Problem 38 - Elementary Algebra

Question

Simplify. $$ 120 \div(-5)(-3)(-2) $$

Step-by-Step Solution

Answer

The simplified result is -4.

Step 1: Understand the Expression

We have to simplify the expression \(120 \div (-5)(-3)(-2)\). This means we need to find the value of this expression by performing the division and multiplication operations in the correct order.

Step 2: Evaluate the Multiplication in the Denominator

First, calculate the product of the numbers in the denominator: \((-5)\times(-3)\times(-2)\).\ - Multiplying the first two numbers: \((-5)\times(-3) = 15\) because the product of two negative numbers is positive. - Now multiply this result by \(-2\): \(15\times(-2) = -30\), as multiplying a positive number by a negative number results in a negative number.

Step 3: Perform the Division

Now that we've simplified the multiplication in the denominator to \(-30\), we can rewrite the original expression as \[120 \div (-30)\].Now, divide \(120\) by \(-30\): - Since division of a positive number by a negative number results in a negative number, we have:\[ \frac{120}{-30} = -4 \]

Key Concepts

Multiplication of Integers

Multiplication is a fundamental arithmetic operation that helps in determining the total number when combining equal-sized groups. In the context of integers, it's important to remember the rules for handling positive and negative numbers:

• Multiplying two positive integers results in a positive product.
• Multiplying two negative integers also results in a positive product.
• Multiplying a positive integer with a negative integer results in a negative product.

In the exercise given, \[(-5) \times (-3) = 15\] illustrates the multiplication of two negative numbers, leading to a positive result. This is because the signs are the same, and when the factors have the same sign, the product is positive.
Next, taking this positive result,\[ 15 \times (-2) = -30\] shows multiplying with a negative factor flips the sign of the product to negative. Understanding these rules helps simplify expressions by determining the correct signs of the final products.

Division of Integers

Division of integers is similar to multiplication in terms of considering the signs of the numbers involved. Here are the rules:

• Dividing two positive integers results in a positive quotient.
• Dividing two negative integers also results in a positive quotient.
• Dividing a positive integer by a negative integer (or vice versa) results in a negative quotient.

In our exercise, the final simplification required dividing:\[ 120 \div (-30)\] Since 120 is positive and 30 is negative, the result is:\[ \frac{120}{-30} = -4\]
Negative results occur because the signs of the dividend and divisor are opposite. Mastering division requires being comfortable with these sign rules to determine the correct sign in the quotient.

Order of Operations

The Order of Operations is a crucial concept in mathematics, dictating the sequence in which calculations are performed. This is often remembered through the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).

In the problem,\[120 \div (-5)(-3)(-2)\],the order is significant. First, address the operations within the parentheses, \[(-5) \times (-3) \times (-2)\], as multiplication must be completed before moving on to division. This aligns with the PEMDAS rule where multiplication and division are handled left to right.
After simplifying the denominator with multiplication, \(-30\) becomes the divisor for 120, at which point the division step proceeds. By following these rules, operations can be correctly executed for the desired result.