Problem 78 - Prealgebra

Question

Work mentally, without pencil and paper or a calculator. The sum of -77 and 22 is closest to which of the following numbers? a. -100 b. -60 c. 60 d. 100

Step-by-Step Solution

Answer

The sum of -77 and 22 is closest to -60.

Step 1: Understanding the Elements

We need to add two numbers: -77 and 22. Recognize that one number is negative and the other is positive.

Step 2: Perform the Addition

To add -77 and 22, start with -77 on the number line. Moving 22 units towards the positive side (since 22 is positive), we reach -55. Hence, the sum is -55.

Step 3: Compare to Given Options

Compare -55 with the options provided: a. -100, b. -60, c. 60, d. 100. The number closest to -55 is -60.

Key Concepts

Understanding the Number Line

The number line is a visual representation that helps us understand numbers and their relationships. It resembles a straight line, where numbers are placed at equal intervals. Zero sits right in the middle, separating positive numbers to the right and negative numbers to the left. This simple tool makes it easier to add or subtract numbers, especially when dealing with negative values.

When we think about the number line:

• Moving right indicates adding or positive numbers.
• Moving left suggests subtracting or negative numbers.

For example, if you start at -77 and move 22 units to the right, as addition of positive numbers suggests, you will land at -55. This concept is crucial for visualizing addition and subtraction tasks without needing paper or a calculator.

Adding Negative Numbers Made Simple

Addition with negative numbers can seem tricky at first. But once you understand the basic idea, it becomes quite simple. Adding a negative number is similar to subtracting a positive number. It means you're moving left on the number line, going further into the negatives.

Consider adding -77 and 22. You start at -77. Instead of moving further left by 77, you move right by 22. This positive move on the number line decreases the negative amount, returning you closer to zero. Again, it's helpful to think of this like taking away part of what you owe, effectively bringing you to -55. This mental math skill is essential in many real-life situations.

Effective Problem Solving Strategies

Problem solving in mathematics often includes estimating and comparing results. When faced with options, like in this exercise, the goal is to find which choice is closest or most accurate. This requires evaluating each result and selecting the most reasonable answer.

In our problem, we calculated that -77 plus 22 equals -55. To select the closest choice from the options -100, -60, 60, 100, you consider how each compares to -55. Focus on numerical proximity. Here, -60 is only 5 units away from -55, making it the nearest choice.

Key tips for solving similar problems include:

• Estimating results before calculating exactly.
• Visualizing numbers on a number line to check proximity.
• Practicing mental math regularly to enhance quick thinking.

These strategies build confidence and accuracy in tackling math problems without external aids.