Perimeter of a Triangle with Integer Sides is Equal to 15 GMAT Problem Solving

Question: Perimeter of a triangle with integer sides is equal to 15. How many such triangles are possible?

1. 7
2. 6
3. 8
4. 5
5. 4

Approach Solution (1):

Concept: If a, b, and c are 3 integers sides of a triangle, “Difference of any 2 sides < Third side < Sum of the 2 sides”.

Assume \(a\leq{b}\leq{c}\),

If a = 1, Possible triangle is (a, b, c) = (1, 7, 7)
If a = 2, Possible triangle is (a, b, c) = (2, 6, 7)
If a = 3, Possible triangle is (a, b, c) = (3, 6, 6) and (3, 5, 7)
If a = 4, Possible triangle is (a, b, c) = (4, 4, 7) and (4, 5, 6)
If a = 5, Possible triangle is (a, b, c) = (5, 5, 5)

So, Total 7 triangles are possible.

Correct option: A

Approach Solution (2):

When perimeter is given as P

If P = odd

Number of triangle = \(\frac{(p+3)^2}{48}\)

No. of triangle = \(\frac{(p)^2}{48}\)

Where [] is nearest integer function

So, number of triangle = \(\frac{18*18}{48}=[6.75]=7\)

Correct option: A

“Perimeter of a triangle with integer sides is equal to 15. How many such triangles are possible?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

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