Problem 96 - Elementary and Intermediate Algebra

Question

Add: \(\quad \frac{3}{10}+\frac{2}{15}\)

Step-by-Step Solution

Answer

The sum of \(\frac{3}{10}\) and \(\frac{2}{15}\) is \(\frac{13}{30}\).

Step 1: Find the Least Common Denominator

First, identify the denominators of the fractions: 10 and 15. Find the least common multiple (LCM) of these denominators to act as the least common denominator (LCD). The multiples of 10 are 10, 20, 30, and the multiples of 15 are 15, 30. The smallest common multiple is 30, so the LCD is 30.

Step 2: Adjust the Fractions

Next, convert each fraction to an equivalent fraction with the denominator of 30. For \(\frac{3}{10}\), multiply the numerator and the denominator by 3 to get \(\frac{9}{30}\). For \(\frac{2}{15}\), multiply the numerator and the denominator by 2 to get \(\frac{4}{30}\).

Step 3: Add the Fractions

Now that both fractions have the same denominators, add their numerators. So, \(\frac{9}{30} + \frac{4}{30} = \frac{13}{30}\).

Step 4: Simplify the Result

Check if the resulting fraction \(\frac{13}{30}\) can be simplified. In this case, 13 is a prime number not divisible by 30, so \(\frac{13}{30}\) is already in its simplest form.

Key Concepts

Least Common Denominator

When adding fractions, it is important to have a common denominator. This is where the least common denominator (LCD) comes into play. To find the LCD, we first identify the denominators of the fractions in question. For the example, these are 10 and 15.
To find the LCD, we need to calculate the least common multiple (LCM) of these denominators. To do this, list the multiples of each number:

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

The smallest multiple that is common to both lists is 30, making it the least common denominator.
Using the LCD is essential because it standardizes the denominators across the fractions, allowing you to easily add or subtract them.

Equivalent Fractions

Once the least common denominator is determined, it's time to adjust each fraction so they both share this denominator. This is done by creating equivalent fractions. Equivalent fractions are different fractions that represent the same value.
To convert a fraction to an equivalent one with a new denominator, you multiply both the numerator and the denominator by the same number. In our example:

• For \(\frac{3}{10}\), we need the new denominator to be 30. Since 10 times 3 equals 30, you also multiply the numerator (3) by 3 to get \(\frac{9}{30}\).
• For \(\frac{2}{15}\), you need to multiply both the numerator and denominator by 2, because 15 times 2 equals 30. This gives you \(\frac{4}{30}\).

Both fractions now have the same denominator, 30, which means you can easily add them.

Simplifying Fractions

After adding fractions, the final step is to simplify the resulting fraction. Simplifying fractions means making them as simple as possible while still representing the same amount.
To simplify a fraction, you look for the greatest common factor (GCF) of the numerator and the denominator. You then divide both by this number.
In our case, the resulting fraction from our addition is \(\frac{13}{30}\). Since 13 is a prime number and cannot be divided evenly by 30, the fraction is already in its simplest form.
When simplifying, always check if the numerator and the denominator have any common factors. If they do, divide them by their GCF to simplify the fraction.