there are 3 types of items in a box pencil ,scale and eraser. if the no. of pencil is double the no. of eraser and the no.of scale is square of no. of pencil. find the count of each if the total is 45

To solve this problem, we can set up a system of equations based on the given information.

Let's represent the number of pencils as "P", the number of erasers as "E", and the number of scales as "S".

Based on the given information, we have the following equations:

1) P = 2E (The number of pencils is double the number of erasers)
2) S = P^2 (The number of scales is the square of the number of pencils)
3) P + E + S = 45 (The total count of items is 45)

Now, we can use these equations to find the values of P, E, and S.

From Equation 1, we can substitute P = 2E into Equation 2:
S = (2E)^2
S = 4E^2

Now, we can substitute P = 2E into Equation 3:
2E + E + 4E^2 = 45
6E + 4E^2 = 45
4E^2 + 6E - 45 = 0

To solve this quadratic equation, we can factor it or use the quadratic formula. We find that the factors of 4E^2 + 6E - 45 = 0 are (4E - 9)(E + 5) = 0.

Setting each factor equal to zero, we can solve for E:
4E - 9 = 0
E = 9/4 (not a valid solution as the number of items cannot be a fraction)

E + 5 = 0
E = -5 (not a valid solution as the number of items cannot be negative)

Since both solutions are not valid, it means that there is no possible combination of numbers that satisfies all the given conditions.

Therefore, there is no solution for the count of each item if the total is 45.

If there are x erasers, then

x + 2x + (2x)^2 = 45
x = 3
...