Problem 3 - Prealgebra

Question

Add and subtract the following mixed numbers as indicated. \(4 \frac{3}{10}+8 \frac{1}{10}\)

Step-by-Step Solution

Answer

The sum is \(12 \frac{2}{5} \).

Step 1: Convert Mixed Numbers to Improper Fractions

First, convert each mixed number into an improper fraction. For \(4 \frac{3}{10}\), multiply the whole number 4 by the denominator 10, then add the numerator 3. Thus, \( 4 \frac{3}{10} = \frac{4 \times 10 + 3}{10} = \frac{40 + 3}{10} = \frac{43}{10} \). Do the same for \(8 \frac{1}{10}\): \( 8 \frac{1}{10} = \frac{8 \times 10 + 1}{10} = \frac{80 + 1}{10} = \frac{81}{10} \).

Step 2: Add the Improper Fractions

Since the fractions \(\frac{43}{10}\) and \(\frac{81}{10}\) have the same denominator, add the numerators directly: \( \frac{43}{10} + \frac{81}{10} = \frac{43 + 81}{10} = \frac{124}{10} \).

Step 3: Convert the Improper Fraction Back to a Mixed Number

Convert \(\frac{124}{10}\) into a mixed number. Divide 124 by 10. The quotient is 12, which is the whole number part, and the remainder is 4, which is part of the fraction. Thus, \(\frac{124}{10} = 12 \frac{4}{10} \).

Step 4: Simplify the Fraction Part of the Mixed Number

Simplify the fraction \(\frac{4}{10}\) by finding the greatest common divisor (GCD) of 4 and 10, which is 2. Divide both the numerator and the denominator by 2: \(\frac{4}{10} = \frac{4 \div 2}{10 \div 2} = \frac{2}{5} \). Thus, the final mixed number is \(12 \frac{2}{5} \).

Key Concepts

Understanding Improper Fractions

Improper fractions can initially seem daunting, but they're actually quite straightforward once you get the hang of them. An improper fraction is a type of fraction where the numerator, or the number on top, is larger than the denominator, the number on the bottom.
For example, in the fraction \(\frac{43}{10}\), 43 is larger than 10, which makes it an improper fraction. Why does this happen? Often, it's because we've converted a mixed number into a fraction all in one piece.

The process goes like this: when you have a mixed number such as \(4 \frac{3}{10}\), you need to convert it into an improper fraction before you can add or subtract it easily with other fractions. You take the whole number, multiply it by the denominator, and then add the numerator. Here, \(4 \frac{3}{10}\) becomes \(\frac{43}{10}\) because you calculate \(4 \times 10 + 3\) and place that total over the original denominator 10.

• Mixed Number \(a \frac{b}{c}\): Multiply \(a\) by \(c\) then add \(b\).
• Result: Numerator becomes \(a \times c + b\); use original denominator.

The Art of Conversion: Mixed Numbers to Improper Fractions and Back

The ability to convert between mixed numbers and improper fractions is crucial when tackling fraction operations. A mixed number combines a whole number with a fraction, example being \(8 \frac{1}{10}\).
To switch from a mixed number to an improper fraction, follow these steps:

• Multiply the whole number by the denominator.
• Add the numerator to this product.
• Place this result above the original denominator.

When changing back—the opposite direction—is also vital after you do your calculations. For instance, if you end up with \(\frac{124}{10}\) through your calculations, you revert it to a mixed number:

• Divide the numerator by the denominator to find the whole number portion.
• The remainder becomes the new numerator, making the fraction part.

Thus, \( \frac{124}{10} \) converts to \(12 \frac{4}{10}\), and with simplification, \(12 \frac{2}{5}\). Conversion lets you shuffle between these number forms so you can tackle whatever operation you need to perform, such as addition here!

Simplifying Fractions: Back to Basics

Fraction simplification is all about making a fraction as simple as possible. It involves reducing the fraction to its lowest terms.
Here's how you simplify a fraction, exemplified by simplifying \(\frac{4}{10}\):

• Find the greatest common divisor (GCD) of both the numerator and the denominator.
• Divide both the numerator and the denominator by this number.

For \(\frac{4}{10}\), the GCD is 2. Dividing both 4 and 10 by 2 gives us \(\frac{2}{5}\). This is the simplified version of the fraction.
Simplification not only makes fractions easier to interpret and work with but often provides a more straightforward reading of quantities. It's a vital step, especially at the end of multi-step problems, ensuring you're presenting the cleanest version of your answer. By practicing this, you enhance your skill in handling and simplifying fractions in any problem!