Problem 29 - Prealgebra

Question

Use any of the rules developed in this chapter and the rule for order of operations to simplify each of the following expressions as much as possible. [Examples 6–9] $$\frac{-3(-10)}{-5}$$

Step-by-Step Solution

Answer

The expression simplifies to \(-6\).

Step 1: Simplify the Numerator

First, focus on the numerator of the fraction \(-3(-10)\). The multiplication of two negative numbers results in a positive number. Therefore, \(-3 \times -10 = 30\).

Step 2: Rewrite the Expression with a Simplified Numerator

Replace the original numerator with the simplified result from Step 1. The expression now becomes \(\frac{30}{-5}\).

Step 3: Simplify the Fraction

Now, divide 30 by -5. Divide 30 by 5 to get 6, and keep the negative sign from -5 to obtain \(-6\). Thus, \(\frac{30}{-5} = -6\).

Key Concepts

Simplifying Fractions

Fractions can be tricky for many students, but simplifying them makes calculations more manageable. Simplifying fractions involves reducing them to their simplest form, where the numerator and the denominator have no common factors other than 1. In this exercise, the fraction \( \frac{30}{-5} \) was simplified using basic division, leading to an integer.

• Identify Factors: Look at both the numerator and the denominator to see if there are any common factors. Here, 30 and 5 have the factor 5 in common.
• Apply Division: Divide both the numerator and the denominator by their greatest common factor to simplify. Dividing 30 by 5 gives 6.
• Sign Handling: Since the denominator is negative, the result of the fraction becomes negative as well, meaning \( \frac{30}{-5} = -6 \).

Simplifying fractions is essential for making complicated problems much more approachable.

Multiplying Integers

Understanding multiplication of integers is key to solving many math problems. When working with integers, it's important to pay attention to their signs, as they influence the result significantly.

• Positive and Negative Integers: Remember that multiplying two negative integers results in a positive integer. This is because the negatives cancel each other out.
• Example: In our problem, \(-3 \times -10\) equals 30. Both numbers are negative, and their multiplication yields a positive result.
• Order of Operations: Always follow the order of operations when dealing with expressions involving integers to ensure accuracy.

By consistently applying these principles, you’ll become proficient in handling integer multiplication.

Division of Integers

Division of integers is straightforward but requires careful consideration of signs. Just like with multiplication, the signs of the integers involved play a crucial role in determining the outcome.

• Positive and Negative Results: When you divide a positive integer by a negative integer, the result is negative. Similarly, if a negative integer is divided by a positive one, the resultant is also negative.
• Example: With \( \frac{30}{-5} = -6 \), you divide 30 (positive) by -5 (negative). The outcome is -6 since a positive number divided by a negative number gives a negative.
• Basic Calculation: Perform the division as usual, but consider the sign of the result. Simplify this process with practice to gain confidence in recognizing how the signs affect division results.

Mastering the division of integers provides a solid foundation for more advanced mathematical concepts.