Problem 32 - Elementary Algebra

Question

For exercises 27-34, evaluate. $$ \frac{5}{8}+\frac{7}{30} $$

Step-by-Step Solution

Answer

\(\frac{103}{120}\)

Step 1: Identify the Fractions

The fractions given are \(\frac{5}{8}\) and \(\frac{7}{30}\).

Step 2: Find a Common Denominator

To add the fractions, they must have a common denominator. The denominators are 8 and 30. The least common multiple (LCM) of 8 and 30 is 120.

Step 3: Convert Fractions to Have the Same Denominator

Convert \(\frac{5}{8}\) and \(\frac{7}{30}\) to fractions with the denominator 120:\[\frac{5}{8} \times \frac{15}{15} = \frac{75}{120}\]\[\frac{7}{30} \times \frac{4}{4} = \frac{28}{120}\]

Step 4: Add the Fractions

Add \(\frac{75}{120}\) and \(\frac{28}{120}\):\[\frac{75}{120} + \frac{28}{120} = \frac{103}{120}\]

Step 5: Simplify the Fraction

The fraction \(\frac{103}{120}\) is already in its simplest form since 103 and 120 have no common factors other than 1.

Key Concepts

least common multiple

When adding fractions, having a common denominator is essential. This is where the concept of the Least Common Multiple (LCM) comes in handy. The **LCM** of two numbers is the smallest number that both of them divide into without a remainder.

For the denominators in our problem (8 and 30), we need to find the LCM. We do this by listing the multiples of each, or using the prime factorization method.

Here's a quick way to find the LCM:

• List the multiples of the larger number until you find one that the smaller number divides into exactly.
  
• Multiples of 30: 30, 60, 90, 120,...
  
• Multiple of 8 in common: 120, ...
  
• Thus, 120 is our LCM.
  

Now that we have found our LCM, we can proceed to convert the fractions so they have the same denominator.

fraction addition

Adding fractions with different denominators can seem tricky at first, but it's straightforward once you have a common denominator.

Here are the steps we follow:

• Convert each fraction to an equivalent fraction with our common denominator (LCM).
  
• For \(\frac{5}{8}\), we convert it to have a denominator of 120 by multiplying both the numerator and the denominator by 15: \[ \frac{5}{8} \times \frac{15}{15} = \frac{75}{120} \]
  
• For \(\frac{7}{30}\), we convert it by multiplying both the numerator and the denominator by 4: \[ \frac{7}{30} \times \frac{4}{4} = \frac{28}{120} \]
  
• Now, both fractions have the same denominator, so we can simply add the numerators: \[ \frac{75}{120} + \frac{28}{120} = \frac{103}{120} \]
  

Notice that we only add the numerators, not the denominators. The common denominator stays the same.

simplifying fractions

Simplifying fractions means reducing them to their simplest form. This occurs when the numerator and the denominator have no common factors other than 1.

Here’s how to simplify a fraction:

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

Let's apply this to our fraction: \[ \frac{103}{120} \]

Find the GCD of 103 and 120.

103 is a prime number, only divisible by 1 and 103.

Checking, we find 103 and 120 have no common factors other than 1.

Hence, \[ \frac{103}{120} \] is already in its simplest form.

Simplifying fractions ensures your final answer is clean and precise. Keep practicing and you'll master these concepts in no time!