Problem 30 - Elementary Algebra

Question

Simplify. $$ -7.32+(-9.3) 2-37.8 \div 1.8 $$

Step-by-Step Solution

Answer

The expression simplifies to \(-46.92\).

Step 1: Distribute the Negative Sign

First, distribute the negative sign on the term \(-9.3\) using parentheses properly to rewrite the expression. The given expression is:\[-7.32 + (-9.3) \times 2 - 37.8 \div 1.8\]

Step 2: Perform Multiplication

Perform the multiplication:\((-9.3) \times 2 = -18.6\).Replace this in the expression. Now, the expression is:\[-7.32 - 18.6 - 37.8 \div 1.8\]

Step 3: Perform Division

Perform the division:\(37.8 \div 1.8 = 21\).Replace this in the expression. Now, the expression is:\[-7.32 - 18.6 - 21\]

Step 4: Perform Addition/Subtraction

Combine all terms by performing subtraction in order:\[-7.32 - 18.6 = -25.92\]Then:\[-25.92 - 21 = -46.92\]Therefore, the simplified expression is:\(-46.92\).

Key Concepts

Order of Operations

The order of operations is essential for simplifying mathematical expressions correctly. When we have expressions with multiple operations, like addition, subtraction, multiplication, and division, there's a standard sequence that we follow to ensure accuracy. A helpful acronym to remember this sequence is **PEMDAS**:

• Parentheses first
• Exponents (i.e., powers and roots, etc.)
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

Follow this order carefully when you simplify any expression to avoid mistakes. In the given example expression \(-7.32 + (-9.3) \times 2 - 37.8 \div 1.8\), you first handle the multiplication and division before moving on to addition and subtraction. Properly managing the order here prevents common errors.

Multiplication of Negative Numbers

Understanding how to multiply negative numbers is important as it often appears in algebra. The general rule is simple:

• A negative number multiplied by a positive number equals a negative number.
• A negative number multiplied by another negative number equals a positive number.

In our example, you multiply \((-9.3) \times 2\), which follows the first rule. Since -9.3 is negative and 2 is positive, their product is a negative number \(-18.6\). Recognizing this rule helps avoid sign errors in your calculations.

Division with Decimals

Dividing with decimals might seem tricky, but understanding the procedure makes it manageable. Let’s imagine that we need to divide \(37.8 \div 1.8\).First, make sure your division is lined up correctly in terms of decimal places. Alternatively, you can convert both the dividend and divisor into whole numbers by multiplying them by 10 (or any multiple of 10) until they are whole numbers, and then proceed with the division. In this example:

• Multiply 37.8 and 1.8 by 10 to eliminate decimals, getting 378 and 18.
• Perform the division \(378 \div 18 = 21\).

Remember, the divisor should not be zero, as division by zero is undefined. Practicing a few examples will make you more comfortable with problems involving decimal division.