MHB Solving the Lawn Mowing Puzzle: Explained!

[amirus15]

So person A takes 1 less hour than person B to mow a lawn, Person C takes twice as long as Person B to mow a lawn. All together they mow a lawn in one hour. How long does it take each of them to mow a lawn working alone? Can someone explain this to me please, thanks.

 

[Evgeny.Makarov]

Hi and welcome to the forum.

Let the productivity of A, B, C, measured in lawns per hour, be $a$, $b$ and $c$, respectively. Then A mows the lawn in $1/a$ hours, B does it in $1/b$ hours and C takes $1/c$ hours. Their combined productivity is $a+b+c$, and by assumption $1/(a+b+c)=1$, i.e.,
\[
a+b+c=1.\qquad(*)
\]
Now write the conditions

1. Person A takes 1 less hour than person B to mow a lawn.
2. Person C takes twice as long as Person B to mow a lawn

in terms of $a$, $b$ and $c$ and express $a$ and $c$ through $b$. Substituting these expressions into (*) will give you one equation in $b$. It has two solutions, but only one of them has physical sense.

 

[HallsofIvy]

Equivalently: let t be the time, in hours, it takes person B to mow the lawn.
so B has rate 1/t "lawns per hour".

person A takes 1 less hour than person B to mow a lawn

So A's time is t- 1 and rate is 1/(t- 1)

Person C takes twice as long as Person B to mow a lawn

So C's time is 2t and rate is 1/(2t)

All together they mow a lawn in one hour.

This last sentence tells us that, together, their rate of work is "1 hour per job".
When people, machines, etc. "work together", their rates of work add.
1/t+ 1/(t-1)+ 1/2t= 1. Solve that for t, the time It takes B to mow a lawn then find the times for A and B.

(Multiplying both sides of the equation by 2t(t-1) will eliminate the fractions.)