Question

Assume you are paid \(\$ 140\) every two weeks, and at the end of the two weeks you have entirely spent your salary. Also, assume that you spend your salary at a constant rate. a) Construct a graph showing your pattern of expenditure for four weeks. b) What would be your average transactions balance during each two week period? Remember, you are spending all your salary at a constant rate every two weeks. c) How much money would you have on hand 2 days after payday, 7 days, 10 days, and 14 days?

Short answer

a) The graph for four weeks (28 days) shows a linear decrease in the amount of money remaining over two-week periods, with a slope of -\$10 per day. b) The average transactions balance during each two-week period is \$70. c) After 2 days, the person would have \$120 remaining; after 7 days, \$70 remaining; after 10 days, \$40 remaining; and after 14 days, \$0 remaining.

Step-by-step solution

Define the linear equation for spending rate

First, we need to establish a linear equation representing the rate of spending. Since the person spends their entire salary of $140 in two weeks (14 days), we can calculate the spending rate per day: Spending rate per day = \(\frac{\$140}{14 \text{ days}} = \$10 \text{ per day}\) The linear equation representing this constant spending rate is: y = 140 - 10x where x represents the number of days passed, and y refers to the amount of money remaining.

Create the graph for four weeks

For part a), we need to construct a graph showing the pattern of expenditure for four weeks (28 days). The graph will consist of two lines for each two-week period. To create the graph, we will first find the y-values for 0, 1, 2, ...28 days. Using the equation mentioned above (y = 140 - 10x), we can do this easily. Then, plot these points on the graph to observe the pattern where x-axis represents the days, and the y-axis represents the amount of money remaining.

Calculate average transactions balance

The average transactions balance would be the average amount of money in the account during the two-week period. To calculate this, we can use the average of the starting balance of \(140 and the ending balance of \)0. Average balance = \(\frac{\$140 + \$0}{2} = \$70\) This means that the average transactions balance during each two-week period is $70.

Determine money on hand after specific days

Finally, we need to determine how much money would be on hand 2 days, 7 days, 10 days, and 14 days after payday. We can find this information by substituting the respective values of x (days passed) into the equation: For 2 days: y = 140 - 10(2) = $120 For 7 days: y = 140 - 10(7) = $70 For 10 days: y = 140 - 10(10) = $40 For 14 days: y = 140 - 10(14) = $0 So, after 2 days, the person would have \(120 remaining, after 7 days \)70 remaining, after 10 days \(40 remaining, and after 14 days \)0 remaining.

Key concepts

Expenditure Pattern

When we talk about expenditure patterns in personal finance, we're referring to how one spends their money over a period. For instance, in the given scenario, the person has a salary of $140 every two weeks and spends it completely within that time frame. This forms a straightforward pattern. By examining this pattern, you can predict how much money will be left at any point in time within the two weeks.
This regular pattern is crucial for budgeting, as it helps in setting financial goals effectively. This consistent spending outflow forms the basis for analyzing your saving's efficiency, allowing you to identify areas where a reduction might be feasible.

Linear Equation

A linear equation in personal finance often helps translate your expenditure into a mathematical model, making it easier to visualize and forecast spending. In this scenario, the given salary is spent at a constant rate, resulting in a linear equation. The equation created here is: \( y = 140 - 10x \), where \( y \) denotes the remaining balance and \( x \) represents the number of days since payday.
This equation simplifies the task of predicting your financial position at any given day in the two-week timeframe. The linear nature means that for every single day that passes (increasing \( x \) by one), \( y \), the remaining balance, decreases by \$10, reflecting the constant daily expenditure rate.

Average Transactions Balance

The average transactions balance provides a snapshot of the mid-point of your spending habits over a specific period, offering insights into your typical financial situation. Calculating this involves averaging the balance from the beginning and the end of the period. In this example, at the start of every two weeks, the person has \\(140 and \\)0 at the end of each two-week period.
So, the average transactions balance is calculated as follows: \[ \text{Average Balance} = \frac{\\(140 + \\)0}{2} = \\(70 \]
This average of \\)70 during the two-week stretch shows that half of the time, the person has more or less this amount as their balance, which can inform decisions about when to schedule payments or make larger purchases.

Spending Rate

The spending rate is a critical aspect of personal finance as it illustrates the speed at which funds are used. In this exercise, the rate is constant, defined by the total earnings divided by the spending duration. Here, it breaks down as \( \\(140 \) over 14 days, equating to \\)10 per day.
This rate can help in predicting not just when funds will deplete, but also in adjusting the rate to align expenses with longer-term financial objectives. Understanding this rate also assists in adopting more sustainable spending habits, as it exposes potential overspending or under-saving risks, guiding towards better financial balancing strategies in the future.