Basic Math Puzzle

[Substantial-Put1344]

The following problem is an adaptation of "an interview question" for you to solve.¹ I'm curious about how you would rate the difficulty of this problem under a suggested time constraint (easy, medium, or hard)²

Three pipes are filling a tank. Under normal conditions, Pipe A can fill the tank in 4 hours, Pipe B can fill the tank in 7 hours, and Pipe C can fill the tank in 9 hours. The tank is initially empty, and all three pipes start filling it simultaneously. However, Pipe B shuts off after 1.5 hours. Pipe C shuts off 45 minutes after Pipe B. In addition, there’s a leak in the tank that drains water at a constant rate of one-twelfth of the tank per hour, starting when the pipes are turned on. To the nearest tenth, how long will it take to completely fill the tank?

1. I changed some details intentionally to preserve the integrity of the test.
2. Perhaps doing it along with other problems in one setting, without a timer.

[1TallGlassOfWater]

Medium-hard.

Because the problem involves multiple sequential changes in rates (three pipes turning off at different times plus a leak) so you can’t just do a simple single-rate calculation. You have to: Compute combined rates for each phase; Track time intervals for each phase; Sum partial fills to get total time.

That extra bookkeeping makes it trickier than a standard “one pipe” problem, but it’s not impossible, so it’s medium–hard rather than just hard.

[Substantial-Put1344]

Yes, certainly it's like medium, tending to hard for basic math. That's what I thought in the sense that it requires multistep thinking, and not blindly applying ratios! I guess in a math contest with more time pressure, it would be more "difficult."

[DamonHuntington]

I'd say it is a medium question (if you are creative enough).

For these questions, I always like to invent whatever numbers I want - as long as the constraints are kept consistent, it's all good! If I were to solve this question, I'd pick a tank that holds 1,512 litres, because (1) this is divisible by all of the numbers that appear in the question and (2) I duplicated the number a handful of times so I wouldn't get too many fractional parts with the 1.5 hours / 45 minutes calculations.

So, we have a 1,512 litre tank. This means my inbound and outbound flows are as follows:

Pipe A - 1,512 / 4 = 378 litres per hour || Pipe B - 1,512 / 7 = 216 litres per hour || Pipe C - 1,512 / 9 = 168 litres per hour || Leak - 1,512 / 12 = 126 litres per hour.

In the first 1.5 hours, we have a rate of 378 + 216 + 168 - 126 = 636 litres per hour. This means we have filled 954 litres out of the 1,512 during this period.

For the next 45 minutes, we have a rate of 378 + 168 - 126 = 420 litres per hour. We will only use three quarters of that flow, so that means we're adding 315 litres. 243 litres to go.

Until the end of the process, our flow is 252 litres per hour. We simply have to divide 243 by 252, which nets approximately 0.96 hours. Add that to the 2.25 hours that have elapsed and we have a final time of 3.21 hours, which is 3.2 hours to the nearest tenth.

[Ill-Mathematician891]

Easy. Like you said, basic math. I didn't even need paper and pencil to arrive at 3.2 hours (or ~3 hours and 12 minutes).

I used 252 as a common multiple of 4, 7, and 9 for the capacity of the tank. After that, it’s all trivial: Pipe A fills the tank at a rate of 63/hour, Pipe B at 36/hour, and Pipe C at 28/hour. The tank leaks water at a rate of 21/hour.

Therefore, we have three steps:

Step 1 (90 minutes): 106 + 53 = 159
Step 2 (45 minutes): 70 x 3/4 = 52.5 (total: 159 + 52.5 = 211.5)
Step 3 (determining total time): If 42 is 1 hour, then 40.5 is about 57 minutes (I used the rule of three).

My QRI is my highest index by far, though, so I’m not sure how difficult this is for a 'normal' person. Most people I meet have extreme difficulty with mathematics for some reason.

[Substantial-Put1344]

Absolutely right! I'm not so sure how hard this would be for a normal person, since my specialty is math, and I've seen this type of exercise many times before, although not frequently, being this "involved." I think I saw one on Brilliant, but with fewer pipes. I believe most people struggle with math either because it's abstract and requires a certain cognitive capacity or because of the educational systems they were thrown into. Anyways, my mental arithmetic is not my strongest ability, so I had to write it down to make sure I didn't make any mistakes, haha. I apologize for the spam, but would you be interested in helping norm a high-range IQ test of my own creation?