Problem 15 - Algebra for College Students

Question

Perform the following operations with real numbers. $$\frac{-112}{16}$$

Step-by-Step Solution

Answer

-7

Step 1: Identify the Numbers

First, identify the two numbers involved in the division, which are -112 (the numerator) and 16 (the denominator).

Step 2: Perform the Division

Divide the numerator by the denominator. Calculate \(-112 \div 16\).

Step 3: Calculate the Quotient

Perform the calculation: \(\frac{-112}{16} = -7\). This means the number -112 divided by 16 equals -7.

Key Concepts

Division of Real Numbers

When we talk about dividing real numbers, we're simply talking about determining how many times one number fits into another. In mathematical terms, this involves dividing a divisor into a dividend.
Real numbers include all the numbers on the number line, such as integers, fractions, and decimals.
Understanding how division affects the sign of the result is crucial.

• If both the dividend and the divisor are positive, the result is positive.
• If one is positive and the other is negative, the result is negative.
• If both are negative, the result is also positive, as the negatives cancel each other out.

In our original exercise, the operation was \(-112 \/ 16\), which involved dividing a negative number by a positive one. Since they have different signs, the outcome is negative. Thus, the result is -7.

Fractions

Fractions represent parts of a whole, and they can take various forms, such as proper fractions, improper fractions, and mixed numbers. In any fraction, there is a top number called the numerator and a bottom number called the denominator.
Fractions are used to perform operations like addition, subtraction, multiplication, and, importantly for us, division. For division, if you're given a fraction form like in our problem, \[\frac{-112}{16}\] becomes a division problem where you divide the numerator by the denominator.
To simplify a fraction means reducing it to its simplest form. Although our problem simplified directly to a whole number, remember that typically, simplifying involves finding the greatest common divisor of the numerator and the denominator.

Numerator and Denominator

A fraction, such as \[\frac{a}{b}\], has two main components:

• Numerator: This is the top number in the fraction. It represents how many parts of the whole are being considered.
• Denominator: This is the bottom number. It indicates how many equal parts the whole is divided into.

Understanding these terms is crucial for solving problems involving fractions. In the original exercise, \(-112\) served as the numerator and 16 served as the denominator. This division of -numerator by denominator gave us the final result of -7. It's essential to recognize that the denominator cannot be zero because dividing by zero is undefined in mathematics.