Problem 49 - Prealgebra

Question

The problems below review some basic concepts of addition of fractions and mixed numbers. Add each of the following and reduce all answers to lowest terms. $$2 \frac{1}{2}+3 \frac{1}{2}$$

Step-by-Step Solution

Answer

The sum is 6.

Step 1: Convert Mixed Numbers to Improper Fractions

First, convert each mixed number into an improper fraction. For \(2 \frac{1}{2}\), multiply the whole number (2) by the denominator (2) and add the numerator (1): \(2 \times 2 + 1 = 5\). So, \(2 \frac{1}{2} = \frac{5}{2}\). Similarly, for \(3 \frac{1}{2}\), \(3 \times 2 + 1 = 7\), so \(3 \frac{1}{2} = \frac{7}{2}\).

Step 2: Add the Improper Fractions

Now that both numbers are improper fractions, add them: \(\frac{5}{2} + \frac{7}{2}\). Since the denominators are the same, simply add the numerators: \(5 + 7 = 12\). The result is \(\frac{12}{2}\).

Step 3: Simplify the Fraction

Divide the numerator and the denominator by their greatest common divisor. \(\frac{12}{2}\) simplifies to \(\frac{12 \div 2}{2 \div 2} = \frac{6}{1} = 6\). Therefore, the simplified answer is 6.

Key Concepts

Understanding Mixed Numbers

Mixed numbers are a combination of a whole number and a fraction. They are often used to represent everyday measurements, like saying two and a half hours. To visualize a mixed number, think of the whole part as a complete unit, while the fraction part adds a bit more. For example, in \(2 \frac{1}{2}\), "2" is the whole number, and "\(\frac{1}{2}\)" is the fraction.

When adding mixed numbers, it's helpful to first convert them into improper fractions for simplicity. This means changing the mixed number into a fraction where the numerator (top part) is greater than the denominator (bottom part). This also makes operations like addition and subtraction easier.

To convert a mixed number to an improper fraction:

• Multiply the whole number by the fraction's denominator.
• Add the result to the numerator of the fraction.
• Use this sum as the new numerator, with the original denominator.

So, \(2 \frac{1}{2}\) becomes \(\frac{5}{2}\), as shown when \(2 \times 2 + 1 = 5\). Repeat the same process for any mixed number you have.

What are Improper Fractions?

Improper fractions are fractions where the numerator is greater than or equal to the denominator, like \(\frac{7}{2}\) or \(\frac{9}{4}\). These fractions often arise when converting mixed numbers or after arithmetic operations like addition and subtraction.

Although they might look unusual, improper fractions are perfectly valid for calculations and can easily be converted into mixed numbers if needed.

With improper fractions, arithmetic operations such as addition become straightforward. For instance, when adding \(\frac{5}{2} + \frac{7}{2}\), you simply add the numerators because the denominators are the same. When the denominators differ, find a common denominator first.

Improper fractions might be converted back into mixed numbers for clarity, especially in situations where mixed numbers provide a better sense of the whole value and the remainder. To convert back, divide the numerator by the denominator: the quotient is the whole number, and the remainder is the new numerator.

Simplifying Fractions Made Easy

Simplifying fractions means reducing them to their simplest form, where the numerator and denominator have no common divisors other than 1. Simplifying makes fractions easier to understand and compare.

To simplify a fraction, like \(\frac{12}{2}\):

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

In this case, the GCD of 12 and 2 is 2, so \(\frac{12}{2} = \frac{6}{1} = 6\).

When simplifying, always check if both the numerator and denominator can be divided by the same number. Continue simplifying until you reach the simplest form. This process ensures that the fraction is as reduced as possible without altering its actual value.

Fractions in their simplest form are often easier to work with and provide clearer insight into the problem you're solving, making them crucial, especially in mathematics and its applications.