Problem 31 - Prealgebra

Question

Add and subtract the following mixed numbers as indicated. $$\begin{array}{r}10 \frac{1}{20} \\\\+11 \frac{4}{5} \\\\\hline\end{array}$$

Step-by-Step Solution

Answer

21 \(\frac{17}{20}\)

Step 1: Convert Mixed Numbers to Improper Fractions

First, we need to convert each mixed number into an improper fraction. The mixed number \(10 \frac{1}{20}\) means \(10 + \frac{1}{20}\). To convert it, multiply 10 by 20 and add 1 to get the numerator: \(10 \times 20 + 1 = 201\). Thus, \(10 \frac{1}{20} = \frac{201}{20}\).Similarly, convert \(11 \frac{4}{5}\) to an improper fraction. Multiply 11 by 5 and add 4: \(11 \times 5 + 4 = 59\). Thus, \(11 \frac{4}{5} = \frac{59}{5}\).

Step 2: Find a Common Denominator

Now we need to find a common denominator to add \(\frac{201}{20}\) and \(\frac{59}{5}\). The least common multiple of 20 and 5 is 20. Convert \(\frac{59}{5}\) to have a denominator of 20 by multiplying the numerator and the denominator by 4: \(\frac{59}{5} = \frac{59 \times 4}{5 \times 4} = \frac{236}{20}\).

Step 3: Add the Improper Fractions

Add \(\frac{201}{20}\) and \(\frac{236}{20}\) now that they have a common denominator: \[\frac{201}{20} + \frac{236}{20} = \frac{201 + 236}{20} = \frac{437}{20}\].

Step 4: Convert the Result to Mixed Number

The improper fraction \(\frac{437}{20}\) can be simplified to a mixed number. Divide 437 by 20 to get the whole number and the remainder: 437 divided by 20 is 21 with a remainder of 17. Thus, \(\frac{437}{20} = 21 \frac{17}{20}\).

Key Concepts

Improper Fractions

Improper fractions are fractions where the numerator is greater than or equal to the denominator. This means the fraction represents a value equal to or greater than 1. They often arise in mathematical processes like converting mixed numbers or performing operations like addition and subtraction. Using improper fractions can simplify calculations involving mixed numbers.

To convert a mixed number to an improper fraction:

• Multiply the whole number by the denominator of the fractional part.
• Add the numerator of the fractional part to the result from the first step to form the new numerator.
• The denominator remains unchanged.

For example, for the mixed number \(10 \frac{1}{20}\), multiply 10 by 20, then add 1, resulting in \(\frac{201}{20}\). Similarly, for \(11 \frac{4}{5}\), multiply 11 by 5, add 4, resulting in \(\frac{59}{5}\). These improper fractions can then be used for other calculations.

Common Denominator

When adding or subtracting fractions, finding a common denominator ensures both fractions have the same bottom part, making the operation possible. The common denominator should be a multiple of the original denominators of the fractions involved.

To find a common denominator, you can:

• Identify a number that is a multiple of both denominators.
• Adjust each fraction to have this common multiple as their denominator by multiplying the numerator and denominator by the same amount.

In our example, we changed \(\frac{59}{5}\) to have a denominator of 20 by multiplying both the numerator and denominator by 4, resulting in \(\frac{236}{20}\). This allows us to perform addition with \(\frac{201}{20}\).

Least Common Multiple

The least common multiple (LCM) is the smallest number that is a multiple of two or more numbers. In fractions, the LCM is used to find the smallest common denominator when performing operations like addition or subtraction.

To determine the LCM:

• List the multiples of each number involved.
• Identify the smallest multiple common to all lists.

For instance, to find the LCM of 20 and 5, list their multiples: multiples of 20: 20, 40, 60...; multiples of 5: 5, 10, 15, 20... The smallest common multiple is 20. Therefore, 20 is used as the common denominator for our fractions.

Addition of Fractions

Adding fractions involves combining their values into a single fraction. Before adding, ensure all fractions have the same denominator by finding a common denominator or LCM.

Here's how to add fractions once they have the same denominator:

• Add the numerators together.
• Keep the common denominator the same.

In the example, we add \(\frac{201}{20}\) and \(\frac{236}{20}\) by adding the numerators: 201 + 236 = 437, giving us \(\frac{437}{20}\). This results in an improper fraction which can be converted into a mixed number if needed. In our exercise, it converts to \(21 \frac{17}{20}\) by dividing 437 by 20, where 21 is the whole number and 17 is the remainder.