Problem 53 - Prealgebra

Question

Work Problems 53–60 mentally, without pencil and paper or a calculator. Is \(397 \div(-401)\) closer to 1 or \(-1 ?\)

Step-by-Step Solution

Answer

The quotient \( \frac{397}{-401} \) is closer to -1.

Step 1: Understanding the Problem

The problem asks us to determine if the quotient \( \frac{397}{-401} \) is closer to 1 or -1. Here, 397 and -401 are the numerator and denominator respectively.

Step 2: Analyze the Magnitudes

To decide if \( \frac{397}{-401} \) is closer to 1 or -1, consider the magnitudes. The absolute value of both 397 and 401 is close to each other, which should result in a fraction close to \( \pm 1 \).

Step 3: Determine the Fraction's Value

Since 397 is slightly less than 401, the fraction \( \frac{397}{401} \) should be slightly less than 1. When you consider the negative sign, \( \frac{397}{-401} \) is slightly more than -1.

Step 4: Compare Closeness to Whole Numbers

\( \frac{397}{401} \) is slightly less than 1, thus \( \frac{397}{-401} \) is slightly more than -1, making it closer to -1 than to 1, because it is a negative number closer to -1.

Key Concepts

Fractions

Fractions are a way to express numbers that are not whole. They consist of a numerator and a denominator. The numerator, located on top, represents the number of equal parts we have. Meanwhile, the denominator, located at the bottom, signifies the total number of equal parts the whole is divided into.
To better understand fractions, let's consider the example of dividing a pizza. If you have a pizza cut into 8 slices and you eat 3 of them, you have eaten a fraction of the pizza—specifically, \( \frac{3}{8} \). Here, 3 is the numerator (slices eaten), and 8 is the denominator (total slices).

• Simple fractions: Whole numbers as numerator and denominator, e.g., \( \frac{3}{4} \).
• Proper fractions: Numerator is less than the denominator, e.g., \( \frac{3}{4} \).
• Improper fractions: Numerator is greater than or equal to the denominator, e.g., \( \frac{5}{4} \).
• Mixed numbers: Combine a whole number and a fraction, e.g., 1\( \frac{1}{4} \).

Understanding fractions is crucial because they allow us to express parts of a whole and compare sizes of different parts efficiently. They are key in flexible arithmetic operations, especially when performing mental math.

Negative Numbers

Negative numbers are integral to math, especially when comparing quantities below zero. They are represented with a minus sign \((-\)). Imagine a thermometer: temperatures below freezing are marked as negative. Similarly, negative numbers are the opposite of their positive counterparts and lie to the left of zero on a number line.
One key aspect of negative numbers is their role in operations:

• Adding a negative number is like subtracting the absolute value of that number. For instance, \( 5 + (-3) = 5 - 3 = 2 \).
• Subtracting a negative number is like adding. Thus, \( 5 - (-3) = 5 + 3 = 8 \).
• Multiplication: Two negatives make a positive (\((-a) \times (-b) = a \times b\)), and a positive and a negative make a negative (\(a \times (-b) = -a \times b\)).
• Division works similarly to multiplication regarding signs.

Negative numbers often indicate debt, loss, or decrease in various contexts. They expand the number system from natural (positive) numbers to integers, allowing for a fuller range of mathematical expressions in equations and inequalities.

Absolute Value

The absolute value of a number reflects its magnitude, ignoring any sign. Represented with two vertical bars like \(|-7|\), it simply measures how far a number is from zero on the number line, whether positive or negative.
Absolute values are vital in mental math as they simplify the practice of comparing numbers without the complication of negative signs.

• The absolute value of a positive number is the number itself: \(|7| = 7\).
• The absolute value of a negative number converts it to positive: \(|-7| = 7\).
• For zero, the absolute value is zero: \(|0| = 0\).

When determining closeness in calculations, especially with fractions like in our exercise, absolute values help assess magnitude. For instance, comparing \( \frac{397}{-401} \) and 1 visually illustrates that its absolute value \( \frac{397}{401} \) is close to 1, but its negative sign leads to a calculated value closer to -1. Absolute value brings a crucial perspective in math, focusing purely on size without the complexity of direction or sign.