Question

Stock Mix You invest \(\$ 5000\) in two stocks. In one year, the value of stock A increases by \(9.8 \%\) and the value of stock B increases by \(6.2 \%\). The total value of the stocks is now \(\$ 5389.20 .\) How much did you originally invest in each stock?

Short answer

Originally, you invested \$2695.37 in stock A and \$2304.63 in stock B.

Step-by-step solution

Formulate the first equation

Let's denote the original investment in stock A as 'a' and in stock B as 'b'. Therefore the total original investment can be represented as: a + b = \$5000.

Formulate the second equation

Stock A has increased by \(9.8\%\) and Stock B by \(6.2\%\), a year later. Thus, the total value of the stocks now is: \(1.098a + 1.062b = \$5389.20\).

Solve the system of equations

Now solve the system of equations formed in step 1 and 2 using substitution or elimination method. The solution to this system will provide the original investments 'a' (stock A) and 'b' (stock B).

Calculation

By substituting the equation from Step 1 \(a = \$5000 - b\) into equation from Step 2 \(1.098a + 1.062b = \$5389.20\), we get \(1.098(\$5000 - b) + 1.062b = \$5389.20\). Solving this gives the value of 'b', which equals \$2304.63. Substituting this solution back into the first equation, we get \(a = \$5000 - \$2304.63 = \$2695.37\).

Key concepts

Systems of equations

When you encounter a financial problem involving multiple unknowns, systems of equations can come to your rescue. In our exercise, we are given two unknowns: the original amounts invested in stocks A and B. We can solve this using two equations.

• The first equation sums up the original investments as \( a + b = 5000 \).
• The second equation considers the growth of each stock, shown by the equation \( 1.098a + 1.062b = 5389.20 \).

By solving these equations together using methods like substitution or elimination, we discover the individual values of 'a' and 'b'.
Systems of equations are powerful because they can solve for multiple variables at once, especially in financial mathematics where multiple investments or components contribute to a total.

Investment growth

Investments change in value over time, often due to percentage increases such as interest or stock value hikes. In our problem, stock A grows by 9.8%, and stock B increases by 6.2%.
Unlike constant changes, percentage changes impact the invested amount based on its size, meaning larger investments grow more in absolute terms. This is why knowing the exact percentage growth is crucial for modeling and understanding investment gains.
To account for this, we use the formula:
\[ \text{New Value} = \text{Original Value} \times (1 + \text{Percentage Increase}) \]
For stock A, \(1.098a\) and for stock B, \(1.062b\), show how the original investment grows, reflecting both the starting investments and their respective growth rates.

Percentage increase

The percentage increase is a way to describe growth over time in relative terms. This concept is essential in finance to evaluate the performance of investments.
In our situation, the original price of each stock changes due to respective percentage increases. These are calculated by finding what percentage of the original value has increased.

• Stock A: It's original value multiplies by 9.8%, changing the expression to \(1.098\).
• Stock B: Similarly, it grows by 6.2%, depicted by \(1.062\).

More fundamentally, understanding percentage increases helps convert basic investment values into compounded amounts, essential for predicting future investment performance.

Financial mathematics

Financial mathematics is a cornerstone of making informed investment decisions and solving related problems. This field applies mathematical methods to financial contexts.
With a grasp of financial mathematics, one can interpret and predict investment behavior, like changes in stock value. For instance, the gain in stock A and stock B due to their respective percentage increases was assessed using this branch of math.
This involves identifying the relationships between initial investments, percentage increases, and resulting total values. Through mathematical modeling, such as the use of equations in our problem, investors can forecast financial outcomes and manage risk effectively.
Financial mathematics thus turns complex financial data into comprehensible insights, aiding in strategic planning and decision-making.