Problem 60 - Elementary and Intermediate Algebra

Question

Perform the indicated operation and, if possible, simplify. If there are no variables, check using a calculator. $$ \frac{4}{5}+\frac{8}{15} $$

Step-by-Step Solution

Answer

\frac{4}{3}

Step 1: Find a Common Denominator

In order to add the fractions \(\frac{4}{5}+\frac{8}{15}\), a common denominator must be found. The least common multiple (LCM) of the denominators 5 and 15 is 15.

Step 2: Convert Fractions

Rewrite \(\frac{4}{5}\) with the common denominator 15. \(\frac{4}{5} = \frac{4 \times 3}{5 \times 3} = \frac{12}{15}\)

Step 3: Add the Fractions

Add the two fractions with the same denominator: \(\frac{12}{15} + \frac{8}{15} = \frac{12+8}{15} = \frac{20}{15}\)

Step 4: Simplify the Result

Simplify \(\frac{20}{15}\). Both 20 and 15 have a common divisor of 5. \(\frac{20 \div 5}{15 \div 5} = \frac{4}{3}\)

Step 5: Verify with a Calculator

Use a calculator to check the addition: \(\frac{4}{5} + \frac{8}{15} \approx \frac{20}{15} \approx 1.333...(or 1.3333)\), which simplifies to \(\frac{4}{3}\).

Key Concepts

Common Denominator

To add fractions, you need to have the same denominator. This is called finding a common denominator. The denominator is the number at the bottom of the fraction. For the given example, we need to add \(\frac{4}{5} + \frac{8}{15}\). Here, the denominators are 5 and 15. Finding a common denominator allows you to add the fractions easily. A common denominator is just a number that both denominators can divide into evenly. Let's move to the next section to understand this process better!

Least Common Multiple (LCM)

The Least Common Multiple (LCM) is the smallest number that is a multiple of both denominators. In our example, we are looking for the LCM of 5 and 15. Think about the multiples of these numbers:

• Multiples of 5: 5, 10, 15, 20, 25, ...
• Multiples of 15: 15, 30, 45, ...

The smallest common multiple is 15. So, the LCM of 5 and 15 is 15. This will be our common denominator. Now we can convert the fractions to have this common denominator. Let’s see how!

Simplifying Fractions

After adding fractions, you might need to simplify the result. Simplifying a fraction means reducing it to its simplest form. You do this by dividing the numerator and the denominator by their greatest common divisor (GCD). In our example, we added the fractions and got \(\frac{20}{15}\). To simplify, we divide both the numerator (20) and the denominator (15) by their GCD, which is 5. So, \(\frac{20 \div 5}{15 \div 5} = \frac{4}{3}\). Simplifying makes fractions easier to understand and use. And there you have it – the whole process from adding fractions to simplifying them!