If \(\$ 10,000\) is borrowed now at 6 percent interest, how much will remain to be paid after a \(\$ 3,000\) payment is made four years from now? a) \(\$ 7,000\) c) \(\$ 9,625\) b) \(\$ 9,400\) d) \(\$ 9,725\)

Understanding how to calculate a loan balance after a series of payments is a pivotal skill in financial mathematics. Let's take an instance where you've borrowed \(\$ 10,000\) at a 6 percent interest rate per annum. Suppose four years down the line, you decide to make a payment of \(\$ 3,000\). How will you determine what you still owe?

To compute this, you must take into account the compound interest that has accrued over the four years. Using the compound interest formula \(A = P(1 + \frac{r}{n})^{nt}\), where \(A\) is the amount owed including interest, \(P\) symbolizes the principal amount, \(r\) denotes the annual interest rate, \(n\) is the compounding frequency, and \(t\) signifies the time in years. After applying the formula and considering the payment made, you'll arrive at the new balance. It’s crucial to remember any payment made will first cover the interest accrued before reducing the principal loan amount.

In our example, an accrued amount of \(\$ 12,620\) is calculated after four years. When we subtract the payment of \(\$ 3,000\), we find that the remaining loan balance is roughly \(\$ 9,620\), allowing us to tackle the initial question with accuracy.

Financial mathematics is a cornerstone of economic and business decisions, providing the tools necessary to analyze various financial problems. This field converges upon concepts like the time value of money, where money available in the present is worth more than the same amount in the future due to its potential earning capacity.

One key concept within this domain is compound interest, which is interest calculated on the initial principal and also on the accumulated interest of previous periods. Think of it as 'interest on interest,' which can lead to a rapid increase in a loan or investment balance over time. The mathematical expression for compound interest is encapsulated by the formula \(A = P(1 + \frac{r}{n})^{nt}\), where \(A\) is the future value of the investment/loan including interest, \(P\) is the principal investment amount, \(r\) is the annual interest rate (decimal), \(n\) is the number of times that interest is compounded per year, and \(t\) is the time the money is invested or borrowed for, in years.

By mastering these principles and formulas, you can accurately predict how investments grow or how debt accumulates over time, which is crucial for making informed financial choices.

Engineering economics, also known as engineering economy, is all about making strategic economic decisions within the scope of engineering projects. It combines financial tactics with engineering processes to enhance project management.

When engineers are faced with financial decisions, they must consider the time value of money. For instance, what's the future worth of today's investment? Alternatively, how much would we need to invest today to achieve a specific financial goal in the future? Techniques from financial mathematics, including the compound interest formula, play a vital role in these analyses.

Whether evaluating the economic viability of a project, estimating costs, or analyzing an investment, the principles of compound interest and loan balance calculations are frequently used. Understanding these can help engineers anticipate costs and assess potential returns on projects, ensuring that financial resources are utilized efficiently and effectively.