Problem 63 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Online retail sales stood at $$\$ 23.5$$ billion for the year 2000 . For the next 2 yr, they grew by \(33.2 \%\) and \(27.8 \%\) per year, respectively. For the next \(6 \mathrm{yr}\), online retail sales were projected to grow at \(30.5 \%, 19.9 \%\), \(24.3 \%, 14.0 \%, 17.6 \%\), and \(10.5 \%\) per year, respectively. What were the projected online sales for 2008 ?

Step-by-Step Solution

Answer

The projected online sales for 2008 are approximately \( \$ 88.70\) billion.

Step 1: Understand the compound interest formula

To calculate the final amount after successive growth rates, we use the formula: Final amount = Initial amount × (1 + growth rate₁) × (1 + growth rate₂) × ... × (1 + growth rateₙ) Here, n is the number of years, and growth rateₙ is the growth rate for the nth year. In our case, we have 8 years of growth rates.

Step 2: Convert the growth percentages to decimal

Before applying the formula, we need to convert the given growth percentages to decimal. To do this, we divide each percentage by 100: 33.2% = 0.332 27.8% = 0.278 30.5% = 0.305 19.9% = 0.199 24.3% = 0.243 14.0% = 0.140 17.6% = 0.176 10.5% = 0.105

Step 3: Apply the compound interest formula and calculate the projected sales for 2008

Now that we have the growth rates expressed as decimal values, we can use the compound interest formula to find the projected online sales for 2008: Projected online sales for 2008 = Initial amount × (1 + growth rate₁) × (1 + growth rate₂) × ... × (1 + growth rate₈) Plugging in the initial sales and growth rates: Projected online sales for 2008 = $23.5 × (1 + 0.332) × (1 + 0.278) × (1 + 0.305) × (1 + 0.199) × (1 + 0.243) × (1 + 0.140) × (1 + 0.176) × (1 + 0.105)

Step 4: Calculate the projected online sales for 2008

Performing the calculations: Projected online sales for 2008 = $23.5 × 1.332 × 1.278 × 1.305 × 1.199 × 1.243 × 1.140 × 1.176 × 1.105 Projected online sales for 2008 ≈ $88.70 billion So, the projected online sales for 2008 are approximately $88.70 billion.

Key Concepts

Compound Interest Formula

When we talk about compound interest, we refer to the process of earning interest on both the initial amount of money and the interest that has been accumulated over previous periods. This is a typical scenario for investments and loans, where money grows over time due to being periodically compounded.

The compound interest formula is a powerful tool for predicting the future value of an investment or loan. It's represented by the equation: \[Compound\ Interest\ = P \times (1 + r)^n\]where:\[P\] is the principal amount (the initial amount of money),\[r\] is the annual interest rate (expressed as a decimal), and \[n\] is the number of times interest is compounded per year multiplied by the number of years.

Understanding this formula is essential when you're trying to determine how much money you will have in the future after a certain number of periods with a consistent growth rate. It encompasses the principle of exponential growth, which indicates that as time goes on, the amount of interest accrued increases at an ever-growing rate.

Percentage to Decimal Conversion

Often, compound interest rates are expressed as percentages. To use these values in mathematical formulas, such as the compound interest formula, you need to convert these percentages into decimal form. This process is simple but essential for accurately calculating interest.

Conversion from percentage to decimal is done by dividing the percentage value by 100. For example:

• 5% becomes 0.05 (5 ÷ 100 = 0.05)
• 20% becomes 0.20 (20 ÷ 100 = 0.20)

After converting to decimals, these numbers can be easily used in the compound interest formula. This step ensures that the growth rate is correctly factored into the calculation. This is an important mathematical skill that applies beyond just compound interest, as the conversion is often necessary in various fields such as statistics, finance, and economics.

Exponential Growth

Exponential growth refers to an increase that occurs at a rate which becomes ever more rapid in relation to the growing total size or number. In finance, this concept plays a crucial role in understanding how investments grow over time through the compounding of interest.

The idea behind exponential growth is that as time progresses, the amount of new growth is proportional to the already existing quantity. This creates a snowball effect where the quantity grows faster and faster as time goes by. One of the most recognizable examples of exponential growth in the financial context is the compounding of interest, where the amount of interest each period is added to the principal balance, resulting in a higher amount of interest in the next period.

Understanding exponential growth is vital, not just in finance but in many areas of life, from population studies to biology, as it explains patterns where changes occur at rates proportional to their current value.