# Answer to Riddle #88: Work Rate Puzzles

88. Adam can mow a field in 1 hour, in the same time Bob can mow 2 fields, if they work together how long does it take them to mow a field?

And a generalised discussion of work-rate problems.

And a generalised discussion of work-rate problems.

Philip Jones emailed me with the suggestion that I include a work-rate based puzzle and so I did:

Before reading the answer can I interest you in a clue?

The key to solving this kind of puzzle is to realise it's just a velocity calculation, or speed more accurately. Speed is distance over time.

In Bob's case for example he can do two fields in one hour. This is no different to 2 miles in 1 hour or 2mph. Adam can do 1 field in 1 hour.

Combined they can do 3fph (fields per hour,) so using our equations of motion, time equals distance over speed: they can mow 1 field in ⅓ of an hour, or 20 minutes.

Person 1: 1 field, 4 days, speed = ¼fpd

Person 2: 1 field, 3 days, speed = ⅓fpd

Adding the speeds 1/4 + 1/3 = 3/12 + 4/12 = 7/12

Time = distance over speed = 1 / (7/12) = 12/7 = 1 + 5/7 = 1 (12 hour,) day 8 hours 34 minutes 17.1429 seconds or 20 hours 34 minutes

Person 2: 1 house, 2.5 days, speed = ⅖hpd

Person 3: 1 house, x days, speed =

1 = 0.5 + 0.4 + 1/x

x = 10

It takes the third man 10 days to paint a house. Or, since I asked for the rate, he paints houses at a rate of 1 every 10 days

The key to solving this kind of puzzle is to realise it's just a velocity calculation, or speed more accurately. Speed is distance over time.

In Bob's case for example he can do two fields in one hour. This is no different to 2 miles in 1 hour or 2mph. Adam can do 1 field in 1 hour.

Combined they can do 3fph (fields per hour,) so using our equations of motion, time equals distance over speed: they can mow 1 field in ⅓ of an hour, or 20 minutes.

## Time for some more complicated problems

I promise not all of these will involve fields. So work out the speeds, add them, and solve for time.If one man can mow a wheat field with a scythe in four twelve-hour days and another man takes three such days, how long will it take them to mow the field working together?

Person 1: 1 field, 4 days, speed = ¼fpd

Person 2: 1 field, 3 days, speed = ⅓fpd

Adding the speeds 1/4 + 1/3 = 3/12 + 4/12 = 7/12

Time = distance over speed = 1 / (7/12) = 12/7 = 1 + 5/7 = 1 (12 hour,) day 8 hours 34 minutes 17.1429 seconds or 20 hours 34 minutes

Person 1: 1 house, 2 days, speed = ½hpdIf one man can paint a house in two days, another man can do it in 2.5 days at what rate would a third man have to paint houses for them, working together, to paint a house in a day?

Person 2: 1 house, 2.5 days, speed = ⅖hpd

Person 3: 1 house, x days, speed =

^{1}/_{x}hpd1 = 0.5 + 0.4 + 1/x

x = 10

It takes the third man 10 days to paint a house. Or, since I asked for the rate, he paints houses at a rate of 1 every 10 days

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