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.

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.

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.

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.

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.

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.

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.

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.

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

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.

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.

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

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.

Actually....after the first scoop of dirt you have a hole.

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'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 😊

The time would double, after Jon hit Dan over the head with a shovel for not pulling his weight.

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.

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

It will take longer because they will keep getting in each other's way. Depending on how close they are, it could turn violent.

Whoever constructed this has never tried to dig a single hole with multiple shovels.

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% (:

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.

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 .

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!

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 ..

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

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…

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.

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.

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.

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.

Dan leans on his shovel and watches as Jon digs the hole in 40 minutes.

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.

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.

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!

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.

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. . .)

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!!

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.

Their individual efficiency will not be the same as working together because they will get in each other's way.

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.

It will take all day because Jon and Dan disagree about everything and one will end up throwing his shovel and shouting vulgarities.

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.

They will obviously take twice as long because they will be in each other's way

Also estimated 22,5min. Averaged their times then divided by 2.

You know you work in consulting when you think, “Well that depends on what Dan & Jon’s bottleneck points are.”

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 added half of 50 and half of 40 and divided by 2=22.5.

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.

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.

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

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.

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.

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.

Dan digs at a speed of 1/50 holes/min or 0.02 holes/min. Jon digs at a speed of 1/40 holes/min or 0.025 holes/min. combined speed is 0.045 holes/min. It would therefore take them 1/0.045 minutes to complete a hole together = 22.2222' minutes :-)

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.

90 minutes, of course. They'll get in each other's way.

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.

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.

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.

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.

Good math teacher encouraging words and tip to solve these kind of math problems.

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.

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?

Dan digs at a rate of 1/50 hole/minute, Jon at a rate of 1/40 hole/minute. Together they're going to work for x minutes to get the job done. So (1/50)x + (1/40)x = 1 hole. That can be rewritten as (4/200 + 5/200)x = 1 (9/200)x = 1 x = 200/9 = 22.222222 minutes

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

I haven't watched the vid, but you don't need algebra. Dan digs 1/50th of a hole per minute. Jon digs 1/40th of a hole per minute. When they're working together, they the rates at which they dig are added: 1/50+1/40 hole per minute. You want to know minutes per hole. That's just the reciprocal. (Same as if you know seconds per mile and want to know miles per second.) So it's 1/(1/50 + 1/40).

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.

This is the same formula as that used to work out parallel resistors in an electronic circuit.

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?

Use the 'product over sum formula' that I learned at Tech school : product of 50*40 (200) divided by sum of 40*50 (90)

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.😊

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.

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.

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.

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.

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.

This is a really good example of imprecise use of language in maths. The question has me picturing Dan, a hole digger and Jon, a job doer, now collaborating on the joint project of hole digging and job doing. There’s no info about how they’ll work together, what the job is etc. etc. This might seem pedantic but consider every time students get asked a lateral thinking question and how they are worded.

I always argued with teachers giving questions like this. The data is incomplete and it's impossible to answer. 9 women can't make a baby in 1 month. 2 men can't dig a hole faster than 1 because they would interfere with each other at some point which will slow them down. And if the hole were the size of a shovel but deep they would interfere 100% of the time and if it were wide and shallow then the time for a single person would need to include moving around so the formula still isn't correct. And yes, project management became a big part of my career.

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

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!

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.

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.

I solved this in my head within seconds.....using this method Dan takes 50 minutes to complete 100% of the work , which = 2 units Jon takes 40 minutes to complete 100 % of the work , which = 2.5 units therefore together they take 4.5 units to complete the work which equals 100% divided by 4.5 = 22.22 minutes

Thanks for inspiring all these vastly creative comments on postulating the results for Dan and Jon's work. .... Or lack thereof.😊

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

I got 22.22222..... I gave the hole an arbitrary number of units (100) and had Dan removing those 100 units in 50 minutes as 100/50 (which is 2 units per minute). Likewise, Jon was 100/40, which is 2.5). So those same 100 units from both would be 100/(2+2.5), or 100/4.5. I'm honestly not sure if it's right or not. Now to find out.... YAY!

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%.

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

Title for Video - "How to turn a 4-minute video into 17 minutes."

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.

45 minutes plus another 10 since they only have one shovel and have to fight over how the other is doing wrong.

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).

I remember 25+ years ago taking the SAT and this type of problem being the only question I did not know. It happened two years in a row. So I went ahead and programmed my Calculator to assist with the answer. In doing so, I learned how to get the solution. It became my favorite type of problem to this day.

Man i thought it was 45

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.

How I remember how to do this is to think of it in per minute intervals. Dan can dig one hole per 50 minutes. Another way of looking at this is Dan can dig 1/50th of a hole per one minute . Likewise Jon can dig at a rate of 1/40th of a hole per minute. Now that you've written their rates as how much of a hole they can dig per minute you can combine their rates together to figure out how much of a hole is dug every minute. Combining the work they do the rate would be (1/50 + 1/40)th of a hole per minute. Remember you multiply the rate by minutes to get number of holes so the over all equation is (# of minutes) * (1/50 + 1/40) = (# of holes). Let X be the # of minutes and we just want to know the time for 1 hole so: x * (1/50 + 1/40) = 1

I read several of the comments and have an approach I did not find yet: After 20 minutes John has dug half the hole. He needs 40 minutes for the full hole, so he's got half of it after 20. Dan is not finished yet after the 20 minutes, he would need 25 minutes for his half. If 25 minutes is the full time he needs for his half of the job, 100%, then 20 minutes equals 80% of his job. So he has only dug out 80% of his half after these 20 minutes, which is 10% of the whole hole, pun intended. So we have left 10% of the hole after 20 minutes - and now John starts helping Dan with the rest of the digging, which conveniently is 10% - and the whole process starts again, reduced by the factor of 10: So this time after 2, not 20 minutes we are at the same point, John has done HIS half of the rest, John's got a little leftover, namely another 10% of the 10%. This ends up in a result of 2.2222 period. - Ok, they'll probably fight over the last shovel of sand - and the period stops at that particular decimal point.. I btw totally agree with those of you, who are uncomfortable with just theoretical formulas being applied - without explaining and understanding the meaning behind them.