Question

Add the following fractions and mixed numbers. Reduce to lowest terms. \(10 \frac{1}{6}+12 \frac{4}{6}=\) ______

Short answer

The sum is \(22 \frac{5}{6}\).

Step-by-step solution

Convert Mixed Numbers to Improper Fractions

Convert each mixed number into an improper fraction. For \(10 \frac{1}{6}\), multiply the whole number 10 by the denominator 6, and then add the numerator 1: \(10 \times 6 + 1 = 61\). So, \(10 \frac{1}{6} = \frac{61}{6}\). Similarly, for \(12 \frac{4}{6}\), multiply 12 by 6, and then add 4: \(12 \times 6 + 4 = 76\). So, \(12 \frac{4}{6} = \frac{76}{6}\).

Add the Improper Fractions

Now, add the two improper fractions \(\frac{61}{6}\) and \(\frac{76}{6}\). Since they have the same denominator, you can simply add their numerators: \(61 + 76 = 137\). Thus, \(\frac{61}{6} + \frac{76}{6} = \frac{137}{6}\).

Convert to Mixed Number and Reduce

Finally, convert the improper fraction \(\frac{137}{6}\) back into a mixed number. Divide 137 by 6, which gives 22 with a remainder of 5. This converts to \(22 \frac{5}{6}\). Since \(\frac{5}{6}\) is already in its lowest terms, the final answer is \(22 \frac{5}{6}\).

Key concepts

Understanding Mixed Numbers

Mixed numbers are utilized to express a combination of a whole number and a fraction. This form makes it easier to comprehend quantities that exceed a whole number. For instance, if you have 10 whole pizzas and a third of another, you could express this as the mixed number \(10 \frac{1}{3}\). The mixed part indicates more than just the whole number itself, without breaking it down into a less intuitive improper fraction.
Think of mixed numbers in everyday life, such as when you're baking and need \(2 \frac{1}{4}\) cups of flour. It's more convenient to think about it in terms of cups and fractions of a cup rather than purely in fractional form.
To perform operations like addition or subtraction with mixed numbers, they can often be easier to handle in improper fraction form at first. This is why knowing how to convert mixed numbers is crucial for solving mathematic problems efficiently.

What are Improper Fractions?

Improper fractions occur when the numerator, or the top part of the fraction, is greater than or equal to the denominator, which is the bottom part. For example, \(\frac{9}{4}\) is an improper fraction because 9 is greater than 4.
Improper fractions represent amounts greater than one whole. They can sometimes look daunting but are quite simple to interpret when you understand their meaning. A real-life example could be if you ate \(\frac{7}{3}\) of a pie. It means you ate more than 2 full pies since \(\frac{7}{3}\) translates to \(2 \frac{1}{3}\) when converted to a mixed number.
To convert mixed numbers to improper fractions, follow a straightforward process:

• Multiply the whole number by the denominator.
• Add the numerator to this product.
• This sum becomes the new numerator, while the denominator stays unchanged.

This conversion simplifies operations between fractions, especially when adding, subtracting, or multiplying.

Reducing Fractions to Lowest Terms

Reducing fractions to their lowest terms means making them as simple or small as possible while retaining their original value. This means the numerator and denominator should be divided by their greatest common divisor (GCD).
When a fraction is in its lowest terms, it looks neater and is often easier to work with, especially in further calculations. For example, reducing \(\frac{10}{15}\) by dividing both parts by 5 gives \(\frac{2}{3}\). This form is simpler and immediately understandable.
Why is it important? Reducing fractions helps to:

• Simplify calculations, making addition, subtraction, multiplication, or division easier.
• Understand proportions better visually - \(\frac{2}{3}\) is more intuitive than \(\frac{10}{15}\).
• Avoid errors in more complex mathematical processes or proofs.

Always check after operating on fractions if they can be simplified. This is often one of the final steps in fraction-based math problems, ensuring that the solution is both accurate and presentable.