Question

What is a constant of Variation?

Answer 1

And can you provide an example of a mathematical problem involving constant of variation then explain to me how to solve it?

Answer 2

Consider this situation:

\(1\) laborer takes \(9\) hours to finish making a wall.
\(2\) laborers take \(4.5\) hours to finish making the same wall.
\(3\) laborers take \(3\) hours to finish making the same wall.

Here, do you see that the product of number of laborers and the time taken is always \(9\)? The constant of variation is \(9\). But remember that this happens in inverse relations only, where one value increases (number of laborers) and the other decreases (the time taken).

Answer 3

I do indeed.

Answer 4

When both values increase, we have the constant of variation equal to their ratio.

For example,

Buying one candy costs \(\$ 5\).
Buying two candies costs \(\$ 10\).
Buying three candies costs \(\$ 15\)

Here, the ratio of the cost to the number of candies remains the same: \(\dfrac{5}{1} = \dfrac{10}{2} = \dfrac{15}{3} = 5\)

Answer 5

If y varies directly as x and y = 15 when x = 24, find x when y = 25.

Find the constant of variation.
k=y/x
=15/24
=5/8
y=5/8x
To find x, substitute 25 for y.
25=5/8x
x=40

Answer 6

I would like to provide a much harder example.

Answer 7

I am not stopping you

Answer 8

There is a ship where food is stored and 120 people travel. The food will take 200 days to get finished for 120 people. However, 5 days later, 30 people die due to a mishap. How many days will it take the people who are left to finish their food supplies?

Answer 9

Can you do this one?

Answer 10

let me see.

Answer 11

I know how it is done, but I just don't know how to do it myself, can you provide an easier one? I am just learning this for my own good

Answer 12

Okay, let's see this one:

It takes you \(1\) hour to do your homework. You clone yourself and you ask him to do your homework for you along. How much time does it take if you do it with your clone?

Answer 13

it would take half an hour.

Answer 14

That's right, show your work?

Answer 15

Maybe that one was too easy,

Answer 16

Yeah I know not how to show my work for this..?

Answer 17

Okay, how about \(2\) clones?

Answer 18

2 clones any myself? can you show me how the work is represented?

Answer 19

and that wold take 20 minutes would it not?

Answer 20

This is an inverse relation: when the number of clones increase, the time decreases. So we know that the constant of variation is \(\rm number~of~yous \times time\) for all "number of yous" and their corresponding time.

So the constant of variation is always \(1 \times 1 = 1\). If we have two yous, then \(2 \times \rm time = 1\) so \(\rm time = 0.5\), which is half an hour.

Answer 21

Okay, that makes more sense.

Answer 22

Can you explain to me how that realates Constant of variation?