The variation equation can be written as below. Direct variation will put in the numerator, while inverse variation will put in the denominator. is the constant that defines the variation.

To find constant of variation, , substitute the values from the first scenario given in the question.

We can plug this value into our variation equation.

Now we can solve for given the values in the second scenario of the question.

The variation equation is  for some constant of variation .

Substitute the numbers from the first scenario to find :

The equation is now .

If , then

The variation equation is  for some constant of variation .

Substitute the numbers from the first scenario to find :

The equation is now .

If , then

If  is the current and  is the resistance, then we can write the variation equation for some constant of variation :

or, alternatively, 

To find  , substitute :

The equation is . Now substitute  and solve for :

The general formula for inverse proportionality for this problem is

Given that  when , we can find  by plugging them into the formula.

Solve for  by multiplying both sides by 5

So .

The general formula of inverse proportionality for this problem is

where  is the number of days,  is the proportionality constant, and  is number of people.

Before finding the number of people needed to build the house in 14 days, we need to find . Given that the house can be built in 28 days with 7 people, we have

Multiply both sides by 7 to find .

So . Thus,

Now we can find the how many people are needed to build the house in 14 days.

Solve for . First, multiply by  on both sides:

Divide both sides by 14

So it will take 14 people to complete the house in 14 days.

The statement, 'The number of days to construct a house varies inversely with the number of people constructing that house' has the mathematical relationship , where D is the number of days, P is the number of people, and k is the variation constant. Given that the house can be completed in 28 days with 6 people, the k-value is calculated.

This k-value can be used to find out how many days it takes to construct a house with 20 people (P = 20).

So it will take 8.4 days to build a house with 20 people.

Given

Hence 

Interchanging  with  we get:

Solving for  results in .

If lonliness and friends are inversely proportional, we can set up an equation to solve for some missing constant, k. To make things easier to write, let's use the variable L for the lonliness scale, and the variable F for how many friends you have. First we can just set up the equation:

. We know that having 4 friends gives you a 10 on the lonliness scale, so we can plug those values in to start solving for k:

to solve, multiply both sides by 4:

Now that we know the constant is 40, we can figure out how much lonliness, L, corresponds to having 100 friends by setting up a new formula. We are still generally using , and we know k is 40 and F is 100:

We can divide to get

If monsters and lights vary inversely, we can set up an equation to solve for some unknown constant k. To make things easier, we can use the variable M for monsters and L for the number of lights on:

In the first example, there are 15 monsters under the bed and 1 light on:

It's pretty easy to see that our constant, k, is 15. To solve for M when there are 5 lights on, we can go back to our original and plug in 15 for k and 5 for L:

Dividing gives us an answer of 3.