Problem 53 - Prealgebra

Question

What is twice the sum of \(2 \frac{1}{5}\) and \(\frac{3}{6} ?\)

Step-by-Step Solution

Answer

Twice the sum is \(5 \frac{2}{5}\).

Step 1: Convert Mixed Number to Improper Fraction

The first number given is a mixed number: \(2 \frac{1}{5}\). To convert it into an improper fraction, multiply the whole number 2 by the denominator 5 and add the numerator 1. \[2 \times 5 + 1 = 10 + 1 = 11\]So, \(2 \frac{1}{5} = \frac{11}{5}\).

Step 2: Simplify Fraction

The second number is \(\frac{3}{6}\). This fraction can be simplified by finding the greatest common divisor of 3 and 6, which is 3.Divide both the numerator and the denominator by 3:\[\frac{3}{6} = \frac{3 \div 3}{6 \div 3} = \frac{1}{2}\]

Step 3: Find the Common Denominator

To add the fractions \(\frac{11}{5}\) and \(\frac{1}{2}\), we need a common denominator. The least common multiple of 5 and 2 is 10.Convert both fractions to have a denominator of 10:\(\frac{11}{5} = \frac{11 \times 2}{5 \times 2} = \frac{22}{10} \)\(\frac{1}{2} = \frac{1 \times 5}{2 \times 5} = \frac{5}{10}\)

Step 4: Add the Fractions

Add the fractions \(\frac{22}{10}\) and \(\frac{5}{10}\):\[\frac{22}{10} + \frac{5}{10} = \frac{22 + 5}{10} = \frac{27}{10}\]

Step 5: Multiply the Sum by Two

The problem asks for twice the sum. Multiply the result \(\frac{27}{10}\) by two:\[2 \times \frac{27}{10} = \frac{54}{10}\]

Step 6: Simplify the Final Result

Simplify \(\frac{54}{10}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 2:\[\frac{54}{10} = \frac{54 \div 2}{10 \div 2} = \frac{27}{5}\] The result can also be expressed as a mixed number:\[\frac{27}{5} = 5\frac{2}{5}\]

Key Concepts

Mixed Numbers

When you hear the term "mixed numbers," it might seem a bit peculiar, but they're quite straightforward! A mixed number consists of two parts: a whole number and a proper fraction.
For example, in our exercise, we encounter the mixed number \(2\frac{1}{5}\). This simply means you have a whole 2, plus an additional part which is \(\frac{1}{5}\).

• The whole number is the number of complete parts.
• The fraction represents a portion of another whole.

To handle mixed numbers in mathematical operations like addition, you need to convert them to improper fractions. That way, they can be easily manipulated with other fractions. Let's see how to do that in the following section.

Improper Fractions

Improper fractions might sound a bit "improper," but they're actually quite useful! This kind of fraction is where the numerator (the top number) is larger than the denominator (the bottom number).
For instance, if we take the mixed number \(2\frac{1}{5}\), converting it gives us \(\frac{11}{5}\). How? Just follow this quick method:

• Multiply the whole number by the denominator.
• Add the result to the numerator.
• Place that sum over the original denominator.

So, \[2\frac{1}{5} = \frac{11}{5}\]By converting mixed numbers to improper fractions, math operations become uniform and much simpler to perform.

Common Denominator

Finding a common denominator is the key to adding fractions smoothly. Whenever you're dealing with more than one fraction, it helps to rewrite them so they share the same denominator.
This process enables us to add their numerators directly without any confusion. For example:

• If we have \(\frac{11}{5}\) and \(\frac{1}{2}\), we need to find a common denominator.
• The simplest way is to find the least common multiple (LCM) of 5 and 2, which is 10.
• Rewrite both fractions with this common denominator.

Now,\[\frac{11}{5} = \frac{22}{10}, \quad \text{and} \quad \frac{1}{2} = \frac{5}{10}\]Once they share a denominator, adding the fractions becomes a breeze!

Addition of Fractions

Adding fractions may initially appear complex, but once they share a common denominator, it's straightforward. Let's break down the addition process:

• Ensure the fractions you are adding have the same denominator.
• Add the numerators together.
• Keep the common denominator the same for the result.

For example, if we have \[\frac{22}{10} + \frac{5}{10} = \frac{22 + 5}{10} = \frac{27}{10}\]Notice how we only added the numerators, and the denominator remained unchanged. This method lets us focus solely on the parts being added, simplifying the addition process significantly. Remember, having a common denominator is essential for making addition manageable and accurate.