Problem 118 - Beginning and Intermediate Algebra

Question

Write a numerical expression for each phrase, and simplify the expression. Three-fourths of the sum of -8 and 12

Step-by-Step Solution

Answer

The simplified expression is 3.

Step 1: Understand the phrase

Break down the phrase into parts to understand what it is asking. The phrase 'Three-fourths of the sum of -8 and 12' can be interpreted as finding the sum of -8 and 12 first and then taking three-fourths of that sum.

Step 2: Find the sum

Calculate the sum of -8 and 12. a = -8 b = 12The sum is a + b = -8 + 12 = 4

Step 3: Multiply by three-fourths

Now take three-fourths of the sum. a = 4We need \(\frac{3}{4} \times 4\) This equals \( \frac{3}{4} \times 4 = 3 \)

Step 4: Simplify the expression

The simplified expression for \( \frac{3}{4} \times ( -8 + 12 ) \) is 3.

Key Concepts

Simplifying Expressions

When working with numerical expressions, simplifying them means making the expression as easy to work with as possible. Start by identifying and performing any operations within parentheses first. This is part of the order of operations, often remembered by the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). After dealing with the parentheses, move on to any multiplication or division, and finally, addition or subtraction. The goal is to reduce the expression to a single numerical value if possible. For example, in simplifying the given phrase, we first need to determine the sum of -8 and 12 before multiplying by three-fourths.

Fractions

Fractions represent parts of a whole and are written in the form \( \frac{a}{b} \), where \( a \) is the numerator and \( b \) is the denominator. To simplify expressions involving fractions, you typically perform operations like multiplication, division, addition, or subtraction following the fraction rules. When multiplying a fraction by a whole number, as in our exercise, multiply the numerator by the whole number, keeping the denominator the same. For example, multiplying three-fourths by 4 is done by multiplying the numerator (3) by 4, giving \( \frac{3 \times 4}{4} = 3 \). This simplifies the expression since the whole number and the denominator are the same, making it easy to see that the fraction is effectively canceled out.

Addition

Addition is one of the fundamental arithmetic operations. It combines two or more numbers into a single sum. In the given problem, we first find the sum of -8 and 12. When dealing with addition, especially with negative numbers, it's useful to think in terms of adding gains and losses: -8 represents a loss of 8, while 12 represents a gain of 12. Adding these together gives 4, because we offset the loss of 8 with a part of the gain of 12. Simplifying this initial sum makes it easier to apply the fraction to the result, leading to an accurate final answer by following the remaining steps properly. As shown, learning how to handle both positive and negative integers proficiently is crucial for simplifying complex expressions.