word problem help

[girlpower]

Working alone, Pierre can complete a job in 10 hours. Juan can do the same job in 8 hours. How long would it take them to complete the job if they work together?

I solved it by adding 10+8=18 but I don't think it's right but I at least I tried.

 

[Yogi]

First off you would expect 2 people working together to get done faster than either one individually which makes an answer of 18 hours not plausible.

We know Pierre can do 1 job in 10 hours.
How many jobs can Juan do in 10 hours? (hint: use 1 job=8 hours and multiply something to both sides to make it 10 hours)
How many jobs do they do together in 10 hours? (_____=10 hours)
Multiply something on both sides so that it looks like (1job =_____hours)

 

[Lost souls]

Working alone, Pierre can complete a job in 10 hours. Juan can do the same job in 8 hours. How long would it take them to complete the job if they work together?

I solved it by adding 10+8=18 but I don't think it's right but I at least I tried.

amount of work = X
if P takes 10 hrs to complete X,
work done/hr by P= X/10
if J takes 8 hrs to complete X
work done/hr by J= X/8
total work done/hr by both combined= (X/8)+(X/10)=y
total hours taken = X/(y).

 

[TchrWill]

Working alone, Pierre can complete a job in 10 hours. Juan can do the same job in 8 hours. How long would it take them to complete the job if they work together?

I solved it by adding 10+8=18 but I don't think it's right but I at least I tried.

Problems of this type are solvable by either of the following methods.


If it takes one person 5 hours to paint a room and another person 3 hours, how long will it take to paint the room working together?

Method 1:

1--A can paint a room in 5 hours.
2--B can paint a room in 3 hours.
3--A's rate of painting is 1 room per A hours (5 hours) or 1/A (1/5) room/hour.
4--B's rate of painting is 1 room per B hours (3 hours) or 1/B (1/3) room/hour.
5--Their combined rate of painting is therefore 1/A + 1/B = (A+B)/AB = (1/5 + 1/3) = (8/15) rooms /hour.
6--Therefore, the time required for both of them to paint the 1 room working together is 1 room/(A+B)/AB rooms/hour = AB/(A+B) = 5(3)/(5+3) = 15/8 hours = 1 hour-52.5 minutes.

Note - Generally speaking (if the derivation is not specifically required), if it takes one person A units of time and another person B units of time to complete a specific task working alone, the time it takes them both to complete the task working together is T = AB/(A + B), where AB/(A + B) is one half the harmonic mean of the individual times, A and B.

You might like to derive the equivalant expression involving 3 people working alone and together which results in T = ABC/(AB + AC + BC).


Method 2:


Consider the following diagram -
..........I<----------B------->I

..........I____________I_________________

..........I........................../........................../\
..........I..*...................../............................I
..........I.....*................/..............................I
..........Iy.......*........../................................I
..........I................../..................................I
..........I***x******/ ...............................I

..........I............./....*................................(c)
..........I(c-y)..../.........*..............................I

..........I......../...............*...........................I.
..........I....../....................*........................I
..........I..../.........................*.....................I
..........I../..............................*..................l
.........I./...................................*...............\/__
.........I<---------------------A-------------->I


1--Let c represent the area of the house to be painted.
2--Let A = the number of hours it takes A to paint the house.
3--Let B = the number of hours it takes B to paint the house.
4--A and B start painting at the same point but proceed in opposite directions around the house.
5--Eventually they meet in x hours, each having painted an area proportional to their individual painting rates.
6--A will have painted y square feet and B will have painted (c-y) square feet.
7--From the figure, A/c = x/y or Ay = cx.
8--Similarly, B/c = x/(c-y) or by = bc - cx.
9--From 7 & 8, y = cx/a = (bc - cx)/b from which x = AB/(A+B), one half of the harmonic mean of A and B.

I think this should give you enough of a clue as to how to solve your particular problem.