varying directly vs inversely/proportion question

[evanr1234]

Hi Everyone,

If 8 men take 12 days to assemble 16 machines, how many days will it take 15 men to assemble 50 machines?

Answer: 20 days

if x varies directly as y^2, then x =ky^2

if x varies inversely with y^2 then x = k/(y^2)

k is a constant.

The book claims that the number of days varies directly as the number of machines and inversely as the number of men, therefore:

12 days = (k(machines))/men

I don't understand why it has to be that exact way, in my mind 12 days varies directly with 8 men so I could say 12 = 8k (if machines were not involved). this way you multiply a smaller number by a constant to get a bigger number.

or I could say 12 days varies inversely with the 16 machines and find a constant where it is true (12 men = k/16).

Thank you!

 

[Steven G]

8 men take 12 days to assemble 16 machines is the same as 1 man take 12 days to assemble 2 machines
Now 1 man takes 12 days to assemble 2 machines is the same as 15 men takes 12 days to assemble 30 machines is the same as 15 men takes (5/3)12 days = 20 days to assemble (5/3)30 = 50 machines. So 15 men takes 20 days to assemble 50 machines.

Alternative it take 96 man-days to assemble 16 machines. Not that 16*(50/16) = 50
So it takes 96(50/16) = 300 man-days to assemble 50 machines
You want to have 15 men. Note that 15*20=300, so 300 man-days means 15 men for 20 days.
Again, the answer is 15 men take 20 days to assemble 50 machines

 

[Steven G]

The book claims that the number of days varies directly as the number of machines and inversely as the number of men, therefore:

12 days = (k(machines))/men

I don't understand why it has to be that exact way, in my mind 12 days varies directly with 8 men so I could say 12 = 8k (if machines were not

12 days varies directly with 8 men. Think about this. You want to build a swimming pool. It will take 8 men 12 days to build the pool. If you want the swimming pool build faster, say in 6 days, will you now hire 4 men or 16 men?

 

[Steven G]

Hi Everyone,

If 8 men take 12 days to assemble 16 machines, how many days will it take 15 men to assemble 50 machines?

Answer: 20 days

if x varies directly as y^2, then x =ky^2

if x varies inversely with y^2 then x = k/(y^2)

k is a constant.

The book claims that the number of days varies directly as the number of machines and inversely as the number of men, therefore:

12 days = (k(machines))/men

I don't understand why it has to be that exact way, in my mind 12 days varies directly with 8 men so I could say 12 = 8k (if machines were not involved). this way you multiply a smaller number by a constant to get a bigger number.

or I could say 12 days varies inversely with the 16 machines and find a constant where it is true (12 men = k/16).

Thank you!

in my mind 12 days varies directly with 8 men so I could say 12 = 8k (if machines were not involved). this way you multiply a smaller number by a constant to get a bigger number. No no no, that is not always true. Take the number 12 and multiply it by 1/2. The result is NOT get bigger!!

 

[Steven G]

Hi Everyone,

If 8 men take 12 days to assemble 16 machines, how many days will it take 15 men to assemble 50 machines?

Answer: 20 days

if x varies directly as y^2, then x =ky^2

if x varies inversely with y^2 then x = k/(y^2)

k is a constant.

The book claims that the number of days varies directly as the number of machines and inversely as the number of men, therefore:

12 days = (k(machines))/men

I don't understand why it has to be that exact way, in my mind 12 days varies directly with 8 men so I could say 12 = 8k (if machines were not involved). this way you multiply a smaller number by a constant to get a bigger number.

or I could say 12 days varies inversely with the 16 machines and find a constant where it is true (12 men = k/16).

Thank you!

12 days = (k(machines))/men. We KNOW that 12 days = 16machines/8men. So k=16/8=2
12 days * 8 men = 16 machines, so 12 days = 16machines/8men

 

[lev888]

Forget about numbers. "Directly" means the two values go up and down together.

The book claims that the number of days varies directly as the number of machines

That's correct - it takes more days to make more machines.

 

[Deleted member 4993]

Hi Everyone,

If 8 men take 12 days to assemble 16 machines, how many days will it take 15 men to assemble 50 machines?

Answer: 20 days

if x varies directly as y^2, then x =ky^2

if x varies inversely with y^2 then x = k/(y^2)

k is a constant.

The book claims that the number of days varies directly as the number of machines and inversely as the number of men, therefore:

12 days = (k(machines))/men

I don't understand why it has to be that exact way, in my mind 12 days varies directly with 8 men so I could say 12 = 8k (if machines were not involved). this way you multiply a smaller number by a constant to get a bigger number.

or I could say 12 days varies inversely with the 16 machines and find a constant where it is true (12 men = k/16).

Thank you!

In this situation, there are three variables - normally all of those numbers are allowed to change.

The book claims that the number of days varies directly as the number of machines

In that it is implied that the - number of men working on the machines remain constant. More machines to be made, with same crew - more time is needed (simple - correct!).

The book claims that the number of days varies inversely as the number of men.

In that it is implied that the - number of machines remain constant. More men in crew - less time is needed to assemble same number of machines (simple again - correct!).

You can think of it in another way:

16 machines were made in (8*12 =) 96 man-days. Then

1 machine will be made in (96/16 = ) 6 man-days

50 machine will be made in (50*6 = ) 300 man-days

We have 15 men available.

So we need (300/50 = ) 20 days to assemble 50 machines with 15 men in the crew.

 

[evanr1234]

I get it now. This was very helpful! These concepts are so important, Im glad I have time to explore them.