Problem 52 - Fundamentals of Mathematics

Question

Perform the addition: \(5 \frac{3}{8}+2 \frac{1}{6}\).

Step-by-Step Solution

Answer

The sum is \(7 \frac{13}{24}\).

Step 1: Convert mixed numbers to improper fractions

The mixed number \(5 \frac{3}{8}\) can be converted to an improper fraction by multiplying the whole number 5 by the denominator 8 and adding the numerator 3. \[5 \frac{3}{8} = \frac{5 \times 8 + 3}{8} = \frac{43}{8}\]Similarly, convert \(2 \frac{1}{6}\):\[2 \frac{1}{6} = \frac{2 \times 6 + 1}{6} = \frac{13}{6}\].

Step 2: Find a common denominator

To add \(\frac{43}{8}\) and \(\frac{13}{6}\), we need a common denominator. The least common multiple (LCM) of 8 and 6 is 24. We will rewrite both fractions with the common denominator 24.

Step 3: Rewrite fractions with common denominator

Adjust \(\frac{43}{8}\) to have the denominator 24:\[\frac{43}{8} = \frac{43 \times 3}{8 \times 3} = \frac{129}{24}\]Do the same for \(\frac{13}{6}\):\[\frac{13}{6} = \frac{13 \times 4}{6 \times 4} = \frac{52}{24}\].

Step 4: Add the fractions

With both fractions having the same denominator, add them:\[\frac{129}{24} + \frac{52}{24} = \frac{129 + 52}{24} = \frac{181}{24}\].

Step 5: Simplify the result into a mixed number

Convert \(\frac{181}{24}\) back into a mixed number. Divide 181 by 24, which goes 7 times with a remainder:\[181 \div 24 = 7 \, R13\]Thus, \(\frac{181}{24} = 7 \frac{13}{24}\).

Key Concepts

Converting Mixed Numbers to Improper Fractions

Understanding how to convert mixed numbers into improper fractions is an essential skill for adding or subtracting these types of numbers. A mixed number consists of a whole number and a fraction, such as \(5 \frac{3}{8}\). To convert a mixed number to an improper fraction, follow these steps:

• Multiply the whole number by the denominator of the fraction.
• Add the numerator of the fraction to the result from the above multiplication.
• Write this sum as the numerator of a new fraction, using the original denominator.

Following these steps, for \(5 \frac{3}{8}\), multiply 5 by 8 to get 40, then add the numerator 3, resulting in 43. Thus, \(5 \frac{3}{8} = \frac{43}{8}\). This same process applies to any mixed number, making it simple to handle in any mathematical operation.

Least Common Multiple

When adding fractions with different denominators, finding the least common multiple (LCM) is crucial. The LCM of two numbers is the smallest number that is a multiple of both. This number will become the common denominator for the fractions.
To find the LCM of two denominators, such as 8 and 6 in our exercise, list the multiples of each number until you find the smallest multiple they have in common.

• Multiples of 8: 8, 16, 24, 32, ...
• Multiples of 6: 6, 12, 18, 24, ...

The first common multiple is 24, so 24 is the LCM. This allows us to rewrite \(\frac{43}{8}\) and \(\frac{13}{6}\) with 24 as the new denominator, making them \(\frac{129}{24}\) and \(\frac{52}{24}\) respectively, ensuring they can be added easily.

Simplifying Fractions

Simplifying fractions is about reducing them to their simplest form, where the numerator and denominator have no common factors other than 1. After adding fractions together and obtaining a result, this step ensures clarity and reduces the fraction to its simplest terms.
For instance, when you add \(\frac{129}{24} + \frac{52}{24}\) and get \(\frac{181}{24}\), you attempt to simplify by checking if 181 and 24 share any divisors besides 1.
To further simplify or convert back into a mixed number:

• Divide the numerator by the denominator to find the whole number part.
• The remainder becomes the new numerator, with the original denominator.

In our example, \(181 \div 24\) gives 7 remainder 13, providing the mixed number \(7 \frac{13}{24}\). If there were any common divisors between 13 and 24, further simplification could be performed.