Problem 78 - Algebra 1: Concepts and Skills

Question

Find the reciprocal of the mixed number. Write your answer in lowest terms. $$ 6 \frac{1}{2} $$

Step-by-Step Solution

Answer

The reciprocal of \(6 \frac{1}{2}\) is \(\frac{2}{13}\).

Step 1 : Convert Mixed Number to Improper Fraction

To convert a mixed number to an improper fraction, first multiply the integer by the denominator of the fractional part, then add the numerator. This becomes the numerator of the improper fraction, while the denominator remains the same. In this case, \(6 \frac{1}{2}\) is converted to \(\frac{13}{2}\).

Step 2 : Find the Reciprocal

The reciprocal of a fraction is found by interchanging the numerator and the denominator. The reciprocal of \(\frac{13}{2}\) is \(\frac{2}{13}\).

Step 3 : Simplify Reciprocal

Given that the reciprocal is \(\frac{2}{13}\), it already is in its simplest form, hence there is no step required to simplify this. As a rule of thumb, always simplify reciprocals where possible by factoring them to their lowest terms.

Key Concepts

Mixed Numbers

Mixed numbers are a combination of a whole number and a proper fraction. They are often used in everyday math to represent quantities greater than a whole. For example, when you have something like six and a half pizzas, that's a mixed number: 6 1/2. Working with mixed numbers involves certain procedures to simplify calculations.

• The whole number part indicates complete units.
• The fractional part indicates how much more you have beyond the whole number.

When converting a mixed number to an improper fraction, you're basically repackaging this information to make it easier to manipulate, especially in operations like division or multiplication.
To convert, multiply the whole number by the fraction's denominator, add the numerator, and place this value over the original denominator. So for the mixed number 6 1/2:

• Multiply 6 by 2 to get 12.
• Add the numerator 1 to 12, yielding 13.
• Write this as the improper fraction 13/2.

Improper Fractions

An improper fraction is where the numerator (the top number) is greater than or equal to the denominator (the bottom number). This means the fraction represents a value equal to or greater than one.
Improper fractions are particularly useful in algebra and calculus where operations are often easier when all numbers are in fractional form. Imagine you're baking and need to multiply several fraction amounts. Keeping them as improper fractions avoids awkward calculations.
For example, the improper fraction 13/2 represents the same value as the mixed number 6 1/2, but is formatted in a way that makes it easier to work with mathematically.
To find the reciprocal of an improper fraction, just switch the places of the numerator and the denominator. If we take the improper fraction 13/2, the reciprocal will be 2/13.

Simplifying Fractions

Simplifying fractions is the process of reducing them to their simplest form, where the numerator and the denominator have no common factors other than 1. This process is important because it makes fractions easier to understand and use.
Here’s a simple way to simplify:

• Identify the greatest common divisor (GCD) of the numerator and denominator.
• Divide both the numerator and the denominator by their GCD.

The simplified fraction retains the same value but is easier to read and interpret.
For the reciprocal of our fraction 13/2, which is 2/13, you would check for common factors. In this case, 2 and 13 are both prime numbers, which means 2/13 is already as simple as it gets.
Always remember that simplifying makes fractions cleaner and often easier to work with, ensuring you have the most direct form possible.