.