Question

Add the following fractions and mixed numbers. Reduce to lowest terms. \(1 \frac{4}{5}+7 \frac{9}{10}+3 \frac{1}{2}=\) ______

Short answer

The sum is \(13 \frac{1}{5}\).

Step-by-step solution

Convert Mixed Numbers to Improper Fractions

The first step is to convert each mixed number into an improper fraction. \(1 \frac{4}{5}\) is equivalent to \(\frac{9}{5}\) since \(1 \times 5 + 4 = 9\). Similarly, \(7 \frac{9}{10}\) becomes \(\frac{79}{10}\) (because \(7 \times 10 + 9 = 79\)) and \(3 \frac{1}{2}\) becomes \(\frac{7}{2}\) (as \(3 \times 2 + 1 = 7\)).

Find a Common Denominator

The denominators are 5, 10, and 2. The least common denominator (LCD) of these numbers is 10. We will need to convert all fractions to have this common denominator.

Convert Fractions to the Common Denominator

Convert \(\frac{9}{5}\) to \(\frac{18}{10}\) by multiplying numerator and denominator by 2. \(\frac{79}{10}\) already has the common denominator, so it remains \(\frac{79}{10}\). Convert \(\frac{7}{2}\) to \(\frac{35}{10}\) by multiplying both the numerator and the denominator by 5.

Add the Improper Fractions

Add the fractions \(\frac{18}{10} + \frac{79}{10} + \frac{35}{10}\) by adding the numerators together: \(18 + 79 + 35 = 132\). The sum is \(\frac{132}{10}\).

Simplify the Resulting Fraction

Convert the improper fraction \(\frac{132}{10}\) to a mixed number by dividing 132 by 10. The quotient is 13 and the remainder is 2, so \(\frac{132}{10} = 13 \frac{2}{10}\). Simplify \(\frac{2}{10}\) to \(\frac{1}{5}\) because the numerator and the denominator can both be divided by 2.

Key concepts

Mixed Numbers

Mixed numbers are a combination of a whole number and a fraction. For example, in the exercise, we encountered numbers like \(1 \frac{4}{5}\), \(7 \frac{9}{10}\), and \(3 \frac{1}{2}\). Each of these represents a whole part and a fraction part.

This concept is often used when dealing with quantities that are more than a whole but less than two wholes, for example.

• The whole number represents the complete, full portions of the quantity.
• The fraction indicates the additional part beyond these wholes.

To work effectively with mixed numbers in an operation like addition, we generally convert them into improper fractions. This makes the arithmetic operation easier to manage, as all numbers will be treated uniformly as fractions.

Improper Fractions

An improper fraction is a fraction where the numerator (the top number) is greater than or equal to the denominator (the bottom number). For example, \(\frac{9}{5}\) and \(\frac{79}{10}\) from the exercise.

Improper fractions can seem a bit unconventional because we are used to seeing the numerator smaller, as in proper fractions.

• To convert a mixed number to an improper fraction, we multiply the denominator by the whole number and add the numerator.
• For instance, to convert \(1 \frac{4}{5}\) to an improper fraction, we calculate \(1 \times 5 + 4 = 9\), giving us \(\frac{9}{5}\).

Using improper fractions can simplify mathematical operations like addition because it avoids dealing with both whole numbers and fractional parts separately. They present quantities in a purely fractional form which can be standardized to ease arithmetic operations.

Simplifying Fractions

Simplifying fractions involves reducing them to their lowest terms, making them easier to understand and work with. This is an important final step in many fraction problems.

For example, in our exercise, after adding the fractions, we got \(\frac{132}{10}\).

• To simplify \(\frac{132}{10}\), we can divide both the numerator and the denominator by their greatest common factor (GCF), which, in this case, is 2.
• Dividing both 132 and 10 by 2, we get \(\frac{66}{5}\), providing a clearer result.

Simplifying a fraction does not change its value, it just presents the same quantity in a more compact form. This process helps in making results more interpretable and efficient for subsequent operations. Reducing fractions, therefore, is not just a mathematical exercise; it's a tool for clarity and simplicity.