Problem 80 - Introductory Algebra for College Students

Question

The bar graph shows that in 2000 and 2001 , the U.S. government collected more in taxes than it spent, so there was a budget surplus for each of these years. By contrast, in 2002 through \(2009,\) the government spent more than it collected, resulting in budget deficits. (GRAPH CAN NOT COPY) a. In \(2006,\) the government collected 2407 dollar billion and spent 2655 dollar billion. Find \(2407+(-2655)\) and determine the deficit, in billions of dollars, for 2006 b. In \(2007,\) the government collected 2568 dollar billion and spent 2730 dollar billion. Find the deficit, in billions of dollars, for 2007 . c. Use your answers from part (a) and (b) to determine the combined deficit, in billions of dollars, for 2006 and 2007

Step-by-Step Solution

Answer

The deficit for 2006 is $248 billion, the deficit for 2007 is $162 billion, and the combined deficit for 2006 and 2007 is $410 billion.

Step 1: Calculate Deficit for 2006

Find the deficit for 2006 by subtracting the amount the government spent from the amount it collected. The formula to use is: Collected - Spent = Deficit. Substitute the given values into the formula: \(2407-2655=-248\). So the deficit for 2006 is $248 billion.

Step 2: Calculate Deficit for 2007

Repeat the process for 2007: Substitute the given values into the formula: \(2568-2730=-162\). So the deficit for 2007 is $162 billion.

Step 3: Calculate Combined Deficit for 2006 and 2007

Add the deficits for 2006 and 2007: \(-248 + -162 = -410\) So the combined deficit for 2006 and 2007 is $410 billion.

Key Concepts

Government Budget Analysis

Understanding how governments manage their finances is essential for comprehending the broader economic picture. Government budget analysis involves examining how much money a government collects versus how much it spends. A budget surplus occurs if a government collects more in taxes and other revenues than it spends. Conversely, a budget deficit happens when spending exceeds revenue.

For example, when analyzing U.S. government financial data from 2000 and 2001, we note that the government had a budget surplus. In the subsequent years through to 2009, the government faced budget deficits due to increased spending over revenue. This form of analysis serves as a baseline for economic planning and policy implementation. It's also critical for stakeholders like taxpayers, economists, and policymakers to monitor and understand the implications of these budgets on the national economy.

Subtracting Negatives

When approaching the concept of subtracting negatives, it's important to understand that two negatives make a positive. This is a fundamental principle that often confuses students. However, seeing it in action – such as in government budget calculations – can provide clarity.

In our deficit calculation for the years 2006 and 2007, subtracting a larger number (government spending) from a smaller number (government revenue) results in a negative number. Mathematically, this is denoted as \( 2407 - 2655 = -248 \) for 2006, indicating a deficit. Similarly, for 2007, the calculation yields \( 2568 - 2730 = -162\). Subtraction involving negatives is more than a mathematical operation; it’s indicative of a shortfall, which, in economic terms, represents a deficit.

Algebraic Operations

Algebraic operations, which include addition, subtraction, multiplication, and division of numbers, are foundational to a wide range of mathematical concepts. In the realm of government budget analysis, subtraction is the central operation used to calculate deficits. For a comprehensive understanding, one must view these operations not just as arithmetic but also as a representation of real-world scenarios.

When we look at the combined deficit calculation, we perform an algebraic addition of two negative numbers. The process is \( -248 + (-162) = -410 \), showing a cumulative deficit over the two years. It's imperative to grasp that each algebraic operation conveys specific information – in this case, the state of government finances over a period. Correct application and interpretation of algebraic operations are crucial in fields such as economics, engineering, and beyond.