Problem 100 - Algebra 1

Question

RECIPROCALS Find the reciprocal. $$ 4 \frac{1}{2} $$

Step-by-Step Solution

Answer

The reciprocal of 4 1/2 is \( \frac{2}{9} \).

Step 1: Convert the Mixed Number to an Improper Fraction

To convert a mixed number to an improper fraction, the formula used is: \( \frac{(whole \ number \times denominator) + numerator}{denominator}\). Here, the whole number is 4, numerator is 1 and denominator is 2. So, we will apply the formula: \( \frac{(4 \times 2) + 1}{2} = \frac{9}{2}\). So the improper fraction is \( \frac{9}{2}\) .

Step 2: Find the Reciprocal

Reciprocal of a number is obtained by interchanging its denominator and numerator. So, reciprocal of \( \frac{9}{2}\) is \( \frac{2}{9}\).

Key Concepts

Mixed Numbers

Mixed numbers are a combination of a whole number and a fraction. They are often used to represent quantities greater than one, but not yet a full next whole number, such as "two and a half" or "three and three-quarters."
For example, in the mixed number:

• 4 is the whole number
• \( \frac{1}{2} \) is the fraction

When you have a mixed number, it's sometimes easier to work with it by turning it into an improper fraction, especially when solving mathematical problems. This is because it allows all parts to be easily compared or used in arithmetic operations without dealing with both whole numbers and fractions separately.

Improper Fractions

Improper fractions are a type of fraction where the numerator (the top part) is larger than the denominator (the bottom part). This suggests that the fraction is more than a whole. For example, \( \frac{9}{2} \) is an improper fraction since 9 is greater than 2.
Converting a mixed number to an improper fraction involves a simple calculation:

• Multiply the whole number by the denominator.
• Add the numerator to this product.
• Place the result over the original denominator.

So, for the mixed number 4 \( \frac{1}{2} \), the improper fraction becomes \( \frac{9}{2} \). This transformation is crucial when you need to perform operations such as finding reciprocals or adding fractions.

Fractions

Fractions represent parts of a whole and are expressed as one number placed over a line and another number below it, known as the numerator and denominator, respectively.
The basic types of fractions include:

• Proper Fractions: Where the numerator is less than the denominator, such as \( \frac{1}{4} \).
• Improper Fractions: Already explained above, like \( \frac{9}{2} \).
• Mixed Numbers: A whole number combined with a fraction, discussed earlier.

Understanding fractions is key because they often turn up in various forms in daily life like in recipes or measuring distances. When working with fractions, especially when trying to find reciprocals, converting all numbers into a similar form such as improper fractions can simplify the process. This is because taking the reciprocal just means flipping the numerator and denominator.