Confusing algebraic word problem

[drooble122]

Homework Statement

See attached.

Relevant Equations

Trying to put word problems into algebraic symbols

I have great difficulty understanding the solution. I'll go through it line by line:

c boys can mow a acres in b days. This condition is from the problem and I understand it.

1 boy can mow a acres in bc days. Why not b/c days? After all isn't 1 boy from c/c=1?

n boys, or 1 man, can mow a acres in bc/n days. Why suddenly divide by n?

And then I'm totally lost from here.

 

[PeroK]

c boys can mow a acres in b days. This condition is from the problem and I understand it.

Let's put some numbers in: 2 boys can mow 1 acre in 10 days.

1 boy can mow a acres in bc days. Why not b/c days? After all isn't 1 boy from c/c=1?

Is it: 1 boy can mow 1 acre in 20 days; or 1 boy can mow 1 acre in 0.5 days.

Does one boy do half the work of two boys or, as you suggest, twice the work?

n boys, or 1 man, can mow a acres in bc/n days. Why suddenly divide by n?

We divide by $n$ because a man does n-times more work than a boy and completes a task in n-times less time.

Let's say 1 man does the work of 4 boys. And 1 boy mows an acre in 20 days. 4 boys mow an acre in 5 days, so 1 man mows an acre in 5 days.

If you multiple by $n$, then 1 man takes 80 days to mow an acre. Does that make sense?

 

[Steve4Physics]

c boys can mow a acres in b days. This condition is from the problem and I understand it.

1 boy can mow a acres in bc days. Why not b/c days

II will add a few words to what @PeroK has said.

It can help to use made-up values to understand what is going on. Then, after a bit of practice, you can do it without the made-up values.

c boys can mow a acres in b days

Make up some values, e.g. c = 5, a = 10, b = 3. This is the same as saying:
5 boys can mow 10 acres in 3 days

First, we want to know how many days for 1 boy to mow 10 acres.

Think carefully at this point. If should be clear that it will take 1 boy longer than 5 boys to do the same job.

How many times longer? 5 times longer. So we need to multiply (not divide) 3 days by 5.

To do the same job (mow 10 acres), 1 boy will take 5 x3 days = 15days. In symbols, the number of days is bc.

 

[drooble122]

II will add a few words to what @PeroK has said.

It can help to use made-up values to understand what is going on. Then, after a bit of practice, you can do it without the made-up values.

c boys can mow a acres in b days

Make up some values, e.g. c = 5, a = 10, b = 3. This is the same as saying:
5 boys can mow 10 acres in 3 days

First, we want to know how many days for 1 boy to mow 10 acres.

Think carefully at this point. If should be clear that it will take 1 boy longer than 5 boys to do the same job.

How many times longer? 5 times longer. So we need to multiply (not divide) 3 days by 5.

To do the same job (mow 10 acres), 1 boy will take 5 x3 days = 15days. In symbols, the number of days is bc.

c boys can mow a acres in b days.

Then 1 boy (I assume this means setting c=1) can mow a acres in bc days. If you plug in c=1, then its b days.

But what if I set 10 boys (c=10), then plugging it back in the number of days becomes bc=10b. This is 10 times more days than 1 boy, which makes no sense.

If it was b/c days, then c=1 means it takes b days for 1 boy to mow a acres. For c=2, it takes b/2, or half the time with 2 boys. c=10 it takes b/10 days. 10 boys take only 10% of the time compared to 1 boy. Hence increasing the number of boys proportionally decrease the number of days needed.

Thanks a lot for your help.

 

[Steve4Physics]

Then 1 boy (I assume this means setting c=1) can mow a acres in bc days. If you plug in c=1, then its b days.

No. Working out how long 1 boy takes is not the same as setting c=1. That's not what c means. c can't change. See below.

The important point is: it takes 1 boy bc days (to mow a acres).

But what if I set 10 boys (c=10), then plugging it back in the number of days becomes bc=10b. This is 10 times more days than 1 boy, which makes no sense.

You can't set c=10. c can't be changed (see below).

1 boy can coomplete the job in bc days.
10 boys can do the job 10 times quicker than 1 boy, so the time taken by 10 boys is bc/10.

Do not treat c, a and b as variables. They are constants. For example:
5 boys can mow 10 acres in 3 days
c = 5, a = 10, b = 3. c, a and b are fixed initial values,
c, a, and b do not change. You can't (for example) suddenly say c=1. c must stay equal to 5.

If you want the number of boys (N) to complete a job, you express N in terms of c, a, b and any other values supplied.

 

[Lnewqban]

I have great difficulty understanding the solution. I'll go through it line by line:
...
And then I'm totally lost from here.

This is the way I would solve this problem:
It is basically a work or energy related problem, for which power data is given.
Then, I would assemble equations that consider all the given variables in function of time.

Power of c number of boys = a acres / b days

Power of 1 boy = Work / time = a acres / (c boys x b days) (equation 1)

Power of 1 man = Power of n boys = (n x a acres) / (c boys x b days) (equation 2)

Now, here is where the problem has been made purposely confusing:
1) The same variable a has been assigned to number of acres and number of men.
2) The same variable b has been assigned to number of days and number of acres.

Just to work avoiding that confusion, I replace those two last variables in the question with α for the number of men, as well as with β for the number of acres.
In order to please the questioner, I will revert α and β to a and b in the final response.

Therefore; as
time = work / power
From equation 1 above:
Time 1 boy to mow a acres = a acres / [a acres / (c boys x b days)] = c boys x b days

From equation 2 above:
Time 1 man to mow a acres = a acres / [(n x a acres) / (c boys x b days)] = (c boys x b days) / n

Time α men to mow a acres = (Time 1 man to mow a acres) / α = (c boys x b days) / (α men x n)

Time α men to mow 1 acre = (Time α men to mow a acres) / a = (c boys x b days) / (a x α men x n)

Time α men to mow β acres = (Time α men to mow 1 acre) x β acres = (c boys x b days x β acres) / (a x α men x n)