The iMath college received big quantity of new iPads for all the college students and staff. Students tried to arrange all the iPads in the cabinet of shelves with slots of 80 iPads per shelf but one shelf remained with empty slots and it was not convenient for charging purposes. Then they decided to arrange the iPads in the cabinet with 60 iPads per shelf but they needed eight shelves more than when they tried to arrange in 80 iPads per shelf and still one shelf remained with empty slots. Eventually, they decided to arrange the iPads in the cabinet with 50 iPads per shelf but they needed five shelves more, than when they tried to arrange in 60 iPads per shelf, however all these shelves had no empty slots anymore. How many iPads did the iMath college receive?

9514 1404 393

  1750

Step-by-step explanation:

Setup

The number of iPads (y) is an even multiple of 50. Let that multiple be x, where x is an integer.

Then y = 50x.     [eq1]

__

When there are 60 iPads per shelf, the total number shelves used is x-5. Of those, x-6 are filled, and the number left over is between 0 and 60.

  0 < y -60(x -6) < 60     [eq2]

When there are 80 iPads per shelf, the total number of shelves used is (x-5)-8. Of those, x-14 are filled, and the number left over is between 0 and 80.

  0 < y -80(x -14) < 80     [eq3]

Solution

Substituting for y and solving for x, we get ...

  0 < 50x -60(x -6) < 60     [eq1 into eq2]

  0 < -10x +360 < 60 . . . eliminate parentheses

  0 > x -36 > -6 . . . . . . . divide by -10

  36 > x > 30 . . . . . . . . add 36

__

  0 < 50x -80(x -14) < 80      [eq1 into eq3]

  0 < -30x +1120 < 80 . . . . eliminate parentheses

  0 > 3x -112 > -8 . . . . . . . . divide by -10

  112 > 3x > 104 . . . . . . . . . add 112

  37 1/3 > x > 34 2/3 . . . . . divide by 3

__

The integer value of x must satisfy all of these constraints, so must be ...

  36 > x > 34 2/3

The only integer in that range is x = 35, so the number of iPads is ...

  y = 50(35) = 1750

The college received 1750 iPads.

answer: true

step-by-step explanation:

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