Dan can dig a hole in 50min, Jon can do it in 40min, how long will it take if they work together?
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- Опубликовано: 28 авг 2024
- How to solve an algebra work word problem.
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As a licensed contractor and someone who has been a project manager for for a lot of other contractors I would say it would take a day and a half. Unsupervised they will be yakking on the job, taking long smoke breaks, getting into each other's way, Jon will get distracted texting and shovel dirt where Dan just dug out, they'll get into an argument then decide to cool off at the local titty bar for couple hour lingerie lunch. They'll come back, look at the hole and decide to knock off early and fudge their time cards. I'll see the hole incomplete the next day, jump on their asses and have to send someone to supervise them who'll smoke and joke with them half the next day and they will finally get it done.
Nice! Well put!
Simply , and quickly .
When you add 50 +40
=90 , 90÷2 =45 ???
45× 1/2 = 22.5 ! Close enough for GOVERNMENT work +
Ty 😂
Are they Union?
Less than 40 minutes
I love your reply. This guy doesn't understand real world problems. 30 yrs in road construction here. I come from the engineering part of it. He kind of knows the math but the math doesn't factor in the HUMAN equation! This guy usually screws the pooch with his explanations.
Jon can dig 5 holes in 200 minutes.
Dan can dig 4 holes in 200 minutes.
200 minutes dived by 9 holes = 22,22 minutes to dig a hole.
that would be my method as well
thanks for saving me 16 minutes!!
This is really nice, but only if you can find a common total time (200 minutes). Another way to think about it to consider work done per hour. Jon can do 1 hole in 50 minutes, or j = 1/50 = 0.02 holes/minute. Dan can do 1/40 = 0.025 holes/minute. Together they can do 0.025 + 0.020 = 0.045 holes/minute, or 1/(0.045) = 22.2.. minutes/hole. (Anything in the "wrong order", such as miles/gallon can be turned into the opposite (gallons/mile) by taking the reciprocal.)
is there a scenario where multiplying two random numbers does NOT lead to the same result at some given multiplier?
Multiply the times to get the common total
In 1 min Dan digs 1/50 of a hole (2%)
In 1 min Jon digs 1/40 of a hole (2.5%)
Between them that’s 4.5%.
100% divided by 4.5% = 22.22 min = 22 min and 13 sec.
Agreed. Did exactly that in my head in about 15 seconds. Can’t believe it takes 16 minutes of video to explain that.
@@davidcolver2502 Well, in fairness it can take a while if you're explaining it in detail to people who are intimidated by math. But yeah, he could have used an editor.
Essentially the same way I did it - except I confess I didn't do the actual calculation. Once I thought "this is how I'd do that," I well, left it there. 😕
This actually makes more sense than the unnecessary long weird explanation that he gave going in circles that only made it more confusing.
Having worked with people any number below 50 is unrealistic.
50 minutes. Dan takes two five minute smoke breaks and gets paid the same as Jon.
Jon doesn't smoke and thinks Dan is lazy. Jon tells Dan to dig it himself.
So it takes 3 days to dig it after Jon beats Dans ass and the police are called. Takes all night for them to post bail and for the crew of mexicans to show up to finish the job. I had to use Pi and a slide rule to calculate this. Im going with a standard deviation of +/- 5.9 mins
These two are workaholics, and do the job together in 22.2 min. But because they take no smoke (or just no-smoking) breaks, they have heart attacks at 40 and die 35 years before their normal life expectancy! [j/k]
@@freeguy77 : Even if these two are true workaholics, they would not be able to complete the assignment in 22.2 minutes due to what is known in Economics as "The Law of Diminishing Returns."
@@israteeg752 That does not apply over very short time periods. Yes, it would over an entire day or if each time was 2+ hours. But it was only 50 min and 40 min., not 2 consecutive hours (no break time). So, it would be appropriate over longer time periods than a time less than an hour, unless one or both were very weak, or the environment was either very hot or very cold, etc. The name you give it is more formally known in economics as: diminishing marginal productivity.
😂😂😂😂😂
You don't need math to know that the answer is 60 minutes because Dan is going to slow the process down, Jon will get angry, and the two of them will argue for 15 minutes about Dan not pulling his weight.
I was thinking 60 minutes b/c Jon would dig the hole by himself and Dan would be chatting and slow Jon down.
Hay It was Jon not pulling his weight ,
If Dan throws back the dirt Jon digs out it becomes a division by zero problem and the task would take an infinite amount of time.
60 minutes is the correct answer only with a decent manager.
With an average manger the time is 90 minutes (40+50).
With a bad manager, it is randomly more.
With a good manager, it's 40 min (Jon digs, Dans does some other usefull stuff).
The 22.25 min only applies in specific circumstances, and only if the manager is good enough.
22.(2) min is just theory.
@@rafeurdean 60 minutes? Are they using Agile? The scrum meeting alone with use of most of the hour.
In the amount of time it took for you to explain your solution, Dan and Jon could have dug a hole, lay a concrete foundation, build a library, and fund raise to buy the books to fill the library.
Yeah. Agreed.--all this was good to know, but I found myself impatient with what seemed liked the meandering.
Not really, they would be about 3/4 the way through digging the hole. 😉
Thanks for the tip, I just scrubbed to the middle to start. 😂
And why do math, just half the time of your slowest man. Good enough for any project manager.
Yes😀🤣
Dan digs 4 holes in 200 minutes. Jon digs 5 holes in 200 minutes. They both dig 9 holes in 200 minutes. Together they dig 1 hole in 200/9 minutes.
That's close to my answer I got in 3 seconds by going 50 = 1.25 x 40. 1.0 would be 2x as fast so 20 minutes. Multiply 20 by 1.25 = 25 minutes. That was just a guess obviously and it was close
Took me three seconds to figure they could dig one hole in 45 minutes divided by two...
@@misterfatcat7729 THREE seconds? Were you minding the kids whilst making dinner and watching a sitcom at the same time? I did it in 0.271 seconds
Lol. That's why you're the math man, Dr.Sam : )
@@misterfatcat7729 That's a fine approximation but it's not the right answer. Jon is faster than Dan at digging, so he digs more hole per hole, and his contribution to the average is more significant. So the right answer is 22.2 instead of 22.5.
I thought the easiest way was to express it as work rate. Dan digs 1.2 holes/hour, Jon digs 1.5 holes /hour. Combined, they dig 1.2 + 1.5 holes /hour = 2.7 holes/hour, so 1 hole takes 60/2.7 minutes = 22.2222 minutes
Better approach than just plugging into a magic equation that is never explained.
This is exactly why I was bad at math in school. People just memorize stuff without understanding why it is the way it is. Years after high school, I learned things again, and I refused to just accept things unless they made sense. Math became 100 times easier after that. Units and rates are the only way anything makes sense.
That happens to be exactly my approach as well. Well done.
That's about the way I did it in less than a minute.
This is how I did it. I did it in my head up to the 1/2.7 part where I solicited my calculator. 😁
Only one person can dig for one hole. Jon is faster, so Jon digs while Dan stands back watch. Therefore 40 minutes. That is consistent with any construction job. One person works while other watches.
Dan is also drinking a Snapple while he watches his friend work hard. That doesn't impact the answer but it's important nonetheless.
Yup! Couldn't agree more! But they're gonna chat and goof off together, so Jon's 40 minutes is gonna go up...
Unless there are 3 people. Then 2 will be watching.
Someone must have called in sick cuz usually there is a whole crew just watching
@@blackstock333 - and the fourth, the supervisor, is calling his girlfriend to try to organize a nooner.
I've been an electrical engineer for 43 years and thankfully I still have my math chops and solved this using the same approach and got 22.2 .minutes. yay ! Great video 😊
You forgot to add in the Smoke Breaks, the Carnival time where they talk about the last party they went to, Text time, RUclips time,time spent on grinder,time spent on the internet looking for car parts and the break to go to the shop to buy morning tea. Job will actually take 1.5 days.
I guesstimated 22.5 minutes. I halved each and split the difference. I’m 61 and your videos help keep the mind sharp. Thank you.
Being 61 you should know by now that there is not nearly enough information here to get a real world answer. Are they getting paid hourly? Is Dan a talker? Is it the last job of the day and Jon doesn't want to get home because he hates his wife. Is the ground here the exact same as the ground where they calculated the previous time?
I hate these math questions because a management bean counter will expect certain results without taking into consideration real world variables.
@@aebalc you've been in the game too long 😅
Basically did the same, ((50+40)/2)/2 and it's only about 17 seconds off, then I rounded up because you have to include a buffer cause what if you hit a rock.
@@aebalc THANK YOU!! I gave my 2 cents a couple of times because if the guy would quit with the nonsense, and help us understand how he worked the problem the way he did and explained what the LCD was before giving us that number, I thought 10 would have been right, but 5 works with both 40 and 50. We're on the same page, Dan hardly knows what a shovel is while Jon could spot him 10 minutes and still get done faster, but that doesn't teach us anything either. The guy is a math nerd.
Hi. Your approach is a good approximation when the 2 working times are close (40 vs 50 minutes). As the 2 working times diverge (for the same sum of 90 minutes), so does the error between the approximation and the correct answer.
I’m 62, retired military Aviator with an undergraduate Aeronautics degree, and these are great for pushing the cobwebs away from 40 years of disuse. As a military and commercial pilot we use higher math skills to calculate fuel and cruise speed efficiency, emergency procedures involving power loss at various weights, etc… but in the aggregate we use less than ten formulas. Strangely enough, they’re generally calculated in our heads. These videos are great to get back to the basic all but forgotten.
In the real world... the answer is 8 hours. Dan and Jon BS all day long, and near the end of the day they realize the boss will be here shortly, so they go to the tool rental store to get a hole digger. The hole is done in 5 minutes.
This is very true.
And it is a kind of complex algebra.
Dan and Jon are then FIRED that same day, because their buying a hole-digger cost the boss more than he was going to pay them!
Not that extreme. You need to know how long the Union workbook says it should take the slowest person to dig the hole. Size matters, but there is no reason to dig a hole faster than the slowest person.
@@NicholasSouris Unless you're doing it to cover up the fact that you haven't done anything all day.
In the real world, after fighting, one of them pushed them down the hole and ends up burying them
Useful in a lot of different applications-for example, this is how you calculate the total resistance of two resistors in parallel. 1/(1/R1+1/R2)=Total R
product over sum
As an Electrical Engineer I endorse this method.
Or R1× R2/ R1 + R2
Jon digs a hole in 40 minutes = x (hole) During that same time Dan can dig 80% of the same hole (40 min/50 min)=.8x (hole) if 40 minutes is spent digging the hole by both Jon and Dan then
x + .8x = 40 minutes
1.8x= 40 minutes
x=22.2222 minutes
I took the approach that both could dig two holes in 90 minutes, with the average time for both holes being 45 minutes, with one hole taking half of that time = 22.5 minutes. I appreciate your solution as more useful for learning algebraic tools, which is the purpose of your site. Thank you.
Same. I thought that since their average time was 45min and they worked together then it would be half of that.
@@donnyh3497Same here! Will someone tell me why we're wrong? Please don't just tell me to watch the video again. Thank you in anticipation❤.
@@supalewYou can't devide them by two since both aren't equaly fast. One is working faster than the other. The avarage approach is good to get a decent estimate (or guesstimate) as long as both times are close together.
Let's take a more extreme example:
One needs 10 hours, the other person needs 1 hour. How long would it take for both combined? With the avarage approach we would end up at a number that is greater than person twos times alone, which doesn't make sense. (2.75 hours with the avarage approach).
The equation shown in the video '1 over all combined/together' calculates the avarage time it takes when everyone is working on the task at maximum efficiency. The result will always be lower than the lowest number you were given for a single person doing the task (unless there are other factors given).
@@izaruburs9389 Thank you. "One is working faster than the other". Of course he is! I'm grateful for your patience and commonsense.
The time would double, after Jon hit Dan over the head with a shovel for not pulling his weight.
I was thinking the same thing... LMAO OR Never because Dan was arrested for hitting Jon with the shovel. Jon can no longer use a shovel because of his altercation with Dan.
Dan was on his damn phone for ten minutes pretending to be in the bathroom.
Actually, the job never gets completed because Jon dumped Dan into the hole and filled it in.
What kind of dirt? Sandy? Clay? 🤣
Its a math class not psychology. I can see who went to the office for being the class clown at your school.
I didn't get algebra at school, I was too busy with other stuff like motorcycles and beer and minges, but I can visualise things. The two diggers are using shovels so the hole must comprise a certain number of shovels full. Let's pretend that number is 200 as that's divisible by both Dan and Jon's numbers of minutes to empty that hole, so Dan can dig 4 shovels in a minute (200 divided by 50) whereas Jon is a bit quicker and can do 5 in a minute (200 divided by 40). So obviously together assuming they don't get in each other's way, the could do 9 in a minute. And 200 divided by 9 is 22.22 and not an x or y in sight. Go me.
Brilliantly simple logic sir. Keeping the problem in real terms is most times the best solution.
Another way to think of it is, instead of working together digging just _one_ hole, consider them needing to dig an arbitrarily large number of holes and each one will dig a hole all on his own, repeatedly. One could dig 200 holes at 1 hole per 50 minutes, or 10,000 minutes to do it all on his own. The other can do 200 in 8,000 minutes. And every 200 minutes, they've exactly finished 9 holes between them. So if it takes 200 minutes for them both to dig 9 holes (one digs 5 in the same time as the other digs 4), then you can just divide that 200 by 9 and get your ~22.2 minutes as an estimate for how long they can dig one if working cooperatively on a single hole. Of course, this presumes there's no interference (getting in each other's way and slowing the work) nor synergy (force multiplier yielding more work than the mere sum of independent units).
Having dug many holes by myself and with others, it will take 35 minutes. It is almost impossible for them both to work on the hole at the same time because they get in each others way giving a time of 45 minutes if each digs half of the hole. However, having a rest in the middle of the process does speed things up, giving 35 minutes.
well it depends on the size and shape of the hole. if they are digging a trench they can start in the middle and work outwards so they don't get in each others way.
@@ronblack7870 It specifically states a hole, you can't assume what kind of hole at the end of the day.
Because if you could assume the size of the hole, then one could assume there could be heavy machinery nearby, in which case the hole can be dug in way less time than either Jon or Dan could hope to keep up with.
Depends on the workflow. If one takes care of shoveling the dirt out of the hole and the other takes care of carrying the dirt away from the job site then you are removing quite a lot of travel time (like going in and out of the hole, switching between the dig bucket and the front loader bucket, etc)
wow dude I love reading the same fucking joke in 19 comments!
well, if they equally split the work and work simultaneously, dan would take 25 minutes to finish his half, but jon would have finished his half in 20 minutes, and would have 5 minutes extra to cheer on the other. realistically somewhere between 20-25, like 22-23minutes
I love your explanation, my educated guess was 20 minutes plus.
I hope this is right .. and will check later.
50 minutes + 40 minutes = 90 minutes
and 90 minutes divided by 2 gives you the average time of 45 minutes.
So, working together, it should take half that 45/2 22.5 minutes.
same
@@oilofzion777 It's a great estimate but it's not 100% accurate. At the 20 minute mark, Jon would be completely done with his half since it takes him alone 40 minutes to do both halves = 1 hole. Dan would only be 4/5th done with his half, but Jon is one the way to help with that. Lets assign the dirt that fills the hole = 100lbs for simplicity. At the 20 minute mark, 90% is done and 10% is left. You can use the same math to determine how long it will take both of them to complete the 10% at the rate at which they work. It would take an additional 2.22 minutes to complete.....and .22 minutes = 13.2 seconds so the absolute correct answer how long?? 22 minutes and 13.2 seconds!!! You estimate is only just under 17 seconds off so it's really close!!
@@magicmuzicman6677 It's not a great estimate. The method is totally wrong. I one took 90 minutes and the other took 10 minutes this method would give 25 minutes for a combined effort when the real answer would be 9 minutes. So, off by 278%.
Whoever constructed this has never tried to dig a single hole with multiple shovels.
Uh, they start at either end and work toward the middle.
@@Pteromandias Still won’t be 100% separated/efficient and the diameter of hole and size of shovel impact the level of interference. This is the difference between theory and practice 😉
tunnel boring machines often dig from 2 different directions to speed up the process. would it make you feel better if he said tunnel boring machine 1 and tunnel boring machine 2? the rates are different because the density of the materia / rock they dig through is different. etc.
@@Pteromandias Okay, that is the first YT comment I have actually laughed at today.
Well Done!
Used the same formula for this that applies to resistances in parallel; (R1R2)/(R1 + R2). In this case, though, it should be (D * J) / (D + J). For more workers, the general formula is: 1 / (1/R1 + 1/R2 + 1/R3. . .)
Most people need spreadsheets or calculators to figure it with that electrician's rule!
@@freeguy77 I did these with a slide ru.le.
It will take longer because they will keep getting in each other's way. Depending on how close they are, it could turn violent.
In the 50 minutes Dan has dug a hole, Jon could dig 1 hole and be 1/4 of the way through a second hole, so the answer is 50 minutes divided by 2 1/4 holes, which is 50 / (9/4) = (50 * 4 ) / 9 = 200 / 9 = 22 2/9
THAT is an interesting alternative way of thinking of this. Thanks!
dan can dig 1 hole in 50 min so his rate is 1/50 holes / 1 min
jan can dig 1 hole in 40 min so his rate is 1/40 holes / 1 min
add their rates up to get (1/50)+(1/40)=(9/200) holes / 1 min
rewrite 9/200 holes / 1 min as 1 holes / (200/9) minutes
so 200/9 = 22 2/9 minutes
@@EricLeePiano That's the way I did it too
The problem here is we don't know what the job is? If you assume the job is digging a hole,then you are correct, but if the job is not to dig a hole, then you are wrong. Job is a different variable from hole.
😊
@@EricLeePiano Or, Dan's progress to a complete hole is 2% per minute, while Jon's is 2.5% per minute. Combined they work at 4.5% to completion per minute. So the answer is 100/4.5 (which is what your final formula reduces to)
2 1/2 Days. By working together they managed to milk it for as long as it took to pay the bills for the week. Teamwork is Amazing.
Actually....after the first scoop of dirt you have a hole.
You have a divot! [golf]
Their individual efficiency will not be the same as working together because they will get in each other's way.
you have made the error of including things that are not stated
I think, my way is easier:
Dan has capacity 1, Jon has 1.25, together they have 1+1.25=2.25.
50 ÷ 2.25 = 22.22…
Why 50
@@Kyle-nm1kh Because at a digging capacity of 1 (Dan's alone) the hole needed 50 minutes to be dug. And after adding Jon (with his capacity of 1.25) the total capacity is 2.25.
@@matthiaswolf4472 I see. Thank you for the explanation
Dan can dig 4 holes in 200 min. Jon can dig 5 holes in 200 min. Together they can dig 9 holes in 200 min. Together they can dig one hole in 200/9 min. 200/9=22.22 .
Well, if you've read Brooks' "Mythical Man Month" you'll know in practice it's not linear but in fact adding more people to a project actually slows it down. So the real answer is 2000 minutes to dig one hole, and document their work.
At least a weekend. Jon and Dan are fishing buddies and this hole has taken up their weekend. Consequently they both brought cylindrical cans of digging fluid with them to the job. This led to discussion of how the hole was to be dug and who was in charge and who was the muscle power. After the consumption of the fluid they sub-contracted the job out to a local contractor because their friendship was to important to jeopardize and fishing was more important. A+ 100% (:
This is too funny and original as can be!
Dan digs 1/50 (4/200) hole per minute, Jon 1/40 (5/200) hole per minute. Together they dig 9/200 hole per minute. It takes them 22.222 minutes to dig 200/200 hole.
That is "prezactly" how I figured it.
I worked it out as follows (maybe it’s already been posted in the comments but I haven’t got the patience to run through the list) : x being the time in minutes it takes them to dig their hole in common, x/50 represents the « fraction » of the hole it takes Dan to dig in said time ; x/40 for Jon. So x/50 + x/40 = 9x/200=1 and x =22,22 (about) that is 22 minutes and 13 seconds.
In 1 min Dan does 1/50 work. In 1 min Jon does 1/40 work. Together in 1 min, they complete 1/50+1/40 or 9/200 work. If 9/200 work is done in 1 min, they should complete in 200/9 min 0r 22.22 min. We learnt it as a unitary method in school.
Dan didd .02 holes per minute and Jon .025. Together they dig .045 holes per minute. 1 hole divided by.045=22.2 minutes.
@@joeuser633 yeah the simple solution is 100% of a hole divided by 50 and 100% of a hole divided by 40 = 2 and 2.5 totaling to 4.5% per minute so 100% divided by 4.5 = 22.22..
but a instant guesstimate was that half time from 40 and 50 is 20 and 25 so its somewhere between those 2 numbers since by the time one of them is halfway there the other has some to go and spliting that 5 minutes of work for the slower guy means its less than half of that 5 so less than 22.5 and that is off by less than 18 seconds and took about no time compared to starting to calculate the specifics.
It completely depends on if they are union or not. If they are, it will take approximately 6 hrs and 43 min to dig the hole.
In 1974, while taking Algebra 2, my math teacher in Junior College told me “The odds of you being an Engineer is 150 to 1”. He was an Engineer and Civil Engineering was my Major. Well, I graduated from UC Berkeley in 1979 with a BS in Civil Engineering, with a GPA over 3.0. In 1979, UC Berkeley was rated #1 in the world for Civil Engineering, with Stanford and MIT second and third. I went on to have a very successful 40 year Engineering career. The moral of the story: people are very often wrong and rather than encourage they demean. Always remember, you haven’t failed until you give up. He vastly underestimated my resolve. I love this math teacher, I love math and doing these problems. I wish I had John’s Math Academy when I was struggling with math!!
Go @robcooke1956! That's a good story.
150 to 1 odds is extremely certain. So maybe your teacher was right?
😂, It was hopefully just a mistake. Otherwise the teacher still can't understand why his favourite student could not take a compliment@@eldritchpalmerable
Dan and Jon need help, do you have a shovel !
I added half of 50 and half of 40 and divided by 2=22.5.
Now do the problem when you have 2 water inputs, but 1 water drain all running at the same time! (X-rate + Y-rate - Z-rate) to figure out when the empty pool (sink, bathtub, pond, lake, ocean, etc.) will be filled!
ANSWER: 6 days and four hours!
Two men in the same hole..means one gets hurt or one gets pissed off! The more than six days includes the trip to hospital and the police investigation following, and the release of the uninjured party..... a full six days after his poor friend was released from hospital.... but was not allowed to return to the job-site!
Also estimated 22,5min. Averaged their times then divided by 2.
Ditto
The answer is Dan's half will take 25 minutes and Jon's half will take 20 minutes, so if they work simultaneously, Jon will have a 5 minute break while he waits for slow Dan to finish his half, and the total elapsed time is 25 minutes to dig the hole.
slightly more sense
Jon ain’t trying to do more work just because he’s faster. Good for him.
Yeah that's likely what would happen
And since Jon has finished his half after 20 minutes, he keeps on collaborating for the enitire hole to be dug, which is why the total time is a bit less than 25 minutes.
I came up with 22 min 12 sec and some change. I used an iterative process where I imagined Dan and Jon working side by side on their respective halves of the hole. After 20 minutes, Jon has finished his half of the hole, while Dan has five minutes of work ahead of him, meaning 20% of his dirt still remains, or 10% of the original volume to be removed. They both continue to work, and since they only have ten percent of the dirt it only takes ten percent of the time, or two minutes, to arrive at a similar situation: Jon has completed his half of the remainder, while Dan has left a fifth of his side yet to be dug. This remaining pile of dirt now represents one percent of the original volume to be removed. One percent of twenty minutes is 12 seconds, which is how long it takes to whittle this pile down to one tenth its size... at which point I decided that estimating jobs down to the second is something no sane contractor would ever concern himself with, and said "close enough." ;)
Thanks for the video!
Thank u, I went into the movie with it gotta be less then 22.2 min cause John will finish first and start helping Dan with his part, bit I was to bored to get the calculator - his calculations did not satisfy me as 😅
They will obviously take twice as long because they will be in each other's way
Dan leans on his shovel and watches as Jon digs the hole in 40 minutes.
Thank you John for your brilliant videos - I’m in my 50s and I try to do one of your maths puzzles everyday - love them and I’m getting more and more happy faces and A+; I feel so happy when I get it right ..
Same as calculating the effective resistance of two resistors in parallel: Reff=(R1*R2)/(R1+R2) = (50*40)/(50+40) = 2000/90 = 22.222 or, to do it longhand, 1/Reff = 1/R1+1/R2 = 1/Reff = 1/50 + 1/40 = 1/Reff = 4/200+5/200 = 9/200 , 1/Reff = 9/200 , flip them both over to get Reff=200/9 = 22.2222. Together, they can dig a hole in 22.222 mins or 22 mins and 13 sec.
Same problem presented in electronics when solving for total parallel resistance when TWO resistors of differing values are put in parallel. Simple solution is to just divide the product of the two values by their sum: (R1*R2)/(R1+R2). An interesting footnote is that the total resistance will always be LOWER than the smallest single value alone. Nice tutorial on working through the math, thanks!
I used a different method to solve this.
I used comparative efficiency, with Dan being 100%efficient and Jon being 125% efficient. Then, I turned these values into decimals to get 1.00 + 1.25 and added them together. I then divided Dan's time of 50 minutes by 2.25, which is the sum of Dan's and Jon's efficiencies.
What about one slowing another error? When one takes a dig another obviously have to wait for his turn.
@@robant5578 It's a big hole to dig for, so 4 or 5 could dig simultaneously, as for the 350 G's ($350,000) buried in Santa Rosita Park, CA *, in the wild-and-wacky film, "It's A Mad, Mad, Mad, Mad World" (Nov. 7, 1963) * underneath a BIG W.
The correct answer is 2 hours 10 minutes. I know this by practical experience. The math only applies to an imagined optimum outcome.
FYI. I learned how to reduce a 40m hole to just 30 seconds, working alone. Using optimal technique. With a partner the same 30s hole would take hours.
The CORRECT Solution to this problem -in real life- A hole has only so much head space above it for the town of them to occupy. Typically for a forty minute hole, there is only space for one head above the hole. Allow for swapping each other out, and clearing their respective spades from the hole leads to 40+50+20minutes =2 hours 10 minutes.
40+50+20= 110minutes or 1:50
@johnwaldmann That works for me!!! Better than his explanation!!
Depends upon whether both can work at the same time or not - if not then any time between 40 and 50 minutes depending on the percentage of time each digs.
It will take all day because Jon and Dan disagree about everything and one will end up throwing his shovel and shouting vulgarities.
I did it like this:
in 20 minutes Dan dug 2/5ths (or 4/10ths) of a hole while Jon dug a half a hole (or 5/10ths) so 9/10ths total.
with a 10th left to dig, I divided the time it took to do 9/10ths by 10 ( = 2 minutes ) to do 9/10ths of that final 10th (or 99/100ths altogether in 22 minutes)
with each iteration whats left is always 1/10th of what was left before, therefore the time it takes for the next iteration will be 1/10th of the previous time.
1st - 20 minutes (9/10ths dug by the pair of workers)
2nd - 2 minutes (9/100ths)
3rd - 0.2 minutes (9/1000ths)
4th - 0.02 minutes (9/10000ths)
and so on to infinity, indicating the result must be a recurring decimal. adding all parts up to infinity will give the result 22.2 recurring minutes for both to dig together 1 hole.
convoluted I know but it took me less than 30 seconds in my head to be confident of the result.
same here. How tf did it take 16.48 minutes for the video to explain such a simple thing?
90 minutes, of course. They'll get in each other's way.
3 hours.
Turns out both Dan and Jon are gay and they got distracted.
Exactly. It is as like two cpu's work at the same problem without coordination.
@@seeharvester by a good-looking passing man!
The issue is that they are taking different times to make one hole. So you need to find an equal time where they have both completed a complete number of holes.
So it’s easy to see that Dan can dig 4 holes in 200 minutes, and Jon can dig 5 holes in 200 minutes. So working together they can dig 9 holes in 200 minutes. So they are digging a complete hole every 200/9 minutes, which equals 22.2 minutes. In the other example you gave where it was an hour and ten minutes for Jon, then you just take 350 minutes - Dan can dig 7 holes in 350 minutes, and Jon can now dig 5 holes in 350 minutes. So 12 holes in 350 minutes means 1 every 29.166 minutes.
Yes, common denominator of 1 hole/50 minutes and 1 hole/40 minutes.
Yes! This makes sense. Thank you. The video maker didn't explain how he arrived at the formula, so his way of working it out was useless to me. But this makes perfect sense.
Dan will do 5 Nineths of the work and Jon will do 4 Nineths. If Jon can do 9 Nineths in 40 minutes it will take him 40*5/9 minutes to do his share. This is 22.22 minutes.
You could also work out Dan's effort. He will do 4 nineths of the job. This gives 50*4/9 which is also 22.22 minutes.
You know you work in consulting when you think,
“Well that depends on what Dan & Jon’s bottleneck points are.”
Or as referred to in economics " The law of diminishing returns".
If they were both digging their own holes for half the depth, I would be inclined to agree with your statement. However, since they're both working on the same hole, the hole can only occupy one shovel at a time and so the answer would more closely resemble as if one person was working on the hole than just simply dividing the time between them. A better way to demonstrate a problem like this would be if they were moving blocks, not digging a hole.
I did a really fast guesstimate of 22.5 minutes.. but Jon works faster so he actually did slightly more than half the work. Which would make the total slightly less than the 22.5 . The 200/9 made perfect sense. Thank you.
@Sailor brown noser!!!🙄
@@survivrs No. I learned something. Glad to learn it. And the polite thing to do is thank the person who took the time to teach me. Polite. Politeness. The grease that makes civilization civil.
@@Sailor376also Now I feel guilty for my comments, but then again, I didn't learn from this. I could go through it again without remembering how to do it. I like your attitude though!!
@@survivrs You are a good person. Just to make you laugh, I did the first rough calculation not because of any skill or genius, I am a builder,, dug a lot of holes. You have a great day.
@@Sailor376also My husband spent 30+ years as a construction superintendent for a general contractor, but he couldn't just supervise his employees, he had to work because there were many job sites where it was like babysitting kids who had no clue. They couldn't read blueprints, or understand a thing. He did everything from concrete work to sheetrock, and everything in between. That came in handy at home when we had to install anything, run new wiring, etc. In our first house, we were taking a couple of weeks vacation to work on remodeling. The house was 100 years old, and when we started , we started tearing out lathe from the inner walls, the insulation was nasty rock wool insulation, and the wiring was what he calls "knob and post". That was bad. Then we did the roof. There were 7 layers of shingles to tear off, and there was no tar paper under them, just boards, and you could see where there had been a fire at some point in time because it had gone through the wood. So we put on new wood, tar paper, and shingles. Thankfully we had friends and family who could help with the roof because it was 2 stories and steep. I don't do heights! Thanks for your comment, I'm ready to try and tackle a SIMPLE problem if I can find one. I was so frustrated yesterday just like when I was back in school because if you didn't raise your hand, the teacher moved on which was great if you "got it", but not for students like me who didn't have a clue. There are times from the 70's that I would like to revisit, but not some of my classes.
This is the same formula as
that used to work out parallel resistors in an electronic circuit.
Exactly. For 2 resistors, the formula is (R1xR2)/(R1+R2). A different formula is used if there are 3 or more resistors.
Another posted pointed that out; it is in the same analogous idea of more than 1 thing all working simultaneously, so you want to find the entire rate (speed) to figure out the time needed or in the electrical case, total resistance.
The answer is 40 minutes. Jon is the more efficient worker, therefore Dan is promoted to management, and Jon does all the work. Obviously we do not include the time of the team building exercise Dan had Jon partake in before Jon could do the actual work.
No - Jon would be the one promoted, leaving Dan to do the work.
Nah, the first answer is right. If the tech is good at the work, they aren't gonna lose that efficiency. They would more likely hire someone who has zero experience digging holes, but went to college for (insert random major here) but can effectively tell Jon that he doesn't deserve a raise because the work being done as a team is still insufficient to raise profit margins for the company, but next year will be different. Jon can't ask for a raise for fear of being fired because of the toxic climate culture, and can't quit because he has to provide for his basic survival needs and inflation is creeping and he has no outlet.
If all you want is a quick estimation for multiple choice test, the fast way to get the answer is exactly how he starts getting to 45 minutes. Thats 2 holes. 1 is half that at 22.5 minutes which should get you close enough for multiple choice. Only do it the long way if you need an exact answer.
One of my seventh grade math team non-algebra student explained that this is just a least common multiple problem. Least common multiple of 40 and 50 is 200. So Dan can dig four holes in 200 minutes and John can dig five holes in 200 minutes and together they can dig nine holes in 200 minutes. Just divide 200 by 9.
Anything like constant rates is like this example. Recognizing that it is this type of example gives the person a good jump-start.
Dan and Jon hired Manuel El Obrero, a Mexican guy who came with a crew of 30 for less money. They worked together in what looked like a perfectly rehearsed ballet and dug a perfect hole is 5 minutes.
They were all legal, I swear!
Actually they put fashioned a sign that said Viagra dirt from Nike 750 dollars per shovelful, then they had a nice 30 minute lunch break, they came back the hole was dug and the shovels gone and they enjoyed a nice leisurely lunch for once ! LOL
Thanks for contributing nothing at all!
@@mrosskne you're welcome shtnut 😙
Again my brain works oddly. J +D = 45 minutes but since they are working together then 45 minutes / 2= 22.5.but J is faster than D so he will dig more than D so divide .5 by 2 and round down giving 22.2. Don’t know the algebraic formula or if it works for other numbers but I got 22.2 for this one.😊
If you add their rates together (Dan = 1 hole per 50 min; Jon = 1 hole per 40 min) you get a combined rate of 9 holes per 200 min (9/200). Rate (r) × time (t) = holes (h). Since h =1, then t = 1 / r. t = 1 / (9/200) = 200/9 = 22.2222...min. The caveat being that the two don't a) increase their collective rate exponentially (individually work faster together than alone), or b) don't get in each other's way.
I like how you take "corner cases" into account. Ppl tend to forget or minimize that
@@SnijtraM There are no corner cases. It's a math problem. It doesn't represent anything. The nouns and names are just there to make it "interesting".
@@mrosskne Names and grammar are not just "interesting", they can have multiple meanings. If you, as a reader, are processing all this language, you have a job to deal with these possibilities.
@@SnijtraM Nah.
This seems like an averaging equation. Since each one is digging aproximately half the hole 25 minutes + 20 minutes = 45 minutes divided by 2 people (to average) = 22.5 minutes. However, knowing what I know about teamwork, it could be a little bit quicker.
So when the faster one has finished half the hole, he watches the slower one finish his work?
Good math teacher encouraging words and tip to solve these kind of math problems.
As mentioned in the video, if they both are digging the same hole, they will probably get in each other's way and take longer than 22.2 minutes overall. So, let's assume that 2 holes need to be dug. If each person digs a hole, the project will be done in 50 minutes, unless Jon pitches in when he finishes after 40 minutes and helps Dan complete his hole. If Jon does and they don't interfere with each other, how long will it take to finish both holes? However, let's say they do interfere with each other, so they decide to switch places at some point in the digging operation. Dan's hole will be further along than Jon's. At the end, both holes are to be finished simultaneously. Not counting the time taken to switch places, when should they switch and how long will it take altogether to finish?
22.5 minutes switch, finish in 45 minutes. ?
I think 22.222 reoccurring
Dan and Jon realize that they will get in each other's way digging the hole, so Dan let's Jon dig the hole since he's faster. Answer: 40 minutes.
If they don't get in each other's way ...
Dan can dig half the hole in 25 minutes while Jon can dig half the hole in 20 minutes, averaging them together (since Jon can help Dan after he finishes his 20 minutes) gives an answer of 22.5 minutes.
So the official algebra answer is 22.22 minutes. My 22.5 is well within engineering accuracy, and since I'm an engineer, I'm happy with that.
22.222
In 20 minutes, even without Dan, 50% of the hole would be done. But Dan can do 40% of the hole in that same 20 minutes. Thus in 20 minutes, there's only 10% of the hole left to do.
Since Jon could knock that out in just 4 minutes, we know that in two more minutes he'll have done half of that 10%, while Dan will have done, again, 40% of that 10%. Thus after 22 minutes, there's only 1% of the hole left to do.
Jon could knock that out in 24 seconds, so in 12 seconds, per the pattern, there's now only a tenth of 1% left to do.
That would take Jon 2.4 seconds to do, so in 1.2 seconds there's now only one hundred of a percent left to do.
Jon could do that in .24 seconds, so in .12 seconds there's now only on thousandth of a percent left to do.
It took 22 minutes, 13.32 seconds to dig 99.99% if the hole. After another .012 seconds, that's 99.999% of the hole, for a span of 22 minutes, 13.332 seconds.
The answer, therefore, is 22 minutes, 13 and one third seconds ...because the pattern will continue 13.332, 13.3332, 13.33332, and so on ad infinitum.
You're assuming they're digging at a constant rate ... but they'll get tired over time, so they'll gradually slow down towards the end, they won't dig half the hole in half the time :)
I got 22.5 minutes solving the problem in my head in about 10 seconds. I know I’m off by a little bit, but here’s the logic. If Dan had a twin, it would take Dan half the time. If Jon had a twin, it would take Jon half the time. So that’s 25 min and 20 min respectively. So I split the difference and get 22.5.
Wow that was a lot of unnecessary calculation.
The hole takes 1000 work units to complete.
50 min for 1000 units = 20 u/m.
40 min for 1000 units = 25 u/m.
Total rate for both is 45 u/m.
1000 / 45 = 22.22 min.
@@mrosskne You're a deliberately offensive person. My answer is correct, and there's nothing inelegant about how I approached solving the problem.
@@mariushmedias This like so many math or logic puzzles, calls for relying on any number of 'not real world' assumptions, yes. The weakness of your particular point, however, is that in saying (for example) that it takes Jon 40 minutes to dig a hole, we can assume that this is despite his having gotten tired and been inconsistent in his work rate ...which means he started out at a more than 40 minute per hole pace. There would be no 'error' in saying that Jon works at a 'one and a half percent hole per minute' rate, even though only sometimes that's his actual pace.
I did it by trying to find a set amount of work. I know I was going to be dividing the work so to get a whole number I went with 200 (the LCD).
Then I looked for how much work they would do per minute.
200/x=50
200/y=40
Then I added their work per min.
X+y=9
And last looked for how long it would take to do the 200 work.
200/9=22.2...
It was basically the same as in the video, but a different way to think thru it.
One of the most powerful things I learned in first year chemistry was to include units in calculations and conversions … (Dan is 1 hole per 50 mins = 0.02 holes/min, Jon is 1 hole per 40 mins = 0.025 holes/min, together they are 0.045 holes/min, which inverts to 22.2 mins/hole) … EVERY time I determine the units involved in some problem, I hear that chemistry profs voice in my head 😊
who do the dig the same hole at the same time?
@@davidjones-vx9ju Solution 1: It is pure math and so no one is getting in each others way.
Solution 2: it is not a circular hole, so they each start at one end and work towards the middle where they finish from either side.
I solved it simply by thinking Jon will be the first one to finish half of the hole and it will take 20 minutes. Then Dan will still have 5% of the hole left and Jon will finish half of that in 2 minutes. Then .5% will be left and Jon will finish half of that in .2 minutes. You can keep doing that until one of them gets the last scoop but it's probably around 22.2 minute
Just sum the infinite series.
@@mikemondano3624 whatever that means
@@heavyglassglass It means do what you just did.
hmmm, and how big is half a hole exactly?
@@D0S81 It's the size of the amount of dirt you could move in half the time it takes you to dig a hole.
A problem designed for a perfect world situation is a bigger problem than saying optimistically that it would take 45 minutes. I realized many years ago that one of the basic rules of life is that Murphy drives the bus and we're just the passengers. We all know Murphy and he's an SOB. Take your best possible guess and add at least 20%.
@jeffrey I like the way you think!! I don't feel like the only idiot here like the guy constantly points out in his videos about how 90% of you won't get this right. Well of course I'm already going to be in that 90% just because of the power of suggestion. Yeah, just tell me I'm stupid and I won't even try. Great concept teacher. NOT
@@survivrs The problem with too many "educated" people is that they haven't gotten out in the real world of work. Over the years I've occasionally helped out guys with PhD's who knew all the theory but not the practical in my field. I learned the hard way with a high school education and some correspondence classes. I've had interns working for me, the first thing I tell them and other youngsters is get your paper education, then before you do anything else, get a job where you're going to get your ass kicked, your nose bloodied and your hands dirty. After that you might be ready to settle in at a desk.
@@jeffreywhitmoyer860 I like what you have done, it's something that all high school and college kids should do before moving forward. So many have no clue. Sort of like having book smarts or having street smarts. You truly need both to be successful.
When humans are involved math is irrelevant to the answer, they argue for an hour over what shovel is right for the job and leave never digging the hole at all.
Use the 'product over sum formula' that I learned at Tech school : product of 50*40 (200) divided by sum of 40*50 (90)
Jon and Dan are clearly digging holes to get rid of bodies. They have a bright future ahead of them.
It'll never get finished. Jon will get angry that he's getting more done than Dan, and will stop digging to demand higher pay. Then Dan will claim that he's working just as hard, and should get paid the same - and then add that unlike Jon he's willing to just do the job so he deserves to get paid more. Pretty soon they're hitting each other with their shovels, and then you need paramedics, and the job never gets done.
It will get done when Juan shows up and does it himself in fifteen minutes.
45 minutes plus another 10 since they only have one shovel and have to fight over how the other is doing wrong.
not forgetting stopping to talk and debate about the previous days football match so it would probably take about four hours to dig the hole
After 20 mins Jon has dug half the hole and Dan has dug 4/5 of half leaving them to then finish the 1/5 of 1/2 (1/10) the hole.
... after 2 mins Jon has dug half the 1/10 of the hole and Dan has dug 4/5 of 1/10 of the hole leaving 1/100 hole still to dig
. after .2 mins Jon has dug half the 1/100 of the hole and Dan has dug 4/5 of 1/100 of the hole leaving 1/1000 hole still to dig
Recurring........
One digs 2% per minute. The other digs 2.5% per minute.
100/4.5 = 22.22222222
Another equally good method is x/40+x/50=1, multiply to get a common denominator: becomes 5x/200+ 4x/200=1, 9x/200=1, so 9x=200 and x=22.22.
Theoretically, and assuming there is no delay due to two people attempting to work in the same space at the same time, Dan digs half the hole in 25 minutes and Jon digs half the hole in 20 minutes. When Jon finishes his half of the hole, he will keep digging to help Dan finish the last half. Jon's half of the hole is complete while Dan's half of the hole is 4/5 complete. The total completed at this point is 90%, leaving 10% left to be completed. This gives us a rate of 90% complete in 20 minutes. From this we make a formula to get to 100%. 90%/20 mins = 100%/T. T=100*20 mins/90. T=22.22 minutes. It will take 22.22 minutes for them to complete the hole.
That’s a long way of just saying 50+40=90/2=45/2=22.5
@@cinforammi8543 No, it is not. My answer is 22.22 minutes. Your answer is 22.5 minutes. That is not the same answer.
@@wesbaumguardner8829 the amount of time I saved on my solution more than makes up for time you spent finding yours.
Wrong question.
Over a 40 hour week, Dan has dug 48 holes & Jon has 60 holes.
Jon should be paid 25% more salary than Dan.
Dan should learn to be more efficient.
End of analysis.
Jon always takes 40 minutes, then he goes to help Dan finish up. But Dan's hole is still finished in 50 minutes. Two people cannot work on the same hole.
Although, if Dan "The Wimp" sits aside, then Jon can finish the last 20% of the hole in 10 minutes.
40 minutes because Dan will stand back and watch Jon do it if he’s a clever man!!!😂😂😂
😂😂😂
This may be fastest if the hole is small and deep.
A good heuristic is that the answer tends to be CLOSE to the average divided by the number of people, or 40 + 50 = 90 minutes / 2 people = average time 45 minutes. Divide that by two people AGAIN, you get 22.5 minutes, which is fairly close to 22.2 - the actual answer.
So if Alice, Bob, and Charlie can dig a hole in 20, 30, and 40 minutes respectively... the total is 90 minutes, the average is 30 minutes, and dividing by the number of people suggests it will take 10 minutes. The actual answer will be CLOSE to ten minutes; if you're very far away from that, you messed up.
Then when you actually dig into it, for each minute of work they will actually dig 1/20 + 1/30 + 1/40 = 6 / 120 + 4 / 120 + 3 / 120 = 13 / 120 of a hole. That's 9.2 minutes, which is close to ten minutes.
In one minute, Dan does 1/50 of the work. In one minute, Jon does 1/40 of the work. (1/50) + (1/40) = 4.5% of the work done in a minute. 100/4.5=22.2.
Least Common Multiple is 200 minutes. Dan can do 4 jobs in that time, Jon can do 5. Together they do 9 jobs, so the answer is 200/9.
Very neatly done! Question: what do women wear in regards to mathematics: they all (maybe none these days) are concerned with alge-bra!
Man i thought it was 45
If Dan is smart, it will take 40 minutes. It will take Dan 1 minute to realize that Jon is the MAN, and that he should assume a supervisory role and sit down and observe while Jon continues to dig for 39 more minutes.
Rate x time = amount. Dan's rate = 1 hole / 50 minutes, Jon's rate = 1 hole / 40 minutes. Quick check (1 hole / 50 minutes ) x 50 minutes = 1 hole. Perfect works great, so now when they work together (1 hole/50min) x time + (1 hole / 40 min) x time = 1 hole. Time is in minutes so units cancel, (1/50) t + (1/40) t = 1 ; find the common factor (4t + 5t)/200 = 1, 9t = 200, t = 200/9 minutes
You're explaining it the right way
@@billetem5868 Hey I'm just happy that I could get it done. Half the time I see this recommended math videos and my mind is instantly "Say WHAT!? High schoolers do this?"
As a kid I had issues with this type of problem because it makes you assume a number of variables as you noted at the end. Point one, you have to assume both men can work in the hole at the same time, ( In my adulthood, I would also be looking at OH&S). You have to assume that both men have the same tools, you have to assume the material in the planned hole is consistent. You have to assume that Jon will continue to work beyond the 50% of the hole he clears, which means he works more than Dan.
If the problem had been vehicle a travels at 80klm to a town 50klm away. And vehicle b travels at 70klm that is 45klm away. Who will get to their destination first. It requires less assumptions and presents more math.
By the way another way to view this if I accepted all of these assumptions would be true, with half the volume to move each, Jon will have cleared his half after 20 mins, while Dan will still have 5 mins of work left to do. Then if they divide that between the two of them you get the final equation of how much of the last five minutes of work will Jon and Dan complete?
the productivity of the first person is 1/50 part of the job per minute. The productivity of the second one is 1/40. The sum of their productivity is 1/50 + 1/40 = (50 + 40) / (50 * 40). But, what we need to check is how many minutes would it take to do the job which basically mean that we should divide a whole job 1 over their common productivity. 1 over any fraction simply swaps the numerator and denominator in places. Thus the answer is 40*50 / (40 + 50).
to quote Jim bowers " The chalk if you please. Using the simple formula A * B over A + B we get" 2000 over 90 or 22.22 repeating decimal minutes.
"Long have I been familiar with the exactitudes of the mathematical world. And Mat the horses name is Friday"
My solution: (paste into notepad using monospace font):
We ask the question: What time (how) will they finish?
The answer is: "At the same time" which is true, even is we do not know when.
We can call this time "t".
We multimply work rate by time taken to get the work done.
However each is doing only a proportion of the work, compared to the work they normally do ,
by this we mean that they do the work compared to themselves, no matter who else can also do that work.
If they only get to finish a proportion of the work we need to see (i) the total work rate and (ii) what proportion of the total work rate each would contribute, in order to see how ling it would take:
d = 1/50 (work rate of Dan) = 0.02 [1]
j = 1/40 (work rate of Jon) = 0.025 [2]
Combined work rate: 0.02 + 0.025 = 0.045 [3]
Proportional work rate of d -> a = d / (d+j) [4]
Proportional work rate of j -> b = j / (d+j) [5]
When we multiply work rate by time we should get the full job done, but we see that each will
only contribute a proportion of the total combined work rate in order to finish the work.
However, they would need still need to finish at the same time no matter what proportion of the work they manage to accomplish in order to do their part of the job when working together, which we agreed would be "t":
Therefore a * d = b * j meaning when we multiply the work rate proportion of d with d's own work rate (i.e. a) or that of j with his own work rate (i.e b),
they should be the same as they will finish at the same time = t given below as:
LHS: a * d = t and RHS: b * j = t { or [4] * [1] and [5] * [2] }
Lets do LHS -> d (d / (d+j)) = 1/50 ( (1/50) / (1/50 + 1/40))
= 1/50 (1/50 / (9/200))
= 1/50 (4/9) = 9/200 = 22.222 minutes
The RHS produces an identical answer, which is what is expected:
b * j = t
j * (j / (d+j)) = 1/40 (1/40 / (1/50 + 1/40) )
= 1/40 (1/40 / (9/200))
= 1/40 (5/9) = 9/200 = 22.22 minutes, confirming our expectations.
Final answer: 22.22 minutes
Work project: Hole = H
Work Performance (Work per minute)
Jon wpm : H/40
Dan wpm: H/50
t = time both work together
H = t x H/40 + t x H/50 divide by H
1 = t/40 + t/50
40 = t+ 40t/50
2000 = 50t + 40t
t = 2000/90
t = 22,2 min
We can compare with resistor value in electeonics, ie. paralell connection with formula:
1/Rp = 1/R1 + 1/R2+....
In this case, R1=50 and R2=40, then Rp = (50x40) /(50+40) = 2000/90 = 22.22
a quick and fast estimate is to add the 2 numbers together then half it twice, 90/2 45/2 22.5, 22 mins and 30 seconds, which isn't far off the actual answer.
Dan's productivity is 1/50 holes/minute. Jon's productivity is 1/40 holes/minute. The sum of the two productivities (1/50+1/40) is 90 holes in 2000 minutes. 2000 minutes divided by 90 holes results in 22,2... minutes
Realistic answer is about 25-30 minutes, because you're going based off of hypothetical peak performance, and you won't have hypothetical peak performance with an extra body in the way, it will be faster, yes, but not the 180.18% increase you described. Realistically it could even take 35 minutes because if you account for added idle chit-chat because workers working alone don't talk to themselves (usually), however two workers will typically talk and get distracted for approximately 2.34 minutes on average at least once every 23 minutes, adding that time up you get a solid approximation of anywhere between 27.34 minutes to 37.68 minutes based on when the conversations diverge off topic and get into debates. Also, are you adjusting for potential inclement weather? Because then the time could go from the aforementioned times to 3-5 business days.
To an electrical engineering student you would have said: "One resistor is 50 ohms, it's in parallel with another resistor that is 40 ohms. How much is the combined resistance?" And it's the same way to find the answer. Another method is x=(50*40)/(50+40) which I prefer to use better than 1/x=1/50+1/40 to avoid the extra reciprocal calculation step to get the x.
The solution in the video is the standard way. But using the average time also makes obvious sense. Ta = (50 + 40)/2 = 45 min. So it takes between them on average 45 min to dig 1 hole each. That’s 2 separate holes in 45 min. Together it would take 45/2 min to dig 1 hole.