Problem 17 - Prealgebra

Question

Add and subtract the following mixed numbers as indicated. \(10 \frac{5}{6}+15 \frac{3}{4}\)

Step-by-Step Solution

Answer

The sum is \(26 \frac{7}{12}\).

Step 1: Convert Mixed Numbers to Improper Fractions

First, convert each mixed number into an improper fraction. For the first number, \(10 \frac{5}{6}\), multiply the whole number 10 by the denominator 6, and add the numerator 5. This gives \((10 \times 6) + 5 = 60 + 5 = 65\). So, \(10 \frac{5}{6} = \frac{65}{6}\). For \(15 \frac{3}{4}\), multiply 15 by 4, then add 3: \((15 \times 4) + 3 = 60 + 3 = 63\), giving \(15 \frac{3}{4} = \frac{63}{4}\).

Step 2: Find a Common Denominator

To add the fractions, find a common denominator. The denominators are 6 and 4. The least common multiple is 12. Convert \(\frac{65}{6}\) to a denominator of 12 by multiplying both the numerator and denominator by 2: \(\frac{65 \times 2}{6 \times 2} = \frac{130}{12}\). Convert \(\frac{63}{4}\) by multiplying by 3: \(\frac{63 \times 3}{4 \times 3} = \frac{189}{12}\).

Step 3: Add the Fractions

Now, add the two fractions \(\frac{130}{12}\) and \(\frac{189}{12}\). Since they have the same denominator, simply add the numerators: \(130 + 189 = 319\). So, \(\frac{130}{12} + \frac{189}{12} = \frac{319}{12}\).

Step 4: Convert Improper Fraction Back to Mixed Number

Convert the improper fraction \(\frac{319}{12}\) back to a mixed number. Divide 319 by 12. The quotient is 26, and the remainder is 7. Thus, \(\frac{319}{12} = 26 \frac{7}{12}\).

Step 5: Simplify the Result (if necessary)

Check if the fraction \(\frac{7}{12}\) is in its simplest form. It is, because 7 and 12 have no common factors other than 1. Thus, the final result is already simplified.

Key Concepts

Improper fractions

When working with mixed numbers, such as in our exercise, it's often easier to first convert them into improper fractions. An improper fraction is when the numerator (the top number) is greater than or equal to the denominator (the bottom number). Converting mixed numbers helps simplify operations like addition and subtraction.

Here's how you can do it:

• Multiply the whole number by the denominator of the fractional part.
• Add the numerator to this product.
• Place the result over the original denominator to get the improper fraction.

For example, converting the mixed number, 10 \(\frac{5}{6}\), to an improper fraction involves these steps:

• Multiply 10 by the denominator, 6: \(10 \times 6 = 60\).
• Add the numerator, 5: \(60 + 5 = 65\).
• So, \(10 \frac{5}{6}\) becomes \(\frac{65}{6}\).

Converting to improper fractions makes it easier to manipulate and calculate with fractions of different denominators.

Common denominators

Adding or subtracting fractions requires that they share a common denominator. The denominator reflects how many parts the whole is divided into, so for the parts to be comparable, these numbers need to be the same. This step involves finding the least common denominator.

The least common denominator is the least common multiple (LCM) of the denominators involved. Here’s how you do it:

• List the multiples of each denominator.
• Identify the smallest multiple that appears in both lists.

In our exercise, the denominators are 6 and 4. The multiples of 6 are 6, 12, 18, etc., and for 4, they are 4, 8, 12, etc. The LCM is 12, which both numbers share. Once you have a common denominator, convert each fraction:

• Multiply both the numerator and denominator of \(\frac{65}{6}\) and \(\frac{63}{4}\) by necessary values to make the denominator 12.
• \(\frac{65}{6} = \frac{130}{12}\) and \(\frac{63}{4} = \frac{189}{12}\).

Now, with common denominators, adding or subtracting becomes straightforward.

Simplifying fractions

The last essential step in fraction problems—after performing the core arithmetic—is simplifying the result. Simplifying means reducing the fraction to its smallest possible form where the numerator and denominator have no common factors other than 1.

To simplify a fraction like \(\frac{7}{12}\):

• Find the greatest common divisor (GCD) of the numerator and the denominator.
• If the GCD is 1, the fraction is already in its simplest form.
• For our fraction, since there are no common factors other than 1, it cannot be simplified further.

Simplifying makes the fraction easier to work with and understand. It's the final touch that ensures your answer is as neat as possible. After simplifying, fractions become clearer, and comparisons between different fractions more intuitive.