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Difficulty: HardAlgebraic Word Problems and Modeling

An investment firm allocated a total principal of $120,000\$120,000 between two accounts, Account X and Account Y. Account X earned a simple annual interest rate of 6%6\%, and Account Y earned a simple annual interest rate of 10%10\%. At the end of one year, the combined total interest earned from both accounts was $9,200\$9,200. An annual administrative fee was then deducted from the interest: a 25%25\% fee on the interest earned from Account X, and a 10%10\% fee on the interest earned from Account Y. What was the net amount of interest, in dollars, remaining after these administrative fees were deducted?

  1. A
    $7,530\$7,530
  2. $7,650\$7,650Answer
  3. C
    $7,750\$7,750
  4. D
    $8,775\$8,775
  5. E
    $10,750\$10,750

Answer

The net amount of interest remaining after administrative fees were deducted is $7,650\$7,650.
The correct answer is $7,650\$7,650. Solving the system x+y=120,000x + y = 120,000 and 0.06x+0.10y=9,2000.06x + 0.10y = 9,200 gives $70,000\$70,000 invested in Account X and $50,000\$50,000 in Account Y. Account X produces $4,200\$4,200 in interest, from which a 25%25\% fee ($1,050\$1,050) is deducted. Account Y produces $5,000\$5,000 in interest, from which a 10%10\% fee ($500\$500) is deducted. Subtracting total fees of $1,550\$1,550 from total interest of $9,200\$9,200 yields $7,650\$7,650.

Step-by-Step Solution

1
Set up a system of linear equations for the principal amounts invested in Account X and Account Y.
Let xx be the amount invested in Account X and yy be the amount invested in Account Y. The total principal is x+y=120,000x + y = 120,000. The total interest earned is 0.06x+0.10y=9,2000.06x + 0.10y = 9,200.
Modeling the word problem using two variables captures both total principal and total combined interest.
2
Solve the system of equations for xx and yy.
Multiplying x+y=120,000x + y = 120,000 by 0.060.06 gives 0.06x+0.06y=7,2000.06x + 0.06y = 7,200. Subtracting this equation from 0.06x+0.10y=9,2000.06x + 0.10y = 9,200 yields 0.04y=2,0000.04y = 2,000, so y=50,000y = 50,000. Substituting back gives x=70,000x = 70,000.
Determining individual account principals allows calculation of interest generated by each account.
3
Calculate the interest earned from each account and the corresponding administrative fees.
Interest from X = 0.06×70,000=$4,2000.06 \times 70,000 = \$4,200. Interest from Y = 0.10×50,000=$5,0000.10 \times 50,000 = \$5,000. Fee for X = 0.25×4,200=$1,0500.25 \times 4,200 = \$1,050. Fee for Y = 0.10×5,000=$5000.10 \times 5,000 = \$500. Total fees = 1,050+500=$1,5501,050 + 500 = \$1,550.
Each fee rate must be multiplied by the specific interest amount earned by that account.
4
Subtract the total administrative fees from the total interest earned.
Net Interest = $9,200−$1,550=$7,650\$9,200 - \$1,550 = \$7,650.
Deducting fees yields the net interest remaining.

Key Concept

System of Linear Equations for Mixture and Interest Word Problems
Estimated Time:2m 0s
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Topics

Algebraic Word Problems and Modeling

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