you have some marbles. when you make a shape of an equilateral triangle, you either have 5 extra, or you need 10 more to complete the shape,

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you have some marbles. when you make a shape of an equilateral triangle, you either have 5 extra, or you need 10 more to complete the shape, how many marbles do you have? how long is one side of your triangle?
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  1. Answer:

    110 marbles

    side 14

    Step-by-step explanation:

    Draw triangles using marbles as the sides.

    If the side is 2 marbles long, the triangle uses 3 marbles. 1 + 2 = 3

    If the side is 3 marbles long, the triangle uses 6 marbles. 1 + 2 + 3 = 6

    If the side is 4 marbles long, the triangle uses 10 marbles. 1 + 2 + 3 + 4 = 10

    If the side is n marbles long, the triangle uses n(n + 1)/2 marbles.

    These numbers are called triangular numbers.

    We are looking for the side, n.

    n(n + 1)/2 + 5 = (n + 1)(n + 2)/2 – 10

    n^2 + n + 10 = n^2 + 3n + 2 – 20

    n + 10 = 3n – 18

    -n = -14

    n = 14

    For n = 14

    n(n + 1)/2 =

    = 14(15)/2

    = 105

    For n = 15

    n(n + 1)/2 =

    = 15(16)/2

    = 120

    105 + 5 = 120 – 10

    You have 110 marbles. If you had 5 fewer, 105, you’d have a triangle with sides 14. If you had 10 more, 120, you’d have a triangle with sides 15. Since you have 110, you have 5 too many for side 14 and 10 too few for side 15.

    The sides of the triangle is 14.

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