Problem 71 - Elementary and Intermediate Algebra

Question

Perform the operations. $$ \frac{0}{150} $$

Step-by-Step Solution

Answer

The result is 0.

Step 1: Identify the Numerator

In the fraction \( \frac{0}{150} \), the numerator is 0. It's important to recognize that any fraction with a numerator of 0 will result in a specific outcome.

Step 2: Understand Division by Non-Zero Denominator

The denominator of the fraction is 150, which is a non-zero number. A fraction with a non-zero denominator is defined because it does not involve division by zero, which is undefined.

Step 3: Apply the Rule of Division by Zero Numerator

When the numerator of a fraction is zero (as in this case), the value of the entire fraction is 0, irrespective of the denominator, as long as the denominator is not zero.

Key Concepts

Division by Zero

Division by zero is a concept that puzzles many students. Imagine you have cookies and you want to share them among friends. If you have zero cookies and no friends to share them with (zero people), you might think that would be a valid scenario. However, mathematically, dividing by zero leads to something we call 'undefined.' This is because you can't distribute something into nothing. When confronted with a division where the denominator (the number you divide by) is zero, the operation becomes impossible to define in mathematical terms.

• Division by zero is undefined.
• Always check the denominator.

In fractions, division by zero leads us into nonsensical territory, which is why it's a fundamental rule in mathematics to prevent such operations.

Fractions

Fractions represent parts of a whole and consist of two main parts: a numerator and a denominator. The numerator signifies how many parts we have, whereas the denominator indicates into how many parts the whole is divided.Consider the fraction \( \frac{3}{4} \).

• The "3" is the numerator, representing three parts.
• The "4" is the denominator, showing the whole is divided into four parts.

Fractions can express values less than one, such as \( \frac{1}{2} \), equal to one, like \( \frac{4}{4} \), or more than one, for instance, \( \frac{5}{4} \).
They are incredibly useful in various aspects of math and everyday life, helping us understand division, ratios, and proportions.

Numerator and Denominator

In any fraction, the numerator and denominator play crucial roles. Let's explore these terms further.

• **Numerator**: The number on top, which indicates how many parts we are considering.
• **Denominator**: The bottom number, signifying the total number of equal parts in a whole.

In the fraction \( \frac{0}{150} \), the numerator is 0. This means we have zero parts out of the 150 possible ones. Even though the denominator is a large number, having a zero numerator implies the entire value of the fraction is 0.Always remember that:

• A fraction with a zero numerator equals 0, as long as the denominator is not zero.
• The roles of numerator and denominator are vital for evaluating not just basic fractions, but also more complex mathematical expressions.