Question

Patricia Johnson is 20 years old and plans to make the following investments beginning next year. She will invest $3,200 in each of the next three years and will then make investments of $3,700,$3,800,$3,900, and $4,100 over the following four years. If the investments are expected to earn 7.60 percent annually, how much will Patricia have when she turns 30 ?

Answer 1

To determine how much Patricia Johnson will have by age 30, we calculate the future value of each investment separately using the compound interest formula, taking into account the different amounts of time each has to grow, given a 7.60% annual interest rate. We then sum up these values to find the total future value of her investments.

Step-by-step explanation:

The question involves calculating the future value of a series of investments over multiple years using compound interest. To solve this, we need to calculate the future value of each annual investment separately because they each will have a different amount of time to compound before Patricia turns 30.

Here's how we calculate it step-by-step:

1. For each investment amount, we apply the future value formula: FV = P(1 + r)^n, where P is the principal investment amount, r is the annual interest rate, and n is the number of years the money is invested.
2. We start by calculating for the first investment when Patricia is 21, and this will compound for 9 years. Then the second investment for 8 years, and so on until the last investment at age 27 which will compound for 3 years.
3. After calculating the future value of each investment, we sum them all to get the total amount Patricia will have by age 30.

Using the given interest rate of 7.60% annually, we substitute in the formula for each investment:

• Year 1 investment ($3,200) for 9 years: FV = 3200(1 + 0.076)^9
• Year 2 investment ($3,200) for 8 years: FV = 3200(1 + 0.076)^8
• Year 3 investment ($3,200) for 7 years: FV = 3200(1 + 0.076)^7
• Year 4 investment ($3,700) for 6 years: FV = 3700(1 + 0.076)^6
• Year 5 investment ($3,800) for 5 years: FV = 3800(1 + 0.076)^5
• Year 6 investment ($3,900) for 4 years: FV = 3900(1 + 0.076)^4
• Year 7 investment ($4,100) for 3 years: FV = 4100(1 + 0.076)^3

The sum of these calculations will give us the total amount available when Patricia is 30. To find the exact number, each of these future values should be calculated using a calculator or financial software, and then added together.