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Difficulty: MediumSystems of Linear Equations

A university admissions committee evaluates applicants from two distinct pools, Pool AA and Pool BB. In the first evaluation phase, accepting 40%40\% of the applicants from Pool AA and 60%60\% of the applicants from Pool BB results in a total of 180180 accepted students. In the second phase, accepting 70%70\% of the applicants from Pool AA and 30%30\% of the applicants from Pool BB results in a total of 210210 accepted students. What is the total number of applicants across both pools combined?

  1. A
    240
  2. 380Answer
  3. C
    390
  4. D
    420
  5. E
    195

Answer

The total number of applicants across both pools combined is 380.
Translating the conditions into linear equations gives 0.40A+0.60B=1800.40A + 0.60B = 180 and 0.70A+0.30B=2100.70A + 0.30B = 210. Multiplying both equations to clear decimals yields 2A+3B=9002A + 3B = 900 and 7A+3B=21007A + 3B = 2100. Subtracting the first from the second eliminates BB, giving 5A=12005A = 1200, so A=240A = 240. Substituting A=240A = 240 into 2A+3B=9002A + 3B = 900 gives B=140B = 140. The sum A+B=240+140=380A + B = 240 + 140 = 380.

Step-by-Step Solution

1
Set up a system of linear equations based on the percentage of accepted applicants from Pool A (AA) and Pool B (BB).
Equation 1: 0.40A+0.60B=1800.40A + 0.60B = 180; Equation 2: 0.70A+0.30B=2100.70A + 0.30B = 210.
Converting percentages to decimals forms linear equations relating the total applicants in each pool to the accepted totals.
2
Simplify the equations by multiplying by 10 and dividing by common factors.
Equation 1 becomes 2A+3B=9002A + 3B = 900; Equation 2 becomes 7A+3B=21007A + 3B = 2100.
Simplifying coefficients facilitates easier variable elimination.
3
Subtract the simplified Equation 1 from Equation 2 to eliminate BB and solve for AA.
(7A+3B)−(2A+3B)=2100−900  ⟹  5A=1200  ⟹  A=240(7A + 3B) - (2A + 3B) = 2100 - 900 \implies 5A = 1200 \implies A = 240.
Eliminating BB yields a single linear equation in AA.
4
Substitute A=240A = 240 back into 2A+3B=9002A + 3B = 900 to solve for BB.
2(240)+3B=900  ⟹  480+3B=900  ⟹  3B=420  ⟹  B=1402(240) + 3B = 900 \implies 480 + 3B = 900 \implies 3B = 420 \implies B = 140.
Determines the population of Pool B.
5
Calculate the combined total of applicants from both pools.
A+B=240+140=380A + B = 240 + 140 = 380.
The question asks for the sum of applicants in both pools.

Key Concept

Solving Systems of Two-Variable Linear Equations using Elimination
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Topics

Systems of Linear Equations

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