Problem 57 - Algebra 1

Question

$$ \text {} 8 \frac{9}{10}+1 \frac{1}{5} $$

Step-by-Step Solution

Answer

The sum of \(8 \frac{9}{10} + 1 \frac{1}{5}\) is \(10 \frac{1}{10}\).

Step 1: Convert Mixed Numbers into Improper Fractions

Transform both mixed numbers into improper fractions. The first mixed number, \(8 \frac{9}{10}\), when converted to an improper fraction will be \( \frac{8 \times 10 + 9}{10} = \frac{89}{10} \). The second one, \(1 \frac{1}{5}\), can be converted to \( \frac{1 \times 5 + 1}{5} = \frac{6}{5} \).

Step 2: Add the Improper Fractions

Add the two improper fractions together. To add two fractions, they must have the same denominator. In this case, the denominators are 10 and 5. The least common multiple of 10 and 5 is 10, and 5 can be converted to 10 by multiplying by 2. So we have \( \frac{89}{10} + \frac{6 \times 2}{10} = \frac{89 + 12}{10} = \frac{101}{10} \).

Step 3: Convert Back to Mixed Number

The improper fraction \( \frac{101}{10} \) can be converted back into a mixed number. Divide the numerator by the denominator: \(101 \div 10 = 10\) with remainder 1, so the fraction \( \frac{101}{10} \) is equivalent to the mixed number \(10 \frac{1}{10}\).

Key Concepts

Mixed Numbers

Mixed numbers are numbers that combine both whole numbers and fractions. They are a convenient way to express quantities that exceed whole numbers but also include a fractional part. The whole number and the fraction are written side by side, such as in \(8 \frac{9}{10}\), which is 8 and nine-tenths.
When it comes to adding mixed numbers, you often need to convert them to improper fractions first. This makes calculation easier by working with similar terms.

• Multiply the whole number by the denominator of the fraction.
• Add the numerator of the fraction to the result.
• Write the total over the original denominator.

For instance, to convert \(8 \frac{9}{10}\) to an improper fraction:

• Multiply 8 by 10 to get 80.
• Add 9 to get 89.
• Therefore, it's \(\frac{89}{10}\).

Improper Fractions

Improper fractions have numerators that are larger than their denominators. This simply means the fraction represents a number greater than or equal to 1. For example, \(\frac{6}{5}\) means you have a fraction that is more than "a whole" because 6 is larger than 5.
To understand how these fractions work, remember:

• They are useful for performing operations such as adding, subtracting, multiplying, or dividing fractions.
• They can often be turned back into mixed numbers after calculations for easier understanding.

In exercises involving operations with fractions, converting all mixed numbers to improper fractions is a fundamental first step. It simplifies calculations as improper fractions allow you to work with one kind of number expression. Whenever you end up with an improper fraction after performing operations, it might be helpful to convert it back to a mixed number format to interpret the final value easily.

Least Common Multiple

The least common multiple (LCM) is the smallest multiple that two or more numbers have in common. In the context of fractions, the LCM is particularly helpful when adding or subtracting fractions. You need a common denominator to make the fractions compatible for straightforward calculation.
Here's how to find it:

• List the multiples of each original denominator.
• Find the smallest number that appears on both lists.
• That's your LCM, and it becomes the new denominator for the fractions.

For example, when working with the denominators 10 and 5, the LCM is 10 because it is the first common multiple. Using the LCM makes sure you convert each fraction to an equivalent fraction with that common denominator.
This simplifies the process of adding fractions and ensures accuracy in your final result. In our exercise, you multiplied the denominator of \(\frac{6}{5}\) by 2 to match the LCM of 10, changing it to \(\frac{12}{10}\) for easy addition with \(\frac{89}{10}\).