Problem 25 - Precalculus Mathematics for Calculus

Question

Perform the indicated operations. (a) \(\frac{3}{10}+\frac{4}{15}\) (b) \(\frac{1}{4}+\frac{1}{5}\)

Step-by-Step Solution

Answer

(a) \(\frac{17}{30}\), (b) \(\frac{9}{20}\)

Step 1: Find a Common Denominator for (a)

The fractions \( \frac{3}{10} \) and \( \frac{4}{15} \) have different denominators. We need to find a common denominator. The least common multiple (LCM) of 10 and 15 is 30.

Step 2: Convert Each Fraction to Have the Common Denominator for (a)

Convert \( \frac{3}{10} \) to have a denominator of 30: \( \frac{3}{10} = \frac{3 \times 3}{10 \times 3} = \frac{9}{30} \). Convert \( \frac{4}{15} \) to have a denominator of 30: \( \frac{4}{15} = \frac{4 \times 2}{15 \times 2} = \frac{8}{30} \).

Step 3: Add the Fractions for (a)

Now add the fractions: \( \frac{9}{30} + \frac{8}{30} = \frac{9+8}{30} = \frac{17}{30} \).

Step 4: Find a Common Denominator for (b)

The fractions \( \frac{1}{4} \) and \( \frac{1}{5} \) have different denominators. The least common multiple (LCM) of 4 and 5 is 20.

Step 5: Convert Each Fraction to Have the Common Denominator for (b)

Convert \( \frac{1}{4} \) to have a denominator of 20: \( \frac{1}{4} = \frac{1 \times 5}{4 \times 5} = \frac{5}{20} \). Convert \( \frac{1}{5} \) to have a denominator of 20: \( \frac{1}{5} = \frac{1 \times 4}{5 \times 4} = \frac{4}{20} \).

Step 6: Add the Fractions for (b)

Now add the fractions: \( \frac{5}{20} + \frac{4}{20} = \frac{5+4}{20} = \frac{9}{20} \).

Key Concepts

Understanding Least Common Multiple

When adding fractions with different denominators, the least common multiple (LCM) is essential. The LCM of two numbers is the smallest number that both numbers divide evenly into. For example, if you have fractions like \( \frac{3}{10} \) and \( \frac{4}{15} \), you need the LCM of 10 and 15 to find a common denominator.
To find it:

• List the multiples of each denominator.
• Identify the smallest multiple that both lists share.

So, for 10, the multiples are 10, 20, 30, 40, etc., and for 15, they are 15, 30, 45, etc. The smallest common multiple is 30, which becomes our common denominator for fraction addition. This step is vital as it sets the stage for accurate addition.

Finding a Common Denominator

After determining the least common multiple, the next task is to find a common denominator. This involves converting each fraction so that they both have the same denominator, which is the LCM you found earlier.
Here's how you do it:

• Multiply the numerator and denominator of the first fraction by a number that results in the LCM as the new denominator.
• Do the same for the second fraction.

Using our example again, \( \frac{3}{10} \) is transformed by multiplying top and bottom by 3, resulting in \( \frac{9}{30} \). Similarly, \( \frac{4}{15} \) is multiplied by 2 to give \( \frac{8}{30} \). Both fractions now have a common denominator, allowing you to proceed to addition.

Mastering Fraction Conversion

Converting fractions is a fundamental skill when dealing with addition. Once a common denominator is established, conversion ensures all fractions are expressed with this new uniform base.
Steps for conversion:

• Identify what factor the original denominator needs to be multiplied by to reach the common denominator.
• Multiply both the numerator and the denominator of the fraction by this factor.

Applying this to our second example, \( \frac{1}{4} \) becomes \( \frac{5}{20} \) (multiply by 5), and \( \frac{1}{5} \) becomes \( \frac{4}{20} \) (multiply by 4).
This ensures all fractions are ready for straightforward addition, showcasing the importance of mastering fraction conversion.