Problem 54 - Precalculus: Mathematical for Calculus

Question

A woman invests a total of \(\$ 20,000\) in two accounts, one paying \(5 \%\) and the other paying \(8 \%\) simple interest per year. Her annual interest is \(\$ 1180 .\) How much did she invest at each rate?

Step-by-Step Solution

Answer

$14,000 invested at 5%; $6,000 invested at 8%

Step 1: Define Variables

Let \( x \) be the amount invested at 5% interest, and \( 20,000 - x \) be the amount invested at 8% interest. These expressions represent the total amount invested, because the sum of investments in both accounts is \( \$20,000 \).

Step 2: Set Up the Interest Equation

The interest earned from the two investments must sum to \( \$1180 \). Therefore, we write the equation for total interest: \( 0.05x + 0.08(20,000 - x) = 1180 \).

Step 3: Simplify the Equation

Distribute the \(0.08\) in the second term: \( 0.05x + 0.08 \times 20,000 - 0.08x = 1180 \). This simplifies to \( 0.05x + 1600 - 0.08x = 1180 \).

Step 4: Combine Like Terms

Combine the terms involving \(x\): \( 0.05x - 0.08x = -0.03x \). The equation is now \( -0.03x + 1600 = 1180 \).

Step 5: Solve for x

Isolate \(x\) by subtracting 1600 from both sides: \( -0.03x = 1180 - 1600 \). This simplifies to \( -0.03x = -420 \). Dividing both sides by \(-0.03\) gives \( x = 14000 \).

Step 6: Calculate the Second Investment

Once we know \( x = 14000 \), find the amount invested at 8% interest by calculating \( 20,000 - x = 20,000 - 14,000 \), which equals \( 6000 \).

Key Concepts

Interest Calculation

Interest calculation in investments involves determining the amount of extra money gained from an initial sum, known as the principal, over a period of time. In this problem, simple interest is used, which means the interest is calculated only on the original principal amount.
The formula for simple interest is given by:
\[ \text{Interest} = \text{Principal} \times \text{Rate} \times \text{Time} \]
In the context of our exercise, the woman receives interest from two separate investments, one at a 5% rate and another at an 8% rate. Her total interest for one year amounts to $1,180.

• The interest from the first account is \( 0.05x \), where \( x \) is the amount invested at 5%.
• The second account generates interest as \( 0.08(20,000 - x) \), since the remaining amount \( (20,000 - x) \) is invested at 8%.

The sum of these two interest values gives us the total interest earned, setting up the equation:
\[ 0.05x + 0.08(20,000 - x) = 1180 \]

Investment Distribution

Investment distribution refers to how an individual's total investment is divided across different investment opportunities or accounts. In this exercise, the woman has a total of $20,000 to invest between two different interest-earning accounts.
She allocates part of her capital to an account with a 5% annual interest and the remainder to an account with 8% annual interest.

• We define the amount invested at 5% as \( x \).
• The remaining amount, \( 20,000 - x \), is invested in the account yielding 8% interest.

By effectively distributing her investments, she maximizes potential returns based on the different interest rates offered. The distribution strategy outlined not only helps in achieving a specific interest return but can also align with risk preferences and financial goals. It's critical for investors to consider the balance between potential earnings and financial security.

Variable Definition

Defining variables is crucial in solving financial math problems, as it provides a clear representation of the unknown quantities involved. In this case, we need to determine how the woman distributes her $20,000 investment.
To do this, we assign a variable to the unknown amount invested at 5%. Let this variable be \( x \). Then, the expression \( 20,000 - x \) naturally represents the amount placed at 8% interest. This setup forms the foundation for constructing equations that lead to a solution.
By understanding variable definition:

• We easily identify that \( x \) is the portion invested at the lower rate.
• And \( 20,000 - x \) is what remains for the higher rate investment.

These variables help us write the interest equation and guide the solution path. Correctly defining and manipulating variables simplifies our approach to deducing the investment amounts for each account.