The expression 90y^5 - 54y^4 is equivalent to \(\mathrm{r}\mathrm{y}^4(15\mathrm{y} - 9)\), where r is a constant. What is the value...
GMAT Advanced Math : (Adv_Math) Questions
The expression \(90\mathrm{y}^5 - 54\mathrm{y}^4\) is equivalent to \(\mathrm{r}\mathrm{y}^4(15\mathrm{y} - 9)\), where \(\mathrm{r}\) is a constant. What is the value of \(\mathrm{r}\)?
1. TRANSLATE the problem information
- Given: \(\mathrm{90y^5 - 54y^4}\) is equivalent to \(\mathrm{ry^4(15y - 9)}\)
- Find: The value of constant r
2. INFER the solution approach
- Since these expressions are equivalent, I need to get them in the same form to compare
- The left side is already expanded, so I should expand the right side
- Then I can equate coefficients of like terms
3. SIMPLIFY by expanding the factored form
- Apply distributive property to \(\mathrm{ry^4(15y - 9)}\):
- \(\mathrm{ry^4 \cdot 15y = 15ry^5}\)
- \(\mathrm{ry^4 \cdot 9 = 9ry^4}\)
- So \(\mathrm{ry^4(15y - 9) = 15ry^5 - 9ry^4}\)
4. INFER coefficient relationships
- Now I have: \(\mathrm{90y^5 - 54y^4 = 15ry^5 - 9ry^4}\)
- For polynomial equivalence, coefficients of like terms must be equal:
- \(\mathrm{y^5}\) terms: \(\mathrm{90 = 15r}\)
- \(\mathrm{y^4}\) terms: \(\mathrm{-54 = -9r}\)
5. SIMPLIFY to solve for r
- From \(\mathrm{90 = 15r}\): \(\mathrm{r = 90 \div 15 = 6}\)
- Check with second equation: \(\mathrm{-54 = -9r}\) → \(\mathrm{r = 54 \div 9 = 6}\) ✓
Answer: 6
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak SIMPLIFY execution: Students make mistakes when expanding \(\mathrm{ry^4(15y - 9)}\), particularly with exponent rules.
They might write \(\mathrm{ry^4 \cdot 15y}\) as \(\mathrm{15ry^4}\) instead of \(\mathrm{15ry^5}\), forgetting that \(\mathrm{y^4 \cdot y = y^5}\). This creates the wrong expanded form and leads to incorrect coefficient equations. This causes them to get stuck or guess randomly.
Second Most Common Error:
Poor INFER reasoning: Students don't recognize that they need to equate coefficients of like terms.
Instead, they might try to substitute specific values for y or attempt to solve the equation algebraically as if it were an equation to solve for y. Without the key insight about coefficient equivalence, they abandon systematic solution and guess.
The Bottom Line:
This problem tests whether students can bridge the gap between factored and expanded polynomial forms. Success requires both solid algebraic manipulation skills and the strategic insight that equivalent expressions have matching coefficients.