Question:Consider the expression: \((4775 - 24\mathrm{t}^2) - 5(6\mathrm{t}^2 - 155) + 2(48 - 5\mathrm{t}^2)\).The expression can be written in the...
GMAT Advanced Math : (Adv_Math) Questions
Consider the expression: \((4775 - 24\mathrm{t}^2) - 5(6\mathrm{t}^2 - 155) + 2(48 - 5\mathrm{t}^2)\).
The expression can be written in the form \(\mathrm{a}\mathrm{t}^2 + \mathrm{b}\), where \(\mathrm{a}\) and \(\mathrm{b}\) are constants. What is the value of \(\mathrm{a} + \mathrm{b}\)?
1. TRANSLATE the problem requirements
- Given: \((4775 - 24\mathrm{t}^2) - 5(6\mathrm{t}^2 - 155) + 2(48 - 5\mathrm{t}^2)\)
- Need: Expression in form \(\mathrm{a}\mathrm{t}^2 + \mathrm{b}\), then find \(\mathrm{a} + \mathrm{b}\)
2. SIMPLIFY by distributing each multiplication
- Handle \(-5(6\mathrm{t}^2 - 155)\):
- Distribute the −5: \((-5)(6\mathrm{t}^2) + (-5)(-155) = -30\mathrm{t}^2 + 775\)
- Handle \(2(48 - 5\mathrm{t}^2)\):
- Distribute the 2: \((2)(48) + (2)(-5\mathrm{t}^2) = 96 - 10\mathrm{t}^2\)
3. SIMPLIFY by rewriting the complete expression
- Original: \((4775 - 24\mathrm{t}^2) - 5(6\mathrm{t}^2 - 155) + 2(48 - 5\mathrm{t}^2)\)
- Becomes: \((4775 - 24\mathrm{t}^2) + (-30\mathrm{t}^2 + 775) + (96 - 10\mathrm{t}^2)\)
4. SIMPLIFY by collecting like terms
- Collect all t² terms: \(-24\mathrm{t}^2 - 30\mathrm{t}^2 - 10\mathrm{t}^2 = -64\mathrm{t}^2\)
- Collect all constants: \(4775 + 775 + 96 = 5646\) (use calculator)
- Result: \(-64\mathrm{t}^2 + 5646\)
5. TRANSLATE to identify coefficients and calculate final answer
- In form \(\mathrm{a}\mathrm{t}^2 + \mathrm{b}\): \(\mathrm{a} = -64\), \(\mathrm{b} = 5646\)
- Therefore: \(\mathrm{a} + \mathrm{b} = -64 + 5646 = 5582\) (use calculator)
Answer: 5582
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak SIMPLIFY execution: Students make sign errors when distributing, especially with the negative coefficient −5. They might calculate \(-5(6\mathrm{t}^2 - 155)\) as \(-30\mathrm{t}^2 - 775\) instead of \(-30\mathrm{t}^2 + 775\), forgetting that \((-5)(-155) = +775\).
This arithmetic error cascades through the constant terms calculation, leading to an incorrect value for b and ultimately wrong final answer.
Second Most Common Error:
Poor SIMPLIFY execution: Students correctly distribute but make arithmetic mistakes when combining like terms. For example, they might calculate the t² coefficient as \(-24 - 30 - 10 = -54\) instead of \(-64\), or miscalculate the constants sum.
These calculation errors produce incorrect values for a and b, resulting in wrong final answer.
The Bottom Line:
This problem tests careful algebraic manipulation rather than complex concepts. Success requires methodical attention to signs during distribution and accurate arithmetic when combining multiple terms.