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How many days will a men take to mow b acres if c boys can mow a acres in b days, and each man's work equals that of n boys?

Correct textbook solution:

  1. c boys can mow a acres in b days
  2. 1 boy can mow a acres in bc days
  3. n boys, or 1 man, can mow a acres in $\frac{bc}{n}$ days
  4. a men can mow a acres in $\frac{bc}{an}$ days
  5. a men can mow 1 acre in $\frac{bc}{a^2n}$ days
  6. therefore a men can mow b acres in $\frac{b^2c}{a^2n}$ days

I don't really understand this solution. In statement 2, I assume it sets c=1. Then how come the amount of days become bc? Because if you go back to statement 1, setting c=1 results in b days only.

In statement 3, why is n in the denominator?

In statement 5, when the a acre is set to 1, why is there an additional a in the denominator?

Thanks.

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    If $10$ workers can finish in a day, then $1$ worker will finish in $10$ days. So if $c$ boys are needed for $b$ days, $1$ boy will have to work for $bc$ days. – Snaw Apr 10 '22 at 07:48
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    Re going from statement-2 to statement-3, if $1$ boy can mow $a$ acres in $r$ days, then $n$ boys will take only $~\displaystyle \frac{1}{n}$th as long. That is, $n$ boys work $n$ times faster than $1$ boy. So, because you are changing the number of boys from $1$ boy to $n$ boys, you have to divide the time it will take by $n$. – user2661923 Apr 10 '22 at 08:32
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    In going from statement-4 to statement-5, the number of acres has been reduced from $a$ acres to $1$ acre. This means that the number of acres has been divided by $a$. Therefore, the corresponding time that will be required, for the much smaller task must also be divided by $a$. – user2661923 Apr 10 '22 at 08:35
  • "Because if you go back to statement 1, setting c=1 results in b days only." Yes, if you set c=1 for both the first two statements, then both indeed consistently tell you that $b$ days is needed. If you don't understand the above comments, google "inverse proportion" or take a look at this example. – ryang Apr 12 '22 at 14:13

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