Problem 40 - 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] $$\frac{9(-8)+5(-1)}{12-1}$$

Step-by-Step Solution

Answer

The simplified expression is -7.

Step 1: Simplify the Numerator

First, we focus on the numerator of the fraction: \( 9(-8) + 5(-1) \). Use the distributive property to calculate the individual terms: \( 9(-8) = -72 \) and \( 5(-1) = -5 \). Add these results together to get: \( -72 + (-5) = -77 \).

Step 2: Simplify the Denominator

Next, simplify the expression in the denominator: \( 12 - 1 \). Subtract 1 from 12 to get \( 11 \).

Step 3: Divide the Simplified Numerator by the Simplified Denominator

Now take the result from the simplified numerator, \(-77\), and divide it by the result from the simplified denominator, \(11\). Compute \(-77 \div 11 = -7\).

Key Concepts

Distributive Property

In mathematics, the distributive property is a fundamental rule that lets you simplify expressions by distributing a single term across terms that are within parentheses. This property is especially useful when dealing with expressions that involve multiplication and addition or subtraction.

In the exercise you're working on, the distributive property is crucial. When you see an expression like \( 9(-8) + 5(-1) \), you're working with a combination of multiplication and addition. Here's how the distributive property plays in:

• First, take each term outside the parentheses and multiply it by every term inside, which in this case means multiplying 9 by -8 and 5 by -1.
• This results in two separate products: \(-72\) and \(-5\).
• Add these values together to simplify the expression: \(-72 + (-5) = -77\).

This simplification is a direct application of the distributive property because we are distributing the integer multipliers across each term inside the parentheses to simplify calculations.

Numerator and Denominator

When you're working with fractions, two key terms to understand are numerator and denominator. These components are the essence of any fraction, where the numerator is the top number and the denominator is the bottom number.

In the given problem, the fraction is presented as \( \frac{9(-8) + 5(-1)}{12 - 1} \). Here's a brief explanation on each part:

• **Numerator:** The numerator is \( 9(-8) + 5(-1) \). This part of the fraction represents the total value that you'll either add or subtract above the division line.
• **Denominator:** The denominator is \( 12 - 1 \). This part determines how many parts the numerator is being divided into.

To fully simplify the fraction, you need to work on simplifying both the numerator and the denominator separately.
For example, after simplifying each part (as detailed in our solution), you resolve the complete fraction, leading to the simplified result \(-7\) as calculated when \(-77\) is divided by \(11\). This approach highlights the critical aspects of handling numerators and denominators effectively.

Simplifying Expressions

Simplifying expressions is a core component of algebra that involves breaking down complicated mathematical phrases into simpler, more manageable parts. The aim is to make your calculations straightforward and less error-prone.

When dealing with expressions like \( \frac{9(-8) + 5(-1)}{12 - 1} \), you must use the order of operations to simplify effectively. Here's a simple guide:

• Always start by simplifying expressions inside parentheses or brackets first. In our example, the numerator \( 9(-8) + 5(-1) \) requires simplifying each multiplicative term using principles like the distributive property.
• Next, tackle the arithmetic operations like addition or subtraction. For the numerator, \(-72 + (-5) = -77\) is your next simple step.
• Then, follow suit with the denominator \(12 - 1 = 11\), making it a straightforward calculation.
• Finally, perform the division. Taking the simplified numerator and dividing it by the simplified denominator results in \(-7\).

By breaking down each step, you effectively simplify the original complex expression to a simple integer, allowing for easier interpretation and understanding.