Painting Michael's fence

Let F : 4 , G : 3 , H : 2

In 1 hour's time, the fence can be painted by F is 1/4 , G is 1/3 and H is 1/2

In 20 minutes' time, the fence painted by F is 1/4 * 1/3 = 1/12 , G = 1/9 , H = 1/6

1/12 + 1/9 + 1/6 = 13/36 ( which makes up 60minutes )

Double it and you will get : 26/36 ( which makes up 120 minutes )

Following the sequence F,G,H , the next 1/12 and 1/9 totals up to 33/36

3/36 = 1/12

Since Hazel is the last worker to paint , the time (t) required is :

1/2 * t = 1/12

t = 1/6 = 10 minutes

Hence the total time taken = 2(60) + 2(20) + 10 = 170minutes

Just represent the rates of work as 1/4, 1/3 and 1/2 for Frank, Grace and Hazel, respectively. These numbers represent their rates of work in 1 hour, but since each of them worked for only 1/3 of an hour, their contributions now become 1/12, 1/9 and 1/6. When they finish one cycle of work[ which is equal to 60 minutes ], it means they have finished 1/12 + 1/9 + 1/6 or 39/108 of the work.

Doubling this results to 78/108 of work done, equal to 2 hours. Then, 30/108 of the work is still unfinished. Add 1/12 to 78/108 and you'll obtain 87/108, which means that the work is not finished yet. Add 1/9 to 87/108 and you'll obtain 99/108. Finally, just add one-half of 1/6 [ which is equal to 9/108 ] to 99/108 and you'll obtain 108/108, which means that the work is completely done.

We have added to 2 hours 3 times, with each time equal to 20 minutes { except for the last one, which is only equal to 10 minutes only, because it's just one-half of the real quantity }. Therefore, the answer is 120 mins. + 50 mins. or 170 mins.

Frank in 20 minutes can paint a fraction of the fence, exactly $\frac{20}{240}=\frac{1}{12}$ of the fence, for 240 minutes are 4 hours. Analogously, Grace and Hazel paint $\frac{1}{9}$ and $\frac{1}{6}$ of the fence respectively; adding them using the l.c.m. 36, we obtain that the sum of 3, 4 and 6 and again 3,4,6 must arrive at the total of 36, for us to obtain $\frac{36}{36}$ . So 3 + 4 + 6 + 3 + 4 + 6 + 3 + 4 = 33, and we miss 3, but Hazel gives 6. So Hazel must work the half to give 3 instead of 6, and so the time is the half. So we have 8 addendi and half an addendum: $20*8+\frac{20}{2}=170$

Let the distance of the fence is X

speed of frank to panit the fence is X/4 speed of grace to panit the fence is X/3 speed of hazel to panit the fence is X/2

so,we have to paint the fence with the help of these 3 workers , first 20 minutes by frank ,then 20 minutes by grace and then 20 minutes by hazel

in first 20 minutes frank will paint = X/4 * 1/3 ( distance =speed * time ) in next 20 minutes grace will paint =X/3 * 1/3 ( distance =speed * time ) in next 20 minutes hazel will paint =X/2 * 1/2 ( distance =speed * time ) in first 20 minutes frank will paint = X/4 * 1/3 ( distance =speed * time ) in next 20 minutes grace will paint =X/3 * 1/3 ( distance =speed * time ) in next 20 minutes hazel will paint =X/2 * 1/2 ( distance =speed * time ) in first 20 minutes frank will paint = X/4 * 1/3 ( distance =speed * time ) in next 20 minutes grace will paint =X/3 * 1/3 ( distance =speed * time )

now ,the distance covered is 33X/36 ,so 3X/36 is required to paint the fence and 20 minutes of hazel is required ,

so we have to paint 3X/36 with speed 0f X/2 so speed * time = distance X/2 * time = X/12 time =1/6

so total time is = 1/3 + 1/3 + 1/3 + 1/3 + 1/3 +1/3 +1/3 + 1/3 + 1/6 ( in hours ) =170 minutes

Frank will do 9/108 of the fence each time he works. Grace will do 12/108, and Hazel will do 18/108.After two rounds or 120 minutes 78/108 are finished, leaving 30 to do. After Frank and grace go once more (40 min. or 160 total) 99/108 are done leaving only 9/108 left. Hazel will do 18 in 20 minutes so will only need to work 10 minutes. 160+10=170

Each painter will paint a certain percentage of the fence in each 20 minute session.

Frank would take 240 minutes to paint the fence by himself. So in 20 minutes he paints 20/240 or .083333 of the fence.

Similarly Grace will paint .1111111 of the fence in 20 minutes. And Hazel will paint .166667 of the fence in 20 minutes.

After the first hour each person will have painted their percent of the fence which adds up to .361111 of the fence.

In the second hour they will also complete .361111 percent of the fence and thus have completed .722222 of the fence.

In the 3rd hour Frank and Grace will not complete the fence but will add there percentage bringing the total after Grace finishes to .9166667 of the fence completed, which leaves .083333 of the fence for Hazel to paint.

If it takes Hazel 20 minutes to do .166667 then it will take her 10 minutes to paint .083333.

Total painting time is 2 hours and 50 minutes ot 170 minutes