Question

Jon invested part of 16,000 at 3% and the rest at 5% for a total return of $680. 
Quantity A: The amount Jon invested at 5% interest 
Quantity B: The amount Jon invested at 3% interest 
 

• A. The two quantities are equal
• B. Quantity A is greater
• C. The relationship cannot be determined from the information given
• D. Quantity B is greater
• E.

Hint:

In mixed investment problems, setting up a system of equations is the standard approach. To avoid decimals, you can multiply the interest equation by 100: \(3x + 5y = 68,000\). This can make the arithmetic easier to handle.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This is an investment problem that can be modeled using a system of two linear equations. One equation will represent the total principal amount invested, and the second will represent the total interest earned.
Step 2: Key Formula or Approach:
Let \(x\) be the amount invested at 3% and \(y\) be the amount invested at 5%.
The interest earned is given by the formula: Interest = Principal \(\times\) Rate.
Equation 1 (Total Principal): \(x + y = 16,000\)
Equation 2 (Total Interest): \(0.03x + 0.05y = 680\)
We need to solve for x (Quantity B) and y (Quantity A) and then compare them.
Step 3: Detailed Explanation:
From Equation 1, we can express \(x\) in terms of \(y\): \[ x = 16,000 - y \] Now, substitute this expression into Equation 2: \[ 0.03(16,000 - y) + 0.05y = 680 \] Distribute the 0.03: \[ 480 - 0.03y + 0.05y = 680 \] Combine the y-terms: \[ 480 + 0.02y = 680 \] Subtract 480 from both sides: \[ 0.02y = 200 \] Solve for y: \[ y = \frac{200}{0.02} = 10,000 \] So, the amount invested at 5% is $10,000. This is Quantity A.
Now find the amount invested at 3% (x) using Equation 1: \[ x = 16,000 - y = 16,000 - 10,000 = 6,000 \] So, the amount invested at 3% is $6,000. This is Quantity B.

Comparison:
Quantity A = $10,000
Quantity B = $6,000
Quantity A is greater than Quantity B.
Step 4: Final Answer:
Quantity A is greater.