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Difficulty: MediumSystems of Linear Equations

A financial advisory firm manages two types of investment accounts: Growth portfolios and Conservative portfolios. Each Growth portfolio generates $5,000\$5,000 in annual dividends and $12,000\$12,000 in annual capital gains. Each Conservative portfolio generates $8,000\$8,000 in annual dividends and $4,000\$4,000 in annual capital gains. Last year, the firm generated a total of $186,000\$186,000 in dividends and $188,000\$188,000 in capital gains from these portfolios. What was the total number of portfolios (Growth and Conservative combined) managed by the firm?

  1. A
    1010
  2. B
    1717
  3. 2727Answer
  4. D
    3131
  5. E
    3535

Answer

The total number of portfolios managed by the firm is 2727.
Setting up the system of equations with xx as Growth portfolios and yy as Conservative portfolios gives 5x+8y=1865x + 8y = 186 and 12x+4y=18812x + 4y = 188. Multiplying the second equation by 22 yields 24x+8y=37624x + 8y = 376. Subtracting the first equation gives 19x=19019x = 190, so x=10x = 10. Substituting x=10x = 10 gives y=17y = 17. The total number of portfolios is 10+17=2710 + 17 = 27.

Step-by-Step Solution

1
Define variables and set up the system of linear equations in thousands of dollars.
Let xx be the number of Growth portfolios and yy be the number of Conservative portfolios.
Dividends equation: 5x+8y=1865x + 8y = 186
Capital gains equation: 12x+4y=18812x + 4y = 188
Dividing all dollar amounts by 1,0001,000 simplifies the coefficients.
2
Solve for xx using the elimination method.
Multiply the capital gains equation by 22:
2(12x+4y)=2(188)  ⟹  24x+8y=3762(12x + 4y) = 2(188) \implies 24x + 8y = 376
Subtract the dividends equation (5x+8y=1865x + 8y = 186) from this new equation:
(24x+8y)−(5x+8y)=376−186(24x + 8y) - (5x + 8y) = 376 - 186
19x=190  ⟹  x=1019x = 190 \implies x = 10
Eliminating yy isolates xx to determine the number of Growth portfolios.
3
Substitute x=10x = 10 into the capital gains equation to solve for yy.
Divide 12(10)+4y=18812(10) + 4y = 188 simplify to 120+4y=188  ⟹  4y=68  ⟹  y=17120 + 4y = 188 \implies 4y = 68 \implies y = 17
Finds the number of Conservative portfolios.
4
Calculate the total number of portfolios.
Total portfolios = x+y=10+17=27x + y = 10 + 17 = 27
The question asks for the sum of both types of portfolios.

Key Concept

Solving Systems of Two Linear Equations with Word Problem Constraints
Estimated Time:1m 30s
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Topics

Systems of Linear Equations

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