Rates

[KWF]

3 people rake a yard in 4 days.
6 people rake the same yard in how many days?

I think the answer is 2 days. I tried to keep this first part of the question simple.

If I add more information to the above, what would be the answer?

3 people rake a yard in 4 day working 8 hours per day.
6 people rake the same yard in how many days working 4 hours per day?

 

[Mrspi]

3 people rake a yard in 4 days.
6 people rake the same yard in how many days?

I think the answer is 2 days. I tried to keep this first part of the question simple.

If I add more information to the above, what would be the answer?

3 people rake a yard in 4 day working 8 hours per day.
6 people rake the same yard in how many days working 4 hours per day?

If it takes 4 eight-hour days, that's 32 hours. 3 people working for 32 hours means it takes 3*32, or 96 person-hours to do the job.

Let x = number of four-hour days it would take 6 people to do the job. The number of hours for the job would be 4x (x days at 4 hours per day), and the number of person-hours for 6 people would be 6*4x, or 24x person-hours.

Now...we know that must be equal to 96 person-hours, so

24x = 96

Solve for x.....

 

[KWF]

Follow-Up Rates

Thanks for the replies and the calculations!

With the solution that Denis provided, how does one know what numbers belong on the left and right hand sides of the equal sign?

Here is an example:

6 men build a wall 96 rods long in 18 days working 11 hours. How many men can build a wall 64 rods long in 2 days working 12 hours?

(A rod is a unit length of 5 yards but this information is not needed to find answer.)

 

[soroban]

Hello, KWF!

I have a very primitive approach . . .

6 men build a wall 96 rods long in 18 days working 11 hours a day.
How many men can build a wall 64 rods long in 2 days working 12 hours?

I solved it by adjusting the "proportions" . . .

The hours required is proportional to the number of rods.
The hours required is inversely proportional to the number of men.

. . \(\displaystyle \begin{array}{ccccc}\text{6 men} & \text{96 rods} & \text{198 hours} \\ \text{1 man} & \text{96 rods} & \text{1188 hours} \\ \text{1 man} & \text{32 rods} & \text{396 hours} \\ \text{1 man} & \text{64 rods} & \text{792 hours}
\end{array}\)

\(\displaystyle \text{To build a 64-rod wall in 24 hours, it will take: }\,\dfrac{792}{24} \:=\:33\text{ men.}\)

 

[HallsofIvy]

3 people rake a yard in 4 days.
6 people rake the same yard in how many days?

I think the answer is 2 days. I tried to keep this first part of the question simple.

Yes, that's good.

If I add more information to the above, what would be the answer?

3 people rake a yard in 4 day working 8 hours per day.
6 people rake the same yard in how many days working 4 hours per day?

Think about it as 3 of those people working 4 hours in the morning and the other 3 working 4 hours in the afternoon. Is that exactly the same as 3 people working the whole day?