**WARNING:** Long rant ahead …

**如有雷同纯属巧合**

(but based on a true story)

Hello, my name is John and I’m a taxi driver.

Contrary to common misperception, I am really nice and will go the extra mile to make your journey a pleasant one. For instance, I will always greet you when you board my cab and will help carry your luggage (if any). Hence there’s absolutely no chance that I will ever be attacked by a celebrity. As a professional, I also take safety very seriously. To demonstrate my commitment, I often make the extra effort to travel at 20 km/h *below* the expressway’s speed limit, regardless of the lane I’m on.

When I’m not being the nice and professional taxi driver on the road, I spend 100% of my free time at my ~~spanking~~ sparkling HDB flat. I really like my flat and I think it’s the best. In fact, I love it so much I’ve done two major renovations on it within this year. Admittedly the resulting noise can sometimes be a little loud. But I’m sure my understanding neighbours (along with their crying babies) will find this minor sacrifice well worth it when they share my joy in seeing the finished product, coz my flat is the best.

One day, however, the world collapsed around me when a little wet patch was spotted on my perfect kitchen ceiling above my perfect kitchen cabinet to ruin my latest perfect renovation job. Though I don’t ever recall seeing that area being wet *before* my renovation began (and since time immemorial), my contractor was adamant the fault lies with my neighbour living above. I know that my honest contractor always tells the truth, and therefore must be trusted. Moreover, my flat is the best so the fault must definitely lie elsewhere.

After calling my neighbour (whom I later found out to be a mathematics tutor) daily once every 2 hours, I managed to arrange for my contractor to visit her home, where I have no doubt the problematic spot will be found. And she would then have to rectify it, so that my renovation job remains perfect, and my flat shall continue to be the best.

To my surprise, they couldn’t find any leak on her kitchen floor. I cannot accept this. Most likely they didn’t check properly due to too many things placed there. It is not supposed to end this way because my flat is the best.

Undeterred, I called the HDB daily once every 2 hours to arrange for an inspector to visit her home, where I have no doubt the problematic spot will be found. And she would then have to rectify it, so that my renovation job remains perfect, and my flat shall continue to be the best.

To my surprise, the HDB inspector couldn’t find any leak on her kitchen floor. I cannot accept this. Most likely he didn’t check properly due to too many things placed there. It is not supposed to end this way because my flat is the best.

Undeterred, I called the HDB daily once every 2 hours to ensure they remind my neighbour daily once every 2 hours to arrange for another visit to perform a series of tests on her kitchen floor. I was told these tests will result in her being unable to use her kitchen taps for 3 days. But I’m sure she won’t mind this minor inconvenience one bit, especially when she sees the completed renovation on my flat, which will be the best.

This time though, the tests (whatever they were) concluded that there *might* be a leak in a hidden pipe beneath her kitchen floor (though the exact location remains a mystery). I am overjoyed. Justice has prevailed! She MUST now rectify it ASAP, or risk not being invited to my ~~show off~~ housewarming party after the renovation is completed on my flat, which is the best.

Unfortunately my neighbour is seldom home so it is difficult to fix a time to resolve this. I cannot understand this. How busy can a mathematics tutor be? How can helping students excel in their mathematics exams be more important than fixing the little wet patch on my kitchen ceiling?

To make things worse she suddenly went missing for a week. I heard from another neighbour that she went on a shopping trip to Japan. I cannot accept this. Staring forlornly at the wet patch for hours end with concurrent thoughts of her happily shopping away in Japan was an experience I would never wish upon anyone. This has affected my work and in turn exposed me to the risk of being attacked by a celebrity.

To cut this long blog post short, I finally managed to fix a time with her upon her return. But not before giving her a lecture on how to be a good neighbour, with an emphasis on her obligation to inform me the next time prior to going overseas. At the same time, I wanted to counsel her to be more prudent with her money, not to waste it on useless things like shoes and clothes (though I’ll secretly admit she looks really sexy in them) when they could have been spent on better things like renovation.

However, I decided to tell this to her on another day as it was obvious she wasn’t on a good mood that day (nothing to do with me though) …

Given that my ~~uncooperative~~ neighbour is only home for a limited time during working hours, the assigned contractor has arranged to fix the leak in 3 days with 3 workers working 2 hours per day.

At the end of the second day, unfortunately, one of the workers fell sick and there was no replacement. On average, how many more hours must each of the remaining men work for the remaining day in order to complete the job?

I immediately called my neighbour and asked her to solve the above problem for me but she said she was busy with her class before abruptly hanging up on me. Subsequent calls once every 2 hours were not picked up. I cannot accept this. How busy can a mathematics tutor be to the extent of not helping me solve a simple mathematical problem on demand? As such, I think she’s a really lousy mathematics tutor and I pity her students.

So if you’re reading this, can you help me solve the above problem so that I can inform that lousy mathematics tutor how much longer she is obliged to be at home for the work to be completed on the last day?

If you can do that, I may even invite you to my housewarming party when everything is done, where you can then have the rare privilege of admiring my flat, which will be the best of the best.

## 16 Comments

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Is this a real story based on Miss Loi life??? Wonder?

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@TJK: Just happens that Miss Loi cannot use her kitchen taps again today 🙁

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@Miss Loi: i had also one such neighbor, thinking his house is number one, and keep on drilling his house once in two weeks, and a lot of debris fell down and had to spent money to repair it. So, it is still better to be on top of one "such" bad neighbor than be below

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@TJK: That sounds very similar to what the neighbour living

belowJohn has been complaining about all these years!曜

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@Miss Loi: Go wreck havoc at his 'housewarming party'. 😛

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You know? I always dread questions like this in elementary school because using fractions to solve always gives me wrong answers 😛 anyway... 3 workers x 2 more days x 2 hours per day = 12 more hours divided by only 2 workers left... so 6 hours more per worker; and if they want to "pang gang" after 2 days, then each day stay 1 more hour lor... unless the mathematics tutor want to see them for one more day for 1 hour 😛

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Since the incident happened on the end of the second day, we can assume that the first and second days were unaffected.

Hence we only need to consider reallocating work on the 3rd day.

Original 3rd day: 3 workers x 2 hours = 6 manhours

New 3rd day: 6 manhours / 2 workers = 3 hours

3 hours - 2 hours = 1 hour

Each worker has to stay for 1 more hour on the 3rd day.

Its so easy, but why do I have a feeling there's a careless mistake somewhere?!

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@mathslover: *A moody-looking Miss Loi speaks*

This must be one of the easiest question ever posted on Jφss Sticks ... if you happen to be someone who understands the concept of Man-hour (or man-day, man-week etc.) used to quantify the amount of time/labour required for a task.

On the other hand, if you're a wide-eyed lower secondary student who cringes at such an industrial-sounding term, then this is actually a typical variations/proportion question where you need to be clear whether the entities involve have

directorinverserelationships.So using the

Unitary Methodas taught in your textbook (please ignore the relativelycheong heiworkings below if you're a man/woman-hour person who will probably find this a waste of time):3 men takes 3 days × 2 hours to fix 1 × leak ⇒

3 men → 6 hours → 1 leak

After two days, we're left with 1/3 of the leak ⇒

3 men → 6/3 hours → 1/3 leak

(less 'leak' requires less time to fix, so we divide both hours and the 'leak' since they are

directlyproportional)3 men → 2 hours → 1/3 leak

(so 3 men would take 2 hours to repair 1/3 leak - notice we keep the number of men constant here)

3/3 men → 2 × 3 hours → 1/3 leak

(less men require more time for the same amount of work, so we if we divide the no. of men we have to multiply the hours by the same factor since they are

inverselyproportional)1 man → 6 hours → 1/3 leak

One men then went missing ⇒

2 man → 3 hours → 1/3 leak

(conversely if we multiply the no. of men we have to divide the hours by the same factor since they are

inverselyproportional)So the poor remaining two workers were told to OT for an extra hour on the third day, but then ...

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Mathematically speaking, with only 2 workers, each will have to work 3 hours on the third day. In other words, 1 more hour than the required 2-hour per day work with the manpower of 3.

Realistically speaking, though, it may take longer as the job (fixing the leak) may not be easily done if it's only carried out by 2 workers.

Heck even the end of the 2nd day may not yield 66.67% (2/3) completion as the worker who falls sick at the end of the 2nd day may already be feeling unwell in the very beginning of the 2nd day. Thus his contribution may be reasonably lesser than the other 2 healthy workers.

In summary, I'd suggest not to let the neighbour know that "mathematically speaking" the 2 workers will work additional 1 hour (to the initial-plan of 2 hours) for the 3rd day to commit. Let them take their own time to ensure the result is truly satisfactory & not merely a rush-up work.

=)

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@a_x: Thought workers and peasants in Singapore are all robots???

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Unfortunately Day Three brought with it a series of misfortunes.

PART II:

Before work even began today (i.e. the third day), another leak (similar to the first one) was discovered on her kitchen floor! I had a heated exchange with one of the remaining workers who dared suggest this new leak could be the result of

mycontinuous renovation activities, which is obviously untrue.Because of this, the worker stormed out of the place leaving only one worker to work today.

To make matters worse, my uncooperative neighbour told me that she can only stay for two hours today, plus up to a maximum of 4 hours in an extra day tomorrow.

I called the contractor and since this scenario is too complicated for his "failed PSLE" brain, so he asked

meto calculate for him how many workers in total he needs to send on the last day (i.e. Day 4) in order to complete the job.I cannot accept this. What have I done to deserve this? Failing PSLE is not an excuse for not knowing proportion and variations to solve my urgent problem.

If you can solve this, I will

confirm chop stampinvite you to my housewarming party, where you can admire my flat, which willconfirm chop stampbe the best of the best.曜

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@John the Neighbour: hmm...

first part: 3days x 3 workers x 2 hours per day => 18 man hours

on the third day originally: 1day x 3 workers x 2hours per day => 6 man hours

on the third day (without sick worker) : 6 man hours / 2 workers => 3 hours per worker

second part: since no work was done remaining labour to be done by worker => 6 man hours + 18 man hours(second leak) = 24 man hours

so the lonely worker completes 2 man hours of work on the third day. leaving 22 man hours of work left.

Since on the 4th day has a maximum time limit of 4hrs, min no. of workers to complete the sealing of miss loi's floor,(neighbor's "best" ceiling) => 5 (hence making 6 workers in total). Then again the neighbor can send as many workers as he can to get things done fast.

Have a nice week Miss Loi ^_^

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@Nash: Oh boy manhours again! But actually, it's a really handy method so it's worth going through mathslover's & Nash's workings to understand it.

Once again, however, for the sake of those who wish to stick to what's been taught in school (or have your cher introduced manhours - anyone?),

From Part I, we know that we're left with 1/3 of the leak after 2 days

⇒ 3 men → 2 hours → 1/3 leak

However, on Day 3:

a new leak has been discover ⇒ 1+1/3 = 4/3 leak

and there's only 1 man to work for 2 hours

⇒ 1 man → 2 hours → 1/3 ÷ 3 = 1/9 leak

(less men do less work in the same amount of time, they are

directlyproportional)So on Day 4:

Amount of 'leak' left: 4/3 − 1/9 = 11/9

We now take

anyexpression from Part I as our base for the final calculation. Since we're stuck with 4 hours on the last day, we chose the following for convenience:3 men → 2 hours → 1/3 leak

1.5 men → 4 hours → 1/3 leak

(we first keep the leak constant to fix the hours)

We have 11/9 leak left to fix and this time we keep the hours constant,

1.5 × 11/3 men → 4 hours → 1/3 × 11/3 = 11/9 leak

(more work require more men in the same amount of time, they are

directlyproportional)5.5 men → 4 hours → 11/9 leak

As it's unlikely the contractor has a mutant half-human in his employee list, he'll need to send 6 men to finish the job in 4 hours on the last day (one of the men can afford slack a little to work at half-pace though 😉 )

Miss Loi hopes this simple exercise will give everyone some practice in playing around with the

Unitary Methodon questions with3 variables, keeping one variable constant at a time in order to achieve your desired results.Having said that,

why are we helping John the Neighbour???!

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@Miss Loi: why are we helping the neighbour? hmm.... considering your neighbours constant nagging its to help miss loi.. did you know.. noise pollution causes lack of concentration, fatigue, increased blood pressure,decreased efficiency and temporary or permanent deafness? so the faster he zips up the better ^_^

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@Nash: Thanks to all your help, Miss Loi thinks it's all settled now.

*fingers tightly crossed*

In other news, your good friend just said Hi.

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then again they can just go fix his ceiling =X.. its probably the holes caused by the drill works from his side..