Problem 119 - Prealgebra and Introductory Algebra

Question

Business Some businesses show a profit as a positive number and a loss as a negative number. During the first quarter of this year, the loss experienced by a company was recorded as \(-12,575 .\) During the second quarter of this year, the loss experienced by the company was \(-11,350 .\) During which quarter was the loss greater?

Step-by-Step Solution

Answer

The loss was greater in the first quarter.

Step 1: Identify Given Losses

In the first quarter, the loss recorded by the company is represented by the negative number \(-12,575\). In the second quarter, the loss is recorded as \(-11,350\).

Step 2: Understand Negative Numbers

In the context of losses, the larger the number (farther away from zero), the greater the loss. Therefore, a number such as \(-12,575\) indicates a greater loss than a number like \(-11,350\) because it is farther from zero on the number line.

Step 3: Compare Given Losses

Comparing \(-12,575\) and \(-11,350\), we can see that \(-12,575\) is farther from zero than \(-11,350\). Therefore, the loss in the first quarter was greater than the loss in the second quarter.

Key Concepts

Negative Number Concepts

When we delve into negative number concepts, it's essential to understand that negative numbers represent values less than zero. They often signify a deficit or a reduction and are crucial in various fields, notably accounting and physics.

Imagine negative numbers as steps below the ground level, where zero is the ground. Each step down represents a decrease and thus, negative numbers get larger in their absolute value as they get further away from zero. This can sometimes be counterintuitive because we must remember that in the realm of negative numbers, a 'bigger' number (with a greater absolute value) actually reflects a smaller value when comparing two negative amounts. For example, (-50) is less than (-30), even though 50 is a bigger number than 30, because it's lower on the number line.

Interpreting Loss and Profit

Understanding loss and profit in business is similar to understanding debits and credits in accounting. A profit, often shown as a positive number, indicates an increase in value or an advantage gained, like a successful outcome from business activities. In contrast, a loss, represented by a negative number, denotes a decrease in value or a shortfall.

When comparing profit and loss, it's key to note that a smaller negative number (closer to zero) can actually mean 'less loss', while a larger negative number indicates a 'greater loss', or in other words, moving further away from profit. For instance, a loss of (-100) is preferable to a loss of (-200), as the former is closer to zero or 'break-even' point, signifying a smaller setback.

Number Line Understanding

The number line is a visual aid that helps in understanding the position and relationship between numbers. It's a horizontal line with zero marked at the center, positive numbers to the right, and negative numbers to the left. The number line is a powerful tool for comparing negative numbers because it allows us to see which numbers are 'less negative' and hence closer to positive territory.

On this continuum, the value of numbers increases as we move to the right and decreases as we move to the left. Negative numbers can be compared by looking at their distance from zero. The further left a number is, the greater its absolute value, and hence the greater it is in terms of being 'negatively large'. It is this intuitive graphical representation that aids in quickly comparing values like the quarterly losses in the example, where visually, (-12,575) is farther left than (-11,350), indicating a greater loss in the first quarter.