Step-by-Step Guide to Adding Fractions and Mixed Numbers
In this article, we will explore the methods of adding fractions and mixed numbers. Whether you are dealing with simple fractions, repeating decimals, or mixed numbers, this guide will provide you with clear, concise steps to perform these calculations accurately.
1. Converting Repeating Decimals to Fractions
Repeating decimals can be converted into fractions by following specific rules. If you have a repeating decimal with a repeating part, you can convert it to a fraction as follows:
Example 1: Single Repeating Decimal
Consider the repeating decimal 0.142857. To convert it to a fraction:
Ignore the decimal point and treat the repeating part as a whole number. Divide the number by as many 9s as the repeating part has digits.Thus, 0.142857 (frac{142857}{999999}).
Example 2: Mixed Repeating Decimal
For a mixed repeating decimal like 0.781, follow these steps:
Subtract the non-repeating part from the whole number including the repeating part. Divide the result by the number of 9s corresponding to the repeating part followed by as many 0s as the non-repeating part has digits.Thus, 0.781 (frac{781-78}{900} frac{703}{900}).
Example 3: More Simplification
Sometimes, you may need to simplify further:
0.127 (frac{127-1}{990} frac{126}{990} frac{63}{495} frac{21}{165} frac{7}{55}).
2. Adding Two Repeating Decimals
Another useful method is to convert the repeating decimals to fractions and then add the fractions:
For instance, consider 0.3428571429 and 0.6:
Convert each repeating decimal to a fraction: 0.3428571429 (frac{7}{21}) and 0.6 (frac{6}{10} frac{3}{5}). Find a common denominator and add the fractions: (frac{7}{21} frac{3}{5} frac{35}{105} frac{63}{105} frac{98}{105} frac{14}{15}).3. Adding Fractions with Different Denominators
When adding fractions with different denominators, the key is to find a common denominator and then add them:
Example:
Consider (frac{1}{5} frac{1}{7}) and a common denominator of 35:
Convert each fraction: (frac{1}{5} frac{7}{35}) and (frac{1}{7} frac{5}{35}). Add the fractions: (frac{7}{35} frac{5}{35} frac{12}{35}).4. Adding Mixed Numbers
Mixed numbers can be converted to improper fractions for easier addition. Here’s how:
Example 1:
Consider 5 2/3 and 3/4:
Isolate the whole numbers: 5 and (frac{3}{4}). Convert 2/3 to a common denominator of 12: (frac{8}{12}). Add the fractions: (frac{8}{12} frac{3}{4} frac{8}{12} frac{9}{12} frac{17}{12}). Add the whole numbers: 5 1 (from (frac{17}{12})) 6. Final answer: 6 5/12.5. Converting Mixed Numbers to Improper Fractions
To add mixed numbers, convert them to improper fractions first:
Example 1:
Consider 1 1/3 and 2 3/4:
Convert to improper fractions: 1 1/3 (frac{4}{3}) and 2 3/4 (frac{11}{4}). Find a common denominator (12): (frac{16}{12} frac{33}{12}). Add the fractions: (frac{16}{12} frac{33}{12} frac{49}{12} 4 frac{1}{12}).Conclusion
Incorporating these methods, you can handle a variety of fraction and mixed number problems efficiently. Whether you are isolating whole numbers, converting to common denominators, or using shortcuts, these techniques will ensure accurate and quick calculations. Happy solving!