There were three times as many books in the top bookshelf as in the bottom one. after 15 books were moved from the top to the bottom, there were still two times as many books in the top than in the bottom. how many books are in the bottom bookshelf now?

  60 books

Step-by-step explanation:

Let x stand for the number of books on the bottom bookshelf now (the answer to the question). This is 15 more than were there to begin with.

The number on the top shelf now is 2x, which is 15 less than were there to begin with. So, you can write the relation ...

  2x +15 = 3(x -15) . . . . there were 3 times as many books on the top shelf to begin with

  2x + 15 = 3x -45 . . . . eliminate parentheses

  60 = x . . . . . . . . . . . . add 45-2x

The number of books in the bottom bookshelf now is 60.

reduce the friction between its moving parts

step-by-step explanation:

in fact, by reducing the friction between the moving parts of the machine, it is possible to reduce the energy wasted due to this friction; therefore, more input energy is converted into useful work, and this will improve the efficiency of the machine.

1) 2/5

2) zero=0/1

3) zero=0/2

4) -4/5

5) undefined=3/0