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Difficulty: Very hardSimple Interest

An investor divides her 24,000savingsintotwodifferentaccounts.Sheplacesaportionintoahigh−yieldcertificateearning924,000 savings into two different accounts. She places a portion into a high-yield certificate earning 9% annual simple interest, and the remainder into a short-term bond earning 6% annual simple interest. After 18 months, the combined simple interest earned from both accounts is exactly 2,430. How much money was originally invested at the 9% rate?

  1. $6,000Answer
  2. B
    $18,000
  3. C
    $14,400
  4. D
    $600

Answer

The amount invested at the 9% rate is $6,000.
The correct answer is derived by modeling the split investment algebraically. By assigning xx to the 9% investment, the 6% investment becomes 24,000−x24,000 - x. Applying the formula I=PrtI = Prt with t=1.5t = 1.5 years to both parts yields 0.09(1.5)x+0.06(1.5)(24,000−x)=2,4300.09(1.5)x + 0.06(1.5)(24,000 - x) = 2,430. Simplifying this to 0.135x+2,160−0.09x=2,4300.135x + 2,160 - 0.09x = 2,430 leaves 0.045x=2700.045x = 270. Dividing 270 by 0.045 confirms that $6,000 was invested at 9%.

Step-by-Step Solution

1
Convert the investment duration from months to years.
18 months = 1.5 years.
The simple interest formula I=PrtI = Prt requires time tt to be expressed in years when dealing with annual interest rates.
2
Define variables for the two investment amounts.
Let xx be the amount invested at 9%. The remaining amount invested at 6% is (24,000−x)(24,000 - x).
This allows us to represent both unknown principals in terms of a single variable, making the equation solvable.
3
Set up the total interest equation using I=PrtI = Prt.
x(0.09)(1.5)+(24,000−x)(0.06)(1.5)=2,430x(0.09)(1.5) + (24,000 - x)(0.06)(1.5) = 2,430
The sum of the simple interest earned from the 9% account and the 6% account must equal the total interest of $2,430.
4
Distribute and simplify the terms in the equation.
0.135x+(24,000−x)(0.09)=2,430⇒0.135x+2,160−0.09x=2,4300.135x + (24,000 - x)(0.09) = 2,430 \Rightarrow 0.135x + 2,160 - 0.09x = 2,430
Multiplying the rate and time constants makes it easier to isolate xx.
5
Combine like terms and solve for xx.
0.045x=270⇒x=2700.045=6,0000.045x = 270 \Rightarrow x = \frac{270}{0.045} = 6,000
Isolating xx provides the original amount invested at the 9% rate.

Key Concept

Systems of Simple Interest
Estimated Time:3m 0s
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Topics

Simple Interest

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