# 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

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

Both AI's did very well with this...

If you're curious what

If you're curious what

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

Both AI's did very well with this...

If you're curious what

**Bard**made of this puzzle...If you're curious what

**ChatGPT**made of this puzzle...© Nigel Coldwell 2004 - – The

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