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.

Write the formula for the indirect proportional relationship.  If one variable increases, the other variable must also decrease.

Using speed and weight as  and  respectively, the equation becomes:

Use the initial condition of the turtle's speed and weight to solve for the  constant.

Substitute this value back into the formula.  The formula becomes:

We want to know the speed of the turtle when it is 50 pounds.  Divide the variable  on both sides to isolate the speed variable.

Substitute the new weight of the turtle.

Inverse variation takes the form:

Plugging in:

Then solve when x=3:

For an indirect proportionality, use the equation

If rate of speed is indirectly proportional to the time, then the speed is y (3 miles per hour) and the time is x (4 hours).

Replace those into the indirect proportionality formula

Solve for k by multiplying by 4 on both sides

The person traveled 12 miles.

Write the relationship for inverse proportionality.  If one variable increases, the other variable must decrease.

  or  

Let  be the turtle's speed, and  is the weight of the turtle since the turtle's speed depends on its weight.

Solve for  by substituting the initial conditions.

Multiply by two on both sides.

Substitute this value back into the original equation.

The relationship is:  

Substitute eight pounds to find its speed.

The turtle will move at a rate of  feet per minute.

Here, we can see that the independent variable is the number of stickers, so we'll call that . Each sticker costs $2 to make, so we'll write that as . The cost of the machine is always the same ($22), so we call that a constant.

So far, we have as the cost of making number of stickers.

Now, we need to describe the revenue gained by selling number of stickers. We know that Carla wants to sell the stickers at $3 each, which we can write as .

"Breaking even" means that your costs equal your income. In other words, it's the point at which Carla's sold exactly enough stickers to make all of her initial investment back. In math, we use an equal sign to describe this. Hence, we end up with the equation:

To set up this equation we need to put the amount of money she needs on one side of the equation and the amount of money she earns on the other side of the equation. It looks like the following

where  represents the number of hours Sally works. To get the dollar amount she earns for the hours she works, we mulitply it by 5.  The 3 represents the amount of money she already has saved. These two added together needs to equal 23 dollars, the amount for the doll.

From here we solve for :

Let us call the number of cards Annie has 

Josh has twice as many so we can call the number of cards Josh has 

Brenda has 3 times as many cards as Josh so we can call the number of cards Brenda has 

The total is 90 so we have 

Which we simplify by adding all the x terms: 

The salesman will sell either more than 5 cars or less than 5.

Therefore, only one set of the equations will be true for any one month.

The word between the expressions needs to be "or."

The total profit is expressed by 30,000x.

The commision can be calculated as .

The higher commision must be matched to the equation for greater amount of cars sold, . 

To make this much easier, translate the word problem into a system of three equations.

We have C for Chai, S for Sora, and N for Newton. To find Sora's age, plug in  into .

Sora is 8 years old. Use this to find Newton's age.

Newton is one year old.  So the answer is:

Sora, 8 years

Newton, 1 year