Problem 77 - Algebra 1: Concepts and Skills

Question

Find the reciprocal. \(1 \frac{3}{4}\)

Step-by-Step Solution

Answer

The reciprocal of \(1 \frac{3}{4}\) is \(\frac{4}{7}\).

Step 1: Convert the Mixed Number into an Improper Fraction

A mixed number is composed of a whole number and a fraction. To convert it into an improper fraction, multiply the whole number by the denominator of the fraction, then add the numerator of the fraction. The result is the numerator of the new fraction, and the denominator remains the same. In this case, \(1 \frac{3}{4}\) becomes \(1*4 + 3 = 7\), so the improper fraction is \(\frac{7}{4}\).

Step 2: Find the Reciprocal

The reciprocal of a fraction is obtained by swapping the numerator and the denominator. Therefore, the reciprocal of \(\frac{7}{4}\) is \(\frac{4}{7}\).

Key Concepts

Improper Fractions

Improper fractions are a type of fraction where the numerator (top number) is greater than or equal to the denominator (bottom number). This means that the fraction is equal to or greater than 1. Understanding improper fractions is essential, as they often appear in mathematics when dealing with mixed numbers or calculations that exceed whole units.

• They can represent quantities that are larger than a whole.
• Improper fractions can be converted back into mixed numbers for easier interpretation or usage.

Converting from a mixed number to an improper fraction is a common skill. To do this, you multiply the whole number by the denominator, add the result to the original numerator, and place this total over the original denominator. For example, the mixed number \(1 \frac{3}{4}\) becomes \(\frac{7}{4}\). This conversion makes further operations, like finding reciprocals, easier to manage.

Mixed Numbers

A mixed number is a number consisting of a whole number and a fraction, such as \(1\frac{3}{4}\). They're useful in representing values that aren't whole numbers but aren't far from being whole either. Mixed numbers show both the magnitude of completeness and the leftover parts in a consistent manner.

• Help in visualizing and comprehending partial quantities.
• While handy, they aren't always practical in arithmetic operations, especially multiplication or division.

To work effectively with mixed numbers, especially in operations requiring fractions, converting them to improper fractions is beneficial. This change of form helps maintain uniformity and simplifies computation, showing the entire value as a numerator and a denominator only.

Converting Fractions

Converting fractions between different forms like mixed numbers and improper fractions is an invaluable math skill. The need to convert arises when solving problems requiring uniformity, such as finding the reciprocal.

• Mixed numbers can be converted to improper fractions by performing a simple arithmetic operation.
• Improper fractions are inverted easily to obtain reciprocals which is necessary in many calculations.

For example, when converting \(1\frac{3}{4}\) to \(\frac{7}{4}\), the focus is on understanding how the whole parts and fractional parts combine into a single standardized fraction form. This remixing makes mathematical operations straightforward and consistent, aiding in better conceptual clarity and computational efficiency.