Deducing a Formula for Compound Interest

Compound interest is interest that is calculated not only on the initial principal but also on the accumulated interest from previous periods. The formula for compound interest is given by:

A = P(1 + r/n)^(nt)

Where:
A = the future value of the investment/loan, including interest
P = the principal investment/loan amount
r = annual interest rate (as a decimal)
n = number of times that interest is compounded per year
t = number of years the money is invested/borrowed for

To deduce this formula, we can follow the steps below:

1. Start with the formula for simple interest:
A = P(1 + rt)

2. Divide the interest rate (r) by the number of times interest is compounded per year (n):
r/n

3. Rewrite the formula for simple interest using the interest rate per compounding period:
A = P(1 + r/n)(nt)

4. Notice that (1 + r/n)^(nt) represents the compound interest factor.

Comparing Quantities

When comparing quantities, it is important to consider the ratios and percentages involved. Here are some key points to keep in mind:

1. Ratios: Ratios compare two quantities and can be in the form of a fraction or a colon. For example, a ratio of 2:3 means that the first quantity is two-thirds of the second quantity.

2. Proportions: Proportions compare two ratios and state that they are equivalent. For example, if one ratio is 2:3 and another ratio is 4:6, they are proportional because both reduce to 2:3.

3. Percentages: Percentages are ratios expressed as fractions out of 100. They are useful for comparing quantities in terms of a common base. For example, if one quantity is 20% and another quantity is 30%, the second quantity is larger.

4. Percentage change: Percentage change compares the difference between two quantities to the original quantity. It is calculated by dividing the difference by the original quantity and multiplying by 100. A positive percentage change indicates an increase, while a negative percentage change indicates a decrease.

5. Unitary method: The unitary method is a technique for solving problems involving ratios and proportions. It involves finding the value of one unit and then using that information to find the value of other units.

Conclusion

By deducing the formula for compound interest and understanding the concepts of comparing quantities, students can solve problems related to these topics effectively. Remember to use the compound interest formula to calculate future values and consider ratios, proportions, and percentages when comparing quantities. Additionally, the unitary method can be utilized to solve problems involving ratios and proportions.