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?
To solve this problem, we need to analyze the information given and find a common factor that represents the total number of iPads.
First, let's consider the situation where there were 80 iPads per shelf, and one shelf remained empty. We can represent this information with the equation: (80 * n) + 1 = x, where n is the number of shelves and x is the total number of iPads.
Next, when there were 60 iPads per shelf, the college needed eight shelves more than before, and one shelf remained empty. This situation can be represented by the equation: (60 * (n + 8)) + 1 = x.
Finally, when there were 50 iPads per shelf, the college needed five shelves more than before, but all shelves were full. This can be represented by the equation: 50 * (n + 5) = x.
We can now solve this system of equations to find the value of x, which represents the total number of iPads.
1. Solving the equation (80 * n) + 1 = x:
We can use trial and error to find a value for n that satisfies this equation.
By substituting values of n, we find that when n = 1, x = 81. This means that if there is only one shelf, there are 81 iPads.
2. Solving the equation (60 * (n + 8)) + 1 = x:
Again, using trial and error, we substitute values of n.
When n = 1, x = 489.
When n = 2, x = 549.
When n = 3, x = 609.
When n = 4, x = 669.
When n = 5, x = 729.
When n = 6, x = 789.
When n = 7, x = 849.
When n = 8, x = 909.
When n = 9, x = 969.
3. Solving the equation 50 * (n + 5) = x:
Again, substituting values of n, we find:
When n = 1, x = 55.
When n = 2, x = 105.
When n = 3, x = 155.
When n = 4, x = 205.
When n = 5, x = 255.
From the information above, we see that the total number of iPads (x) should be a common factor of all the calculated values. By inspecting the calculated values, we can determine that the only common factor is 255. Therefore, the iMath college received a total of 255 iPads.