Question

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

Short answer

\(\frac{7}{10}\)

Step-by-step solution

Identify a Common Denominator

To add fractions, we need a common denominator. The denominators in this problem are 2 and 5. The least common multiple of 2 and 5 is 10, so the common denominator is 10.

Convert Fractions to Equivalent Fractions

We need both fractions to have the common denominator of 10. Convert \(\frac{1}{2}\) to an equivalent fraction: \(\frac{1}{2} = \frac{5}{10}\). Convert \(\frac{1}{5}\) to an equivalent fraction: \(\frac{1}{5} = \frac{2}{10}\).

Add the Fractions

Now that both fractions have the same denominator, add them: \(\frac{5}{10} + \frac{2}{10} = \frac{7}{10}\).

Simplify the Result

The fraction \(\frac{7}{10}\) is already in its simplest form, as 7 and 10 have no common factors other than 1.

Key concepts

Understanding Common Denominator

To successfully add fractions, it’s important to have a common denominator. A common denominator is a shared multiple of the fractions’ original denominators. It ensures you can combine them easily. For example, when you have fractions like \(\frac{1}{2}\) and \(\frac{1}{5}\), their denominators 2 and 5 require you to find a number that both divide into evenly. This number, called the least common multiple (LCM), becomes the common denominator.

• Find the LCM of the original denominators.
• In this exercise, the LCM of 2 and 5 is 10.
• Use this common denominator to convert the fractions so they can be added together.

Equivalent Fractions Explained

Once you have a common denominator, you convert each fraction to an equivalent fraction that shares this new denominator. This means adjusting the numerator so the overall value of the fraction remains the same, just expressed differently.
For \(\frac{1}{2}\) and \(\frac{1}{5}\), you aim to express them as fractions with 10 as the denominator.

• To convert \(\frac{1}{2}\) to an equivalent fraction with 10 as the denominator, multiply both the numerator (1) and the denominator (2) by 5 to get \(\frac{5}{10}\).
• For \(\frac{1}{5}\), multiply both the numerator (1) and the denominator (5) by 2 to get \(\frac{2}{10}\).

These equivalent fractions now share the same denominator, making it straightforward to add them.

Simplifying Fractions

After adding fractions, the result might not always be in its simplest form. Simplifying fractions means reducing them to their lowest terms. This process involves dividing the numerator and the denominator by their greatest common divisor (GCD). This makes the fraction easier to understand and use in further calculations.

• For example, if you ended up with \(\frac{8}{10}\), you would simplify by dividing both parts by their GCD, 2, to get \(\frac{4}{5}\).
• In our exercise, however, \(\frac{7}{10}\) is already in its simplest form as 7 and 10 only share the factor 1.

Simplifying ensures clarity in solutions and is a crucial step in fraction operations.

Finding the Least Common Multiple

The least common multiple (LCM) plays a key role in operations with fractions, particularly in finding a common denominator.
To find the LCM of two numbers, such as the denominators 2 and 5 in our example:

• List multiples of each number: for 2, these are 2, 4, 6, 8, 10, 12, etc., and for 5, they are 5, 10, 15, 20, etc.
• Identify the smallest multiple they share: in this case, it’s 10.

Understanding the LCM allows you to find common denominators quickly, facilitating calculations with fractions. This knowledge is essential when dealing with more complex problems involving multiple fractions.