Problem 13 - Elementary Algebra

Question

Multiply and divide. $$ (-3)(25) \div(-5) $$

Step-by-Step Solution

Answer

The result is 15.

Step 1: Multiplication of Negative and Positive Numbers

To begin, we need to multiply -3 by 25. Recall the rule for multiplying a negative number by a positive number: the result is always negative. Calculate the product: \((-3) \times 25 = -75\).

Step 2: Division of Negative and Negative Numbers

Next, we divide -75 by -5. When dividing two negative numbers, the result is positive. Perform the division: \((-75) \div (-5) = 15\).

Key Concepts

Multiplication of Negative Numbers

When multiplying negative numbers by positive numbers, it is essential to remember a basic arithmetic rule: the result is negative. This rule helps simplify calculations and predict outcomes when dealing with different number types.
A negative number multiplied by a positive number always results in a negative product. For example,

• if you multiply \(-3 \times 25\), the result is \(-75\),
• or if you consider \(-6 \times 4\), you get \(-24\).

The process involves multiplying the absolute values as if they were positive and then attaching a negative sign to the outcome. This holds whenever you encounter a situation that involves one negative and one positive factor.

Division Rules

Division rules are pivotal for understanding how numbers interact when you divide them. A core rule in arithmetic is that dividing two numbers with the same sign (both positive or both negative) yields a positive quotient. Conversely, dividing numbers with different signs gives a negative quotient.
With this in mind:

• If dividing a negative number by another negative number, like \((-75) \div (-5)\), the result is positive, leading to \(15\).
• When dividing a positive number by a negative number, the result is negative. For example, \(50 \div (-5) = -10\).

Always ensure the division maintains the sign rules for accurate arithmetic operations.

Negative and Positive Number Operations

Operations involving both negative and positive numbers require a solid grasp of arithmetic rules. These rules ensure correct calculations:

• When dealing with addition, know that adding a negative number is essentially subtracting its positive counterpart.


• Meanwhile, subtracting a negative number is equivalent to adding its positive form; think of it as negating a negative, resulting in a positive addition.


• Multiplication and division require careful consideration of sign rules, as explained earlier, ensuring products and quotients have the correct signs.

Remember these general guidelines:

• The outcome is positive when the numbers share the same sign (both positive or both negative).
• The outcome is negative when the numbers have differing signs (one positive and one negative).

Following these rules will help maintain accuracy and consistency in calculations involving both negative and positive numbers.