Problem 131 - Prealgebra and Introductory Algebra

Question

Use the following information: When \(-3.54\) is divided into a certain dividend, the result is a positive number less than \(1 .\) Determine whether each statement is true or false. The dividend is a positive number.

Step-by-Step Solution

Answer

The statement 'The dividend is a positive number' is false.

Step 1: Analyze the characteristics of division operation

When we divide a positive number by a negative number, the result or quotient is always a negative number, while division of a negative number by a negative number results in a positive number.

Step 2: Apply division rule to given problem

Since the provided divisor is \(-3.54\), a negative number, and the result after division is a positive number, it implies the dividend must be a negative number itself (based on the division rule mentioned in step 1). Moreover, the value of dividend must be less as compared to divisor in absolute terms as the resultant quotient is less than \(1\).

Step 3: Decision based on division rule

Based on these inferences, the statement 'The dividend is a positive number' is false.

Key Concepts

Negative Numbers

Negative numbers are those numbers that lie to the left of zero on a number line. They are written with a minus sign (-) in front of them, indicating that they are less than zero. In mathematical operations, negative numbers behave differently than positive numbers.
For example, when a negative number is added to another negative number, the result is a more negative number. However, when subtracted, it can lead to a less negative or even a positive result, depending on the magnitude of the numbers involved.

• Negative numbers are often used to represent losses, debts, or decreases.
• A negative number divided by a positive number results in a negative quotient.

Understanding how negative numbers interact with other numbers is crucial, especially when it comes to division.

Quotient

The quotient is the result you get when one number, known as the dividend, is divided by another number, the divisor. In simple terms, if you divide 8 by 2, the quotient is 4. However, when dealing with negative numbers, the sign of the quotient can change.
For instance, when dividing a positive number by a negative number, the quotient will be negative. Conversely, dividing a negative number by a negative number results in a positive quotient.

• The sign of the quotient depends on the signs of the dividend and the divisor.
• If both numbers are negative, the quotient is positive.
• If one number is negative and the other positive, the quotient is negative.

This understanding helps solve division problems involving negative numbers and ensure accurate calculations are made.

Division Rule

The division rule helps determine the sign of the quotient when dividing numbers. It's essential, especially when dealing with negative numbers. The primary rules to remember are:

• Positive ÷ Positive = Positive
• Negative ÷ Negative = Positive
• Positive ÷ Negative = Negative
• Negative ÷ Positive = Negative

These rules are vital when trying to predict the outcome of a division operation. For example, knowing that dividing a negative number by a negative number yields a positive result guides us in solving equations where the quotient is specified as positive. If you understand these rules, you can deduce the sign of any division result easily.

Absolute Value

The absolute value of a number refers to its distance from zero on a number line, without considering its direction. In other words, it's the magnitude of the number. For instance, the absolute value of both -5 and 5 is 5. The absolute value is always a non-negative number, regardless of whether the original number was negative or positive. This concept is especially useful in division, where understanding the size of numbers without regard to their sign can simplify calculations.

• The absolute value helps determine the numerical size of a quotient without considering its sign.
• Understanding absolute value can clarify when a quotient is less than 1, as size is considered without sign.

In problems where the quotient is less than 1, analyzing absolute values can be fundamental in understanding the problem, especially if involving negative numbers.