Problem 40 - Precalculus: Mathematics for Calculus

Question

Cyndee wants to invest \(\$ 50,000 .\) Her financial planner advises her to invest in three types of accounts: one paying \(3 \%,\) one paying \(5 \frac{1}{2} \%,\) and one paying \(9 \%\) simple interest per year. Cyndee wants to put twice as much in the lowest-yielding, least-risky account as in the highest-yielding account. How much should she invest in each account to achieve a total annual return of \(\$ 2540 ?\)

Step-by-Step Solution

Answer

Invest $14,000 in the 9% account, $28,000 in the 3% account, and $8,000 in the 5.5% account.

Step 1: Define Variables

Let \( x \) be the amount invested in the highest-yielding account (9%). Then \( 2x \) will be the amount invested in the lowest-yielding account (3%) because it is stated that twice as much is invested in this account. Let's denote the investment in the third account (5.5%) as \( y \).

Step 2: Set Up Equations

We have two conditions to set up our equations. First, the total investment should be \( \\(50,000 \), so:\[ x + 2x + y = 50,000 \]which simplifies to:\[ 3x + y = 50,000 \]Second, the total interest earned from these investments should be \( \\)2,540 \). Conversion of percentage to decimal yields:\[ 0.09x + 0.03(2x) + 0.055y = 2,540 \]This simplifies to:\[ 0.09x + 0.06x + 0.055y = 2,540 \]or:\[ 0.15x + 0.055y = 2,540 \]

Step 3: Solve the Equations

We have the system of equations:1. \( 3x + y = 50,000 \)2. \( 0.15x + 0.055y = 2,540 \)To solve these, first express \( y \) in terms of \( x \) using the first equation:\[ y = 50,000 - 3x \]Substitute this into the second equation:\[ 0.15x + 0.055(50,000 - 3x) = 2,540 \]Now, expand and simplify:\[ 0.15x + 2,750 - 0.165x = 2,540 \]Combine like terms:\[ -0.015x + 2,750 = 2,540 \]Solving for \( x \):\[ -0.015x = 2,540 - 2,750 \]\[ -0.015x = -210 \]\[ x = \frac{-210}{-0.015} \]\[ x = 14,000 \]

Step 4: Find Investment in Each Account

Using \( x = 14,000 \):- For the 9% account: \( x = 14,000 \)- For the 3% account: \( 2x = 2 \times 14,000 = 28,000 \)- For the 5.5% account (\( y \)): Use \( y = 50,000 - 3x = 50,000 - 42,000 = 8,000 \).Thus, Cyndee should invest \( 14,000 \) in the 9% account, \( 28,000 \) in the 3% account, and \( 8,000 \) in the 5.5% account.

Key Concepts

Simple Interest

Simple interest is the most straightforward way to calculate the return on an investment.This method considers only the principal amount and a fixed interest rate over a certain period.The formula for simple interest is:\[ \text{Interest} = \text{Principal} \times \text{Rate} \times \text{Time} \]In our scenario, each account has a different rate of interest, so we calculate the interest separately for each.For example, the account with a 9% rate means that for every dollar invested, 9 cents are paid annually.This predictable return makes it simpler to plan, but it doesn't account for compounding, which could lead to larger returns.

System of Equations

In this problem, setting up a system of equations is key to finding the correct investment amounts for each account.Cyndee's situation involves multiple constraints:

• The total amount invested should sum up to \(50,000.
• The annual interest should equal \)2,540.

These constraints allow us to create equations with variables representing the amount invested in each account.With the equation \( 3x + y = 50,000 \), it is clear that the distribution of money must meet this total.The second equation expresses total interest from all accounts: \( 0.15x + 0.055y = 2,540 \).Together, these two equations provide a linear system that, when solved, provides the solution to how much goes in each account.

Financial Planning

Financial planning is crucial for making informed investment decisions. For Cyndee, a well-structured plan involves knowing how much to invest in each account to meet her goals. This requires assessing both risk and return.

• The 3% account is low-risk but offers less return.
• The 9% account, though riskier, yields higher returns.

Balancing these factors according to her risk tolerance and financial goals forms the basis of sound financial planning. Effective planning ensures that returns like $2,540 meet her expectations without exceeding her risk appetite.

Investment Strategy

Crafting an investment strategy involves determining how to allocate funds across different assets. For Cyndee, her strategy was guided by a rule: invest twice as much in the lowest-risk account compared to the highest risk. This approach manages potential losses while ensuring satisfactory returns.

• Choosing diverse accounts with different rates helps mitigate risk.
• By allocating more funds to the 3% account, she reduces total risk.

This strategy highlights the importance of aligning investments with financial objectives and risk tolerance. An effective strategy doesn't just aim for high returns; it also safeguards financial stability over time.