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.

If the number of math problems varies inversely with the number of distractions, we can write an equation to solve for some constant, k. To make things easier, let's use the variable P for the number of math problems Ricky can solve, and D for the number of distractions. We can write this in a couple ways, but it will be easier later if we write it as:

. We could just as easily have written it as , and actually those equations are equivalent, or the same.

We know that Ricky can solve 10 problems with 12 distractions, so we can plug in 10=P and 12=D to solve for k:

to solve for k, multiply both sides by 10:

Now that we know k, we can figure out how many distractions Ricky can handle in order to solve 15 problems in the same amount of time. We'll do this by pluggin in 120 for k, and 15 for P: 

Dividing gives us an answer of . This means that he can only handle 8 distractions. The question is asking us how many he needs to remove. Right now he has 12, so to get only 8 he'd have to remove 4. 

To find the inverse, switch the placement of the  and  variables:

Next,  should be isolated, providing the inverse function:

The notation  means a function in terms of .  Replace this notation with .

Interchange the  and  variables.

Solve for .  Subtract six on both sides.

Divide by three on both sides to isolate .

The answer is:  

The notation  is asking for the inverse of the function.

First, replace  with .

Swap the variables.

Solve for .  Subtract seven on both sides.

Divide by four on both sides.

Simplify both sides.

The inverse function is:  

To determine the inverse function, first replace the  with .

Interchange the x and y variables.

Solve for y.  Add nine on both sides.

Divide by two on both sides.

Simplify both sides.

The answer is: