Problem 75 - Elementary Algebra

Question

An investment of \(\$ 500\) earns \(\$ 45\) in a year. At the same rate, how much additional money must be invested to raise the earnings to \(\$ 72\) per year?

Step-by-Step Solution

Answer

You need to invest an additional \(\$300\).

Step 1: Calculate the Initial Rate of Interest

To find the rate of earning, divide the earnings per year by the initial investment. Here, the earning is \(\\(45\) and the investment is \(\\)500\). The rate \(r\) is given by:\[r = \frac{45}{500} = 0.09\,\text{or}\,9\%.\]

Step 2: Determine Total Investment for Target Earnings

Use the rate calculated in Step 1 to determine how much total investment is needed to earn \(\$72\). Set up the equation using the rate:\[72 = I \times 0.09,\]where \(I\) is the total investment required.

Step 3: Solve for Total Investment

Rearrange the equation from Step 2 to solve for \(I\):\[I = \frac{72}{0.09} = 800.\]So, the total investment needed to earn \(\\(72\) at the same rate is \(\\)800\).

Step 4: Find the Additional Investment Needed

Subtract the original investment from the total investment needed to find how much additional money must be invested:\[\text{Additional Investment} = \\(800 - \\)500 = \$300.\]

Key Concepts

Rate of Interest

When you invest money, the rate of interest tells you how much return you can expect from your investment over a period of time, usually a year. It is typically expressed as a percentage.

For instance, if you invest at a 9% interest rate, this means you earn \(9\%\) of the amount you invested as interest each year.

• To find the rate of interest, divide the earnings by the amount initially invested.
• Using the exercise example, an investment of \(\\( 500\) earns \(\\) 45\) in one year.
• The calculation for the rate becomes \(\\( 45\) divided by \(\\) 500\), equating to a 0.09 rate or 9%.

Understanding how to determine and interpret the rate of interest is crucial in ensuring you're making informed financial decisions.

Initial Investment

The initial investment is the amount of money that you put into a financial venture at the very start. This forms the base sum that the rate of interest is applied to.

In our example, the initial investment is \(\\( 500\).

• This is the amount upon which you calculate the earnings of \(\\) 45\) at a 9% rate.
• Your initial investment acts as the foundation for further investment calculations.
• Knowing your initial investment helps track your financial growth over time.

A clear understanding of your initial investment helps plan for future growth in your portfolio.

Total Investment Needed

To achieve a targeted amount of earnings, knowing the total investment needed is essential. This total is the collective sum needed to be invested at a certain rate to achieve desired earnings.

In the exercise, the goal is to earn \(\\( 72\) in interest per year.

• Using the 9% rate, we calculate: \(72 = I \times 0.09\).
• Solving gives \(I = \frac{72}{0.09} = \ 800\).
• Hence, \(\\) 800\) is required to earn \(\$ 72\).

By identifying the total investment needed, you can determine if your investment strategy aligns with your financial goals.

Additional Investment

Sometimes, your initial investment might not be enough to reach a specific earning target. In such cases, figuring out the additional investment needed is necessary.

To increase earnings from \(\\( 45\) to \(\\) 72\), it requires additional investment.

• From the previous calculation, a new total of \(\\( 800\) is needed.
• The original investment was \(\\) 500\).
• Thus, additional investment needed is \(\$ 800 - \ 500 = \ 300\).

Calculating additional investment helps you plan the extra funds you may need to meet your financial targets efficiently.