Problem 35 - Algebra 1: Concepts and Skills

Question

Find the quotient. $$ \frac{\frac{1}{3}}{-3} $$

Step-by-Step Solution

Answer

The quotient is \( \frac{-1}{9} \)

Step 1: Identify the Dividend and the Divisor

First identify the dividend (the number being divided) and the divisor (the number you are dividing by). Here, the dividend is \( \frac{1}{3} \) and the divisor is -3

Step 2: Convert the Divisor to a Fraction

To correctly perform the operation, both numbers need to be in fraction form. Convert the integer to a fraction by putting it over 1. In this case, -3 becomes \( \frac{-3}{1} \)

Step 3: Multiply by the Reciprocal of the divisor

After converting the divisor, flip the divisor fraction (this is called taking the reciprocal). Now, instead of dividing, multiply the dividend fraction by the reciprocal of the divisor fraction. This gives us: \( \frac{1}{3} * \frac{-1}{3} \)

Step 4: Multiplication of Fractions

When multiplying fractions, multiply the numerators together and the denominators together. This gives us \( \frac{-1}{9} \)

Key Concepts

Understanding Fractions

Fractions represent a part of a whole, and understanding them is crucial for solving problems that involve division or multiplication. A fraction consists of two parts:

• The numerator: This is the top number, indicating how many parts we have.
• The denominator: This is the bottom number, showing into how many equal parts the whole is divided.

For example, in the fraction \( \frac{1}{3} \), 1 is the numerator and 3 is the denominator, signifying one part out of three equal parts.

Fractions can be represented in different ways, such as improper fractions where the numerator is larger than the denominator, or mixed numbers, which combine a whole number with a fraction. To work with fractions, especially in division, it's often helpful to understand how they relate to decimals and percentages.

By mastering fractions, students can easily perform operations like addition, subtraction, multiplication, and division, leading to better proficiency in solving mathematical problems.

Exploring the Concept of Reciprocal

The reciprocal of a number is its multiplicative inverse. It means that when you multiply a number by its reciprocal, the result is 1. Finding a reciprocal is essential when dividing fractions.

Here's how to find the reciprocal of a fraction:

• Switch the numerator and the denominator.
• If the fraction is \( \frac{a}{b} \), its reciprocal becomes \( \frac{b}{a} \).

For whole numbers, the reciprocal is simply 1 divided by that number. For instance, the reciprocal of 3 is \( \frac{1}{3} \). In calculations involving division of fractions, we convert the division into multiplication by using the reciprocal of the divisor.

This is what we do in any division problem involving fractions - instead of dividing, we multiply by the reciprocal of the divisor, transforming a seemingly complex operation into a straightforward multiplication task. Understanding reciprocals makes the process of solving such problems much more manageable.

The Multiplication of Fractions

Multiplying fractions may seem challenging at first, but with a clear procedure, it becomes much simpler. When multiplying fractions, follow these steps:

• Multiply the numerators of the fractions to get the numerator of the product.
• Multiply the denominators of the fractions to get the denominator of the product.

For example, if you have to multiply \( \frac{1}{3} \) by \( \frac{-1}{3} \), you will calculate:

• Numerator: \(1 \times (-1) = -1\)
• Denominator: \(3 \times 3 = 9\)

This results in a product of \( \frac{-1}{9} \).

It's important to simplify the fraction to its lowest terms, if possible, to ease understanding and make further calculations easier. Multiplying fractions is a foundational skill that applies to more complex mathematical operations, reinforcing understanding of how parts interact in numerical expressions.