Question:The perimeter of a triangle is given by the expression 10y + 12. The lengths of two of the sides...
GMAT Advanced Math : (Adv_Math) Questions
The perimeter of a triangle is given by the expression \(10\mathrm{y} + 12\). The lengths of two of the sides of the triangle are given by the expressions \(3\mathrm{y} - 2\) and \(4\mathrm{y} + 5\). Which of the following expressions represents the length of the third side of the triangle?
1. TRANSLATE the problem information
- Given information:
- \(\mathrm{Perimeter = 10y + 12}\)
- \(\mathrm{Side\ 1 = 3y - 2}\)
- \(\mathrm{Side\ 2 = 4y + 5}\)
- Need to find: Side 3
- What this tells us: Since perimeter equals the sum of all three sides, we can find the third side by subtracting the two known sides from the perimeter.
2. INFER the approach
- Key insight: For any triangle, \(\mathrm{Perimeter = Side_1 + Side_2 + Side_3}\)
- Therefore: \(\mathrm{Side_3 = Perimeter - Side_1 - Side_2}\)
- We'll substitute our expressions and simplify algebraically
3. Set up the equation
\(\mathrm{Side_3 = (10y + 12) - (3y - 2) - (4y + 5)}\)
4. SIMPLIFY by distributing and combining like terms
- Distribute the negative signs:
\(\mathrm{Side_3 = 10y + 12 - 3y + 2 - 4y - 5}\)
- Group like terms:
\(\mathrm{Side_3 = (10y - 3y - 4y) + (12 + 2 - 5)}\)
- SIMPLIFY to get final answer:
\(\mathrm{Side_3 = 3y + 9}\)
Answer: A
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak SIMPLIFY skill: Sign errors when distributing negative signs
Many students correctly set up \(\mathrm{Side_3 = (10y + 12) - (3y - 2) - (4y + 5)}\) but then make errors like:
- Writing \(\mathrm{10y + 12 - 3y - 2 - 4y + 5}\) (forgetting to flip the sign of -2)
- Getting \(\mathrm{3y + 15}\) instead of \(\mathrm{3y + 9}\)
This may lead them to select Choice B \(\mathrm{(3y + 15)}\).
Second Most Common Error:
Poor TRANSLATE reasoning: Adding instead of subtracting the known sides
Some students think they need to add all the given expressions together rather than recognizing that they need to subtract the known sides from the perimeter.
They calculate: \(\mathrm{(10y + 12) + (3y - 2) + (4y + 5) = 17y + 15}\)
This may lead them to select Choice D \(\mathrm{(17y + 15)}\).
The Bottom Line:
This problem tests whether students can correctly translate a perimeter relationship into algebra and then execute the algebraic manipulation without sign errors. The key insight is recognizing that finding an unknown side requires subtraction, not addition.