Problem 130 - Beginning and Intermediate Algebra

Question

Solve each problem. In \(2000,\) the U.S. federal budget had a surplus of \(\$ 236\) billion. In \(2016,\) the federal budget had a deficit of \(\$ 616\) billion. Express the difference between these amounts as a positive number. (Data from U.S. Office of Management and Budget.)

Step-by-Step Solution

Answer

The difference is \(852\) billion dollars.

Step 1: Identify the amounts

In 2000, the U.S. federal budget had a surplus of \(236\) billion dollars. In 2016, the federal budget had a deficit of \(616\) billion dollars.

Step 2: Understand the terms

A surplus means money left over, and a deficit means money that is overspent. Thus, a surplus is positive, and a deficit is negative. We express these values as +236 billion and -616 billion.

Step 3: Set up the difference

We want to find the difference between these amounts as a positive number. The difference is calculated as: \(Difference = Surplus - (Deficit)\), which results in: \(236 - (-616)\).

Step 4: Simplify the subtraction

Subtracting a negative number is equivalent to adding the positive value of that number. Therefore: \(236 - (-616) = 236 + 616\).

Step 5: Calculate the result

Adding the two values gives: \(236 + 616 = 852\). Hence, the difference between the surplus and the deficit as a positive number is \(852\) billion dollars.

Key Concepts

Understanding Budget Surplus and Deficit

In the context of a federal budget, a surplus refers to the amount by which income exceeds expenditures. This means the government has extra money left over.
On the other hand, a deficit occurs when expenditures exceed income. Here, the government spends more money than it receives. A surplus is generally a positive value, while a deficit is negative.
When solving problems involving surplus and deficit, it's crucial to recognize these terms. They tell us whether we're dealing with extra funds (positive numbers) or overspending (negative numbers). For instance, a surplus of 236 billion dollars is represented as \(236\). A deficit of 616 billion dollars is represented as \(-616\). Knowing these concepts helps in correctly setting up and solving the equations.

Negative and Positive Numbers

In algebra, understanding positive and negative numbers is essential. Positive numbers are greater than zero, representing quantities like income or surplus. Negative numbers are less than zero, representing quantities like debt or deficit.
When working with negative and positive numbers, certain operations can change their properties. Subtracting a negative number is equivalent to adding a positive number. For example, \(-(-616) = +616\). This principle is key for solving algebraic equations involving budget surplus and deficit.
Recognizing these rules allows you to simplify expressions correctly. When asked to find the difference between surplus and deficit, remember that subtracting a deficit (negative) is like adding its positive counterpart.

Basic Arithmetic Operations

Basic arithmetic is foundational to solving algebraic problems. Operations such as addition, subtraction, multiplication, and division are used frequently.
In this exercise, we primarily deal with addition and subtraction. To find the difference between a surplus and a deficit as a positive number, you start by setting up the expression. For example, \236 - (-616)\.
Subtraction of a negative number changes into an addition: \236 - (-616) = 236 + 616\. By then performing addition \(236 + 616\), you get \852\. This result is the positive difference between the surplus and the deficit.