Problem 121 - Prealgebra and Introductory Algebra

Question

Evaluate the expression \(x \div y\) for the given values of \(x\) and \(y.\) $$x=-2 \frac{5}{8}, y=1 \frac{3}{4}$$

Step-by-Step Solution

Answer

The evaluated expression of \(x ÷ y\) for \(x=-2 \frac{5}{8}\), \(y=1 \frac{3}{4}\) is \(-3/2\).

Step 1: Convert the mixed numbers to improper fractions

An improper fraction is a fraction where the numerator is greater than the denominator. We convert mixed numbers to improper fractions to make operations easier. To convert a mixed number to an improper fraction, one multiplies the whole part by the denominator of the fraction part, then adds the numerator. The result is the numerator of the improper fraction and the denominator remains the same. \n Thus, \(x = -2 \frac{5}{8}\) becomes \(-\frac{21}{8}\) and \(y=1 \frac{ 3}{4}\) becomes \(\frac{7}{4}\).

Step 2: Perform the division

When dividing fractions, one multiplies the first fraction by the reciprocal of the second fraction. A reciprocal of a fraction is obtained by swapping the numerator and the denominator. Thus, \(-\frac{21}{8} ÷ \frac{7}{4}\) becomes \(-\frac{21}{8} \times \frac{4}{7}\).

Step 3: Simplify the result

To simplify the resulting fraction, \(-\frac{21}{8} \times \frac{4}{7}\), first simplify the fractions by cancelling out common factors between numerators and denominators before multiplication. After the cancellation, multiply across to get the result. The common factors are 7 and 4, and after simplifying \(-21 ÷ 7 = -3\) and \(8 ÷ 4 = 2\). The result is \(-3/2\).

Key Concepts

Converting Mixed Numbers to Improper Fractions

Understanding how to convert mixed numbers to improper fractions is a vital skill in mathematics. A mixed number consists of a whole number and a fraction, while an improper fraction has a numerator that is larger than its denominator.

To convert, you multiply the whole number by the denominator of the fraction. Add this result to the numerator, and this sum becomes the new numerator of the improper fraction. The denominator remains unchanged. Let's put this into practice. Consider the mixed number \( -2 \frac{5}{8} \). Multiply the whole number 2 by the denominator 8 to get 16. Add the numerator 5 to obtain 21. Since the mixed number was negative, this gives us \( -\frac{21}{8} \).

This process aligns with our focus on fostering a clear understanding by breaking it into comprehensible steps. By mastering this conversion, students can maneuver among different fraction representations, which simplifies more complex operations such as division.

Multiplying by the Reciprocal

When dividing fractions, instead of performing the division directly, one multiplies by the reciprocal of the divisor. The reciprocal of a fraction is simply achieved by exchanging its numerator and denominator. For instance, the reciprocal of \( \frac{7}{4} \) is \( \frac{4}{7} \).

If you have to divide \( -\frac{21}{8} \) by \( \frac{7}{4} \), you would multiply \( -\frac{21}{8} \) by the reciprocal of \( \frac{7}{4} \), which is \( \frac{4}{7} \). This operation transforms division into multiplication, which is generally a more straightforward process to execute. This essential concept is a game-changer for students as it demystifies division of fractions and helps to avoid common pitfalls in this challenging area.

Simplifying Fractions

After converting mixed numbers and multiplying by the reciprocal, the next step is to simplify fractions before finalizing the multiplication. Simplification involves finding common factors in the numerator and the denominator and cancelling them out.

For example, in our solution \( -\frac{21}{8} \times \frac{4}{7} \), number 7 is a common factor of 21 and 7, and 4 is a common factor of 4 and 8. Cancel these out to get \( -3/2 \), which is the simplified result. By simplifying, we reduce fractions to their lowest terms, thus making them easier to understand or further manipulate. This step is crucial, especially when the goal is for students to grasp the foundations of fraction operations and execute them with finesse.