Question

Add the following fractions and mixed numbers. Reduce to lowest terms. \(20 \frac{1}{2}+\frac{1}{4}+\frac{5}{4}=\) ______

Short answer

The result is 22.

Step-by-step solution

Convert Mixed Number to Improper Fraction

The mixed number here is \(20 \frac{1}{2}\). To convert it to an improper fraction, we multiply the whole number 20 by 2 (the denominator) and add 1 (the numerator). That is \(20 \times 2 + 1 = 41\), so the fraction becomes \(\frac{41}{2}\).

Common Denominator Identification

The fractions we have are \(\frac{41}{2}\), \(\frac{1}{4}\), and \(\frac{5}{4}\). The common denominator for these fractions is 4.

Convert Fractions to Common Denominator

Convert \(\frac{41}{2}\) to a fraction with a denominator of 4 by multiplying the numerator and the denominator by 2. This gives us \(\frac{82}{4}\).

Add the Fractions

Now that all fractions have a common denominator, we add them: \(\frac{82}{4} + \frac{1}{4} + \frac{5}{4} = \frac{82 + 1 + 5}{4} = \frac{88}{4}\).

Reduce the Fraction to Lowest Terms

To reduce \(\frac{88}{4}\) to its lowest terms, we divide both the numerator and the denominator by their greatest common divisor, which is 4. \(\frac{88 \div 4}{4 \div 4} = \frac{22}{1} = 22\).

Key concepts

Improper Fractions

Improper fractions might sound complicated, but they are quite simple to understand. An improper fraction is a type of fraction where the numerator (the top number) is larger than the denominator (the bottom number). This means that the fraction represents a value greater than 1. For example, in the fraction \( \frac{41}{2} \), 41 is greater than 2, making it an improper fraction.

Converting mixed numbers to improper fractions is the first step when adding fractions, especially if you're dealing with a combination of mixed numbers and fractions. Take the mixed number \( 20 \frac{1}{2} \). You convert it by multiplying the whole number 20 by 2 (the denominator) and then adding the numerator 1. The formula looks like this: \( 20 \times 2 + 1 = 41 \). So, \( 20 \frac{1}{2} \) becomes \( \frac{41}{2} \).

Mixed Numbers

Mixed numbers are a combination of a whole number and a fraction, which can be very useful in representing values greater than one. They are much friendlier in terms of easy understanding when you're describing things in daily life.

For instance, instead of saying \( \frac{41}{2} \), saying \( 20 \frac{1}{2} \) might feel more intuitive, especially if you're cooking and the recipe says to add \( 20 \) of something and \( \frac{1}{2} \) of another. However, to perform operations like addition and subtraction, you'll need to convert these to improper fractions first so they can be managed with straightforward numeric operations.

Common Denominator

The common denominator is crucial when adding fractions. It means adjusting all the fractions involved so that they share the same denominator. This makes addition (or subtraction) possible without errors.

In our example, the fractions \( \frac{41}{2} \), \( \frac{1}{4} \), and \( \frac{5}{4} \) need a common denominator to combine them. The simplest way to find it is by looking at the denominators of all fractions, which are 2 and 4 here. The smallest number that is a multiple of both 2 and 4 is 4. Therefore, 4 becomes our common denominator.

For \( \frac{41}{2} \), converting it to have 4 as the denominator involves multiplying both the numerator and the denominator by 2: \( \frac{41 \times 2}{2 \times 2} = \frac{82}{4} \). Now all fractions can be easily added together.

Reducing Fractions

Reducing fractions, also known as simplifying, is the process of making a fraction as simple as it can be. This involves dividing both the numerator and the denominator by their greatest common divisor (GCD).

In the provided example, after adding the fractions, we end up with \( \frac{88}{4} \). Both 88 and 4 can be divided by 4, making it the GCD. Simplifying means performing \( \frac{88 \div 4}{4 \div 4} \), resulting in \( \frac{22}{1} \), which simplifies further to just 22.

Always reduce fractions in your final step to ensure clarity and simplicity in your results. A simple, reduced fraction is usually easier to interpret, especially when communicating your findings.