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 the iMath college received?
1750
To solve this problem, we need to find a common multiple among the three given scenarios.
First, let's find the least common multiple (LCM) of 80, 60, and 50.
To find the LCM, we can start by listing the multiples of each number until we find a common multiple.
Multiples of 80: 80, 160, 240, 320, 400, 480, 560, 640, 720, 800
Multiples of 60: 60, 120, 180, 240, 300, 360, 420, 480, 540, 600
Multiples of 50: 50, 100, 150, 200, 250, 300, 350, 400, 450, 500
From the lists above, we can see that the common multiple of 80, 60, and 50 is 240.
Now, we can determine the number of iPads the college received by finding the total number of iPads in each case.
For 80 iPads per shelf scenario:
Number of shelves = (Total iPads) / (Number of iPads per shelf) = (Total iPads) / 80
For 60 iPads per shelf scenario:
Number of shelves = (Total iPads) / (Number of iPads per shelf) = (Total iPads) / 60 + 8
For 50 iPads per shelf scenario:
Number of shelves = (Total iPads) / (Number of iPads per shelf) = (Total iPads) / 50 + 8 + 5
Since the number of shelves is an integer value, we can use this information to solve the problem.
Let's solve the given equations step by step:
(Number of iPads) / 80 = (Number of iPads) / 60 + 8
Simplifying, we get:
(Number of iPads) / 80 - (Number of iPads) / 60 = 8
To find the common denominator, we multiply each fraction by the other denominator:
60 * (Number of iPads) / 80 - 80 * (Number of iPads) / 60 = 8
3 * (Number of iPads) - 4 * (Number of iPads) / 2 = 8
(3 * Number of iPads - 4 * Number of iPads) / 2 = 8
(-1 * Number of iPads) / 2 = 8
-1 * Number of iPads = 8 * 2
Number of iPads = -16
We get a negative number of iPads, which doesn't make sense. Therefore, there is an error in one of the equations or assumptions made.
Please double-check the information and question to see if there are any inconsistencies or mistakes.