Problem 55 - Introductory Algebra for College Students

Question

One apartment is directly above a second apartment. The resident living downstairs calls his neighbor living above him and states, "If one of you is willing to come downstairs, we'll have the same number of people in both apartments." The upstairs resident responds, "We're all too tired to move. Why don't one of you come up here? Then we'll have twice as many people up here as you've got down there." How many people are in each apartment?

Step-by-Step Solution

Answer

The number of people in the downstairs apartment is 4, and the number of people in the upstairs apartment is 6.

Step 1 - Set up the equations

From the first statement, it can be concluded that if one person goes down from upstairs, the number of persons in both apartments will be equal. Hence, the equation is \(U-1 = D+1\), which simplifies to \(U = D+2\). From the second statement, it can be concluded that if one person goes up from downstairs, the number of persons upstairs will be twice the number as downstairs. Hence, the equation is \(D+1 = 2*(U-1)\), which simplifies to \(D = 2U - 3\).

Step 2 - Solve the equations

Now we solve these two equations simultaneously. Substitute \(U\) from the first equation \(U = D+2\) into the second equation to get: \(D = 2*(D+2) - 3\). Simplifying this will lead to \(D = 4\).

Step 3 - Find the value of U

Substitute \(D = 4\) into the first equation \(U = D+2\), so we will get \(U =4 + 2 = 6\).