Problem 17 - Algebra 1: Concepts and Skills

Question

In Exercises 17 and 18, find and correct the error. \begin{equation} -9 \div \frac{1}{3}=-9 \cdot \frac{1}{3} \end{equation}

Step-by-Step Solution

Answer

\(-9 \div \frac{1}{3} = -9 \times 3 = -27\)

Step 1: Analyze the original equation

The original equation: -9 ÷ 1/3 = -9 × 1/3 is being given which seems to involve improper operation.

Step 2: Recognize the proper operation for division by a fraction

Understand that dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of a fraction is obtained by switching the numerator and the denominator.

Step 3: Correct the error in the equation

-9 divided by 1/3 should be written as -9 multiplied by the reciprocal of 1/3, which is 3/1, or simply 3. So, the correct operation is -9 × 3.

Step 4: Calculate the result

Multiply -9 by 3 to get -27.

Key Concepts

Reciprocal of a Fraction

When it comes to dividing by fractions, understanding the concept of the reciprocal of a fraction is crucial. Imagine flipping a fraction upside down — this is what we refer to as the 'reciprocal'. More formally, to obtain the reciprocal, you switch the numerator (the top number) with the denominator (the bottom number). For instance, the reciprocal of the fraction \(\frac{1}{3}\) is \(\frac{3}{1}\), which can be simplified to 3.

If you're dividing a number by a fraction, like \(-9 \text{ divided by } \frac{1}{3}\), you're essentially multiplying the number by this flipped fraction. Understanding this concept helps avoid common mistakes in fraction division, and it is an essential tool that is useful throughout different areas of mathematics.

Let's apply this concept to an exercise where a student initially makes an error by treating division as multiplication without changing the fraction to its reciprocal \( -9 \text{ divided by } \frac{1}{3} eq -9 \times \frac{1}{3} \). Correcting this mistake involves multiplying by the reciprocal of \(\frac{1}{3}\), leading to the correct form: \( -9 \times 3 \).

Multiplicative Inverse

The multiplicative inverse or reciprocal of a number is essentially a way to find the 'opposite' of that number in terms of multiplication - it is the number that, when multiplied with the original number, yields the product 1. In the context of fractions, the multiplicative inverse is simply the reciprocal of the fraction. For example, if you have a fraction like \(\frac{2}{5}\), its multiplicative inverse is \(\frac{5}{2}\).

This principle is crucial when dividing numbers by fractions or by other numbers, as division, can be considered multiplication by the reciprocals. This key idea allows us to simplify division problems into multiplication ones, making them easier to solve, especially when dealing with complex algebraic expressions.

Arithmetic Operations

Arithmetic operations are the foundation of all mathematics. They consist of addition, subtraction, multiplication, and division. Having a strong grasp of these operations, including how they interact with fractions, is vital. When multiplying or dividing fractions, the rules change slightly compared to whole numbers.

For multiplication of fractions, we multiply the numerators together and the denominators together. Division, as we've seen, requires the use of the multiplicative inverse: we multiply by the reciprocal of the fraction instead of 'dividing' in the traditional sense we might with whole numbers. This method helps simplify the process and reduces the likelihood of errors, which could occur if students incorrectly perform these operations with fractions.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like x or y), and the operations of arithmetic. However, when fractions are included in these expressions, they can intimidate students. By remembering that dividing by a fraction is the same as multiplying by its reciprocal, complex expressions become more manageable.

In the context of the given exercise, the error in the algebraic expression was treating division by a fraction as ordinary multiplication. To correct it, we convert the division into multiplication by the reciprocal of the fraction, leading to a simplified and correct expression. This ability to manipulate algebraic expressions by understanding and applying arithmetic operations, including the use of reciprocals, is an invaluable skill in mathematics.