Problem 45 - Prealgebra and Introductory Algebra

Question

What is \(-\frac{7}{12}\) more than \(-\frac{5}{9} ?\)

Step-by-Step Solution

Answer

-\frac{41}{36}

Step 1: Understand the Problem

Again, this is a simple arithmetical problem. We are asked to find the number that is \(-\frac{7}{12}\) more than \(-\frac{5}{9}\). In other words, sum these fractions. Since both numbers are negative, they can be directly added.

Step 2: Find a common denominator

When adding fractions, we need a common denominator. The denominators here are 12 and 9. The smallest common multiplier (least common denominator) of both, is 36. We then adjust both fractions to get the equivalent fractions with the denominator of 36: \(-\frac{7}{12} = -\frac{21}{36}\) and \(-\frac{5}{9} = -\frac{20}{36}\).

Step 3: Add the Fractions

Now that we have common denominators, we add the fractions together. The result is \(-\frac{21}{36} - \frac{20}{36} = -\frac{41}{36}\).

Key Concepts

Common Denominator

When dealing with an arithmetical problem that requires adding or subtracting fractions, finding a common denominator is essential to simplifying the process.

Finding a common denominator refers to finding a common base for the fractions to unify them, which allows for direct addition or subtraction of the numerators while keeping the denominator the same. In the case of \( -\frac{7}{12} \) and \( -\frac{5}{9} \), the least common denominator is 36 because it is the smallest number that both 12 and 9 can divide into without leaving a remainder.

With the right common denominator, we can combine fractions with different denominators in a coherent and simplified manner. It is a critical step that forms the basis for solving fraction problems across various mathematical and real-world applications.

Arithmetical Problem

Arithmetical problems involve the basic operations of mathematics such as addition, subtraction, multiplication, and division. The problem at hand requires adding two negative fractions, which can initially seem daunting but follows the same principles as adding positive fractions.

To tackle this problem effectively, understanding that adding negative numbers is similar to subtracting their positive counterparts is crucial. For \( -\frac{7}{12} \) and \( -\frac{5}{9} \), we look at the problem as taking two amounts away from zero, which means the values are combined into a larger negative amount.

Arithmetic problems like these are fundamental to building a strong mathematical foundation. They help in developing logical thinking and problem-solving skills that are applicable in varied settings, from balancing a budget to calculating rates and measurements.

Equivalent Fractions

Equivalent fractions are different fractions that represent the same part of a whole. To add or subtract fractions with different denominators, we must first convert them into equivalent fractions with a common denominator.

In the process of finding equivalent fractions for \( -\frac{7}{12} \) and \( -\frac{5}{9} \), we multiply the numerators and denominators by the number necessary to achieve the same denominator. For instance, \( -\frac{7}{12} \) is multiplied by 3 to get \( -\frac{21}{36} \) and \( -\frac{5}{9} \) is multiplied by 4 to arrive at \( -\frac{20}{36} \).

Understanding equivalent fractions is critical because it simplifies complex problems and enables easier calculation and comparison of fractions. Mastery of this concept is not only significant in academic pursuits but is also practical in everyday scenarios that involve ratios, proportions, and percentages.