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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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Mathstudent55idn

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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