Problem 65 - Elementary and Intermediate Algebra

Question

Perform the indicated operation and, if possible, simplify. If there are no variables, check using a calculator. $$ \frac{3}{10}+\frac{8}{15} $$

Step-by-Step Solution

Answer

\(\frac{5}{6}\)

Step 1: Find Common Denominator

To add the fractions \(\frac{3}{10}+\frac{8}{15}\), first find a common denominator. The least common multiple of 10 and 15 is 30.

Step 2: Convert Fractions

Convert each fraction to have the common denominator of 30. For \(\frac{3}{10}\), multiply both the numerator and the denominator by 3 to get \(\frac{3 \times 3}{10 \times 3} = \frac{9}{30}\). For \(\frac{8}{15}\), multiply both the numerator and the denominator by 2 to get \(\frac{8 \times 2}{15 \times 2} = \frac{16}{30}\).

Step 3: Add Fractions

Add the converted fractions: \(\frac{9}{30} + \frac{16}{30} \). Since the denominators are the same, simply add the numerators: \(\frac{9 + 16}{30} = \frac{25}{30}\).

Step 4: Simplify the Fraction

Simplify \(\frac{25}{30}\) by dividing both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 25 and 30 is 5. So, \(\frac{25 \div 5}{30 \div 5} = \frac{5}{6}\).

Step 5: Verify with a Calculator

Using a calculator, convert the fractions to decimal to verify the answer: \(\frac{3}{10} \approx 0.3\) and \(\frac{8}{15} \approx 0.5333\). Adding these gives approximately 0.8333, which matches \(\frac{5}{6} \approx 0.8333\). This confirms that the simplified answer is correct.

Key Concepts

common denominator

To add fractions together, you need a common denominator. The common denominator is a shared multiple of the denominators of both fractions.
This allows the fractions to be combined under a single denominator.
Consider the fractions in our example: \( \frac{3}{10} \) and \( \frac{8}{15} \).
Their denominators are 10 and 15. First, find the least common multiple (LCM) of these two numbers.
The LCM of 10 and 15 is 30.

Once you have the common denominator, adjust each fraction so that they both use this common denominator.
This way, you're dealing with parts of the same whole, making addition possible.

simplifying fractions

Simplifying fractions means reducing them to their simplest form.
This happens when the numerator and the denominator have no common divisors other than 1.
To simplify a fraction, divide both the numerator and the denominator by their greatest common divisor (GCD).
In our example, after converting the fractions to a common denominator and adding them, we get \( \frac{25}{30} \).
The GCD of 25 and 30 is 5.
Divide both the numerator and the denominator by 5:
\( \frac{25 \div 5}{30 \div 5} \) = \( \frac{5}{6} \).
Now, \( \frac{5}{6} \) is in its simplest form.

least common multiple

The least common multiple (LCM) of two numbers is the smallest number that is a multiple of both of them.
Finding the LCM is crucial for adding fractions.
In our example, we need to find the LCM of 10 and 15.
List the multiples of each number:

• Multiples of 10: 10, 20, 30, 40, ...
• Multiples of 15: 15, 30, 45, 60, ...

The smallest common multiple between these lists is 30.
By converting both fractions to have a denominator of 30, you can easily add them.

greatest common divisor

The greatest common divisor (GCD) is the largest number that divides both the numerator and the denominator evenly.
To simplify fractions, you need to find the GCD.
In our example, after converting the fractions to a common denominator and adding them, the fraction to simplify is \( \frac{25}{30} \).
The divisors of 25 are 1, 5, and 25.
The divisors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30.

The largest number in both lists is 5. This is the GCD.
Dividing the numerator (25) and the denominator (30) by 5, we simplify the fraction to \( \frac{5}{6} \).
This makes the fraction more understandable and easier to work with.