Question

√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?

Answer 1

1750

Explanation:

You want the number of iPads that completely fills shelves of 50, fills 5 fewer shelves of 60 with some spare slots, and fills 8 fewer shelves of 80 than of 60, still with some spare slots.

Setup

Let x represent the number of iPads, and let 'a' and 'b' represent the numbers of empty slots on shelves of 60 and 80, respectively. The given relations tell us ...

(x +a)/60 +5 = x/50

(x +b)/80 +8 = (x +a)/60

Solution

Solving the first equation for 'a', we get ...

5(x +a) +1500 = 6x

x = 1500 +5a

a = (x/5) -300

Since 'a' is the number of empty slots in a shelf of 60, we have ...

0 < a < 60

0 < x/5 -300 < 60

0 < x -1500 < 300

1500 < x < 1800

Using the expression for 'a' in the second equation and solving for b, we have ...

3(x +b) +1920 = 4(x +a)

3x +3b +1920 = 4x +4(x/5 -300)

3b = 9/5x -3120

b = 3/5x -1040

Since 'b' is the number of empty slots on a shelf of 80, we have ...

0 < b < 80

0 < 3/5x -1040 < 80

1040 < 3/5x < 1120

1733 < x < 1866

Combined with the limits imposed by the value of 'a', the allowed values of x are multiples of 50 in the range ...

1733 < x < 1800

The only such multiple of 50 is 1750

The college received 1750 iPads.

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Additional comment

The number took 22 shelves of 80, 30 shelves of 60, and 35 shelves of 50. There were 10 empty slots on the last shelf of 80, and 50 empty slots on the last shelf of 60.