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Question

Difficulty: MediumSystems of Linear Equations

An investor divides a total of $15,000\$15,000 between two accounts, Account P and Account Q. Account P earns an annual simple interest rate of 6%6\%, while Account Q earns an annual simple interest rate of 8%8\%. If the total interest earned from both accounts combined after one year is $1,020\$1,020, how much more money was invested in Account P than in Account Q?

  1. A
    $1,500\$1,500
  2. $3,000\$3,000Answer
  3. C
    $4,500\$4,500
  4. D
    $6,000\$6,000
  5. E
    $9,000\$9,000

Answer

$3,000\$3,000
Setting up the system of linear equations p+q=15,000p + q = 15,000 and 0.06p+0.08q=1,0200.06p + 0.08q = 1,020 yields p=9,000p = 9,000 and q=6,000q = 6,000. Subtracting the amount in Account Q from Account P gives 9,000−6,000=3,0009,000 - 6,000 = 3,000. Thus, $3,000\$3,000 is the correct answer.

Step-by-Step Solution

1
Define variables and set up the system of linear equations
Let pp be the amount invested in Account P and qq be the amount invested in Account Q. The equations are p+q=15,000p + q = 15,000 and 0.06p+0.08q=1,0200.06p + 0.08q = 1,020.
The first equation represents the total investment amount, and the second equation represents the total annual interest earned.
2
Simplify the interest equation and solve for one variable using elimination
Multiply 0.06p+0.08q=1,0200.06p + 0.08q = 1,020 by 100 to obtain 6p+8q=102,0006p + 8q = 102,000, which simplifies to 3p+4q=51,0003p + 4q = 51,000. Multiplying the first equation by 3 yields 3p+3q=45,0003p + 3q = 45,000. Subtracting this from 3p+4q=51,0003p + 4q = 51,000 gives q=6,000q = 6,000.
Eliminating pp allows for direct calculation of the amount invested in Account Q.
3
Calculate the amount invested in Account P
p=15,000−6,000=9,000p = 15,000 - 6,000 = 9,000.
Substitute the value of qq back into the total investment equation.
4
Determine the requested difference
p−q=9,000−6,000=3,000p - q = 9,000 - 6,000 = 3,000.
The question specifically asks how much more money was invested in Account P than in Account Q.

Key Concept

Solving Systems of 2x2 Linear Equations for Word Problems
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Topics

Systems of Linear Equations

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