Two amateur wrestling squads—Squad A and Squad B—competed against each other. Each squad had the same (nonzero) number of its...

OWNING THE DATASET

Understanding Source A: Text - Problem Statement

Summary: Source A sets up a wrestling competition where equal numbers from two squads compete, and asks for a formula to find the overall participation rate in terms of individual squad participation rates (mathrm{x}) and (mathrm{y}).

Understanding Source B: Text - Concise Solution

Summary: Source B derives the mathematical solution showing that the overall participation rate equals (frac{mathrm{xa+yb}}{mathrm{a+b}}), which can be simplified using the equal-contestants constraint from Source A.

Understanding Source C: Table

• What the table shows: A 5×5 grid displaying calculated values of (frac{mathrm{xa+yb}}{mathrm{a+b}}) for different combinations of (mathrm{x}) and (mathrm{y}) values
• Key patterns observed:
  • Diagonal entries (where (mathrm{x=y})) show values: (frac{1}{4}), (frac{1}{3}), (frac{1}{2}), (frac{2}{3}), (frac{3}{4})
  • ""n/p"" entries appear at opposite corners: ((frac{1}{4}, frac{3}{4}))) and ((frac{3}{4}, frac{1}{4})))
  • All values range between (frac{1}{4}) and (frac{3}{4})
• Linkage insights:
  • To Source A: The table provides practical answers to the problem's question for specific (mathrm{x}) and (mathrm{y}) values
  • To Source B: The table contains pre-calculated results using the formula (frac{mathrm{xa+yb}}{mathrm{a+b}}), and the ""n/p"" entries occur where the constraint (mathrm{xa=yb}) cannot be satisfied with reasonable squad sizes
• Key findings:
  • When participation rates are equal ((mathrm{x=y})), the overall rate equals that same rate
  • Certain (mathrm{x,y}) combinations marked ""n/p"" cannot satisfy the equal-contestants constraint with integer squad sizes

Summary: Source C provides a lookup table for the formula derived in Source B, showing that certain participation rate combinations are impossible due to the equal-contestants constraint from Source A.

Overall Summary

• The wrestling competition problem requires equal numbers of contestants from each squad, creating the mathematical constraint (mathrm{xa = yb})
• This constraint allows us to derive a formula for overall participation rate: (frac{mathrm{xa+yb}}{mathrm{a+b}})
• The lookup table reveals:
  • When squads have equal participation rates, the overall rate equals that same rate
  • Certain combinations (like (frac{1}{4}) and (frac{3}{4})) are impossible because they can't satisfy the equal-contestants requirement with integer squad sizes
  • The overall participation rate is always bounded by the individual squad rates

Question Analysis

• Goal: Find which mathematical operation transforms the expression (frac{2xa}{a+frac{x}{y}a}) into an expression involving only x and y variables, eliminating 'a'
• Key constraints:
  • Must eliminate variable 'a'
  • Result must be expressed only in terms of x and y
  • Result should represent the proportion of contestants
• Answer type needed: Identification of the correct algebraic manipulation procedure

Connecting to Our Passage Analysis

• Analyzing each proposed procedure to determine whether it simplifies the expression to one involving only x and y, without 'a'
• Can be answered from analysis alone through algebraic manipulation of the given expression

Statement Evaluations

Statement 1 Analysis

""Multiply both the numerator and denominator by a.""

• Calculation: (frac{2xa cdot a}{(a + frac{x}{y}a) cdot a} = frac{2xa^2}{a^2 + frac{x}{y}a^2})
• Result: Still contains 'a' and fails to eliminate the variable
• Conclusion: INCORRECT

Statement 2 Analysis

""Multiply both the numerator and denominator by y.""

• Calculation: (frac{2xa cdot y}{(a + frac{x}{y}a) cdot y} = frac{2xay}{ay + xa} = frac{2xy}{x + y})
• Result: Successfully removes 'a' from expression
• Conclusion: CORRECT

Statement 3 Analysis

""Divide both the numerator and denominator by a.""

• Calculation: (frac{2xa / a}{(a + frac{x}{y}a) / a} = frac{2x}{1 + frac{x}{y}} = frac{2xy}{x + y})
• Result: Successfully removes 'a' from expression
• Conclusion: CORRECT

Statement 4 Analysis

""Subtract a and then divide the result a + (x/y)a.""

• This statement is unclear and does not represent a valid algebraic operation on the given expression
• Conclusion: INCORRECT

Statement 5 Analysis

""Divide by a + (x/y)a and then subtract a from the result.""

• This statement describes a sequence of operations that would not eliminate 'a' from the expression
• Conclusion: INCORRECT

Verification

• Options 2 and 3 both simplify the expression to (frac{2xy}{x + y}), eliminating 'a'
• Option 1 keeps 'a' in the expression and therefore fails the requirement
• Option 3 (division by 'a') is the most straightforward and standard method to remove the variable

Answer

Option 3