Confusing algebraic word problem

• drooble122
In summary: This is not what c means. c can't change. See below.The important point is: it takes 1 boy bc days (to mow a acres).
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.

drooble122 said:
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.
drooble122 said:
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?
drooble122 said:
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?

Lnewqban
drooble122 said:
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.

Lnewqban
Steve4Physics said:
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.

drooble122 said:
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).

drooble122 said:
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
drooble122 said:
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)

Delta2

1. What is an algebraic word problem?

An algebraic word problem is a mathematical problem that involves using algebraic equations and variables to represent unknown quantities and solve for a specific solution.

2. Why are algebraic word problems confusing?

Algebraic word problems can be confusing because they often involve translating written language into mathematical expressions, which requires critical thinking and problem-solving skills.

3. How can I approach solving a confusing algebraic word problem?

One approach to solving a confusing algebraic word problem is to first carefully read the problem and identify the given information and what is being asked. Then, create an algebraic equation to represent the problem and solve for the unknown variable.

4. What are some common mistakes to avoid when solving algebraic word problems?

Some common mistakes to avoid when solving algebraic word problems include misinterpreting the given information, using incorrect algebraic rules, and not checking the solution for accuracy.

5. Are there any tips or strategies for solving algebraic word problems?

Yes, some tips and strategies for solving algebraic word problems include drawing diagrams or using visual aids, breaking the problem down into smaller parts, and practicing with different types of problems to improve problem-solving skills.

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