Problem 61 - Algebra A Combined Function

Question

Perform each indicated operation. $$ \frac{1}{12}+\frac{3}{20} $$

Step-by-Step Solution

Answer

\( \frac{1}{12} + \frac{3}{20} = \frac{7}{30} \) after simplification.

Step 1: Find a Common Denominator

To add fractions, we need a common denominator. The denominators here are 12 and 20. The least common multiple (LCM) of 12 and 20 is 60.

Step 2: Convert Fractions to Equivalent Fractions

Convert each fraction to equivalent fractions with the common denominator 60. For \( \frac{1}{12} \), multiply both numerator and denominator by 5, giving us \( \frac{1 \times 5}{12 \times 5} = \frac{5}{60} \).For \( \frac{3}{20} \), multiply both numerator and denominator by 3, giving us \( \frac{3 \times 3}{20 \times 3} = \frac{9}{60} \).

Step 3: Add the Numerators

Now that we have a common denominator, add the numerators: \( 5 + 9 = 14 \).The sum of the fractions is \( \frac{14}{60} \).

Step 4: Simplify the Fraction

Simplify \( \frac{14}{60} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 2.Thus, \( \frac{14 \div 2}{60 \div 2} = \frac{7}{30} \).

Key Concepts

Fractions Addition

Adding fractions might seem a bit tricky at first, but once you understand the process, it becomes much simpler. The key to fractions addition is that you have to add the numerators (the top numbers) together, but before you can do that, the fractions must have the same denominator (the bottom number). This rule applies because fractions represent parts of a whole, and you need to ensure you're working with equally sized portions before combining them.

Let's take a look at an example:

• Consider two fractions: \( \frac{1}{12} \) and \( \frac{3}{20} \).
• To add them, convert them into fractions with the same denominator.

Once the fractions have a common denominator, you can simply add the numerators and keep the denominator constant.

The result of adding the numerators will be a fraction that represents the new combined portion of the whole.

Least Common Denominator

The least common denominator (LCD) is crucial when you need to add fractions with different denominators. It is essentially the smallest number that each of the denominators can divide into evenly. Finding the LCD allows you to convert your fractions into equivalent fractions with a common, shared denominator, which is necessary for addition or subtraction.

To find the LCD of two fractions, \( \frac{1}{12} \) and \( \frac{3}{20} \):

• List the multiples of each denominator. For 12, the multiples are 12, 24, 36, 48, 60, 72, etc. For 20, the multiples are 20, 40, 60, 80, etc.
• Identify the smallest multiple that appears in both lists, which is 60 in this case.

60 is the least common denominator for these fractions, so we will convert each fraction to have a denominator of 60 before proceeding with addition.

Simplifying Fractions

Once fractions are added, simplifying the result makes it easier to understand and use. Simplifying a fraction involves reducing it to its smallest possible form where the numerator and denominator have no common factors other than 1.

After adding \( \frac{1}{12} \) and \( \frac{3}{20} \), we end up with \( \frac{14}{60} \). To simplify:

• Identify the greatest common divisor (GCD) of the numerator and the denominator. In this case, it is 2.
• Divide both the numerator and the denominator by the GCD: \( \frac{14 \div 2}{60 \div 2} = \frac{7}{30} \).

Now, \( \frac{7}{30} \) is the simplest form of the fraction, meaning it can't be reduced any further while remaining a fraction.