Problem 226 - Intermediate Algebra

Question

In the following exercises, simplify. $$ \frac{\frac{1}{6}+\frac{3}{10}}{\frac{14}{30}} $$

Step-by-Step Solution

Answer

1

Step 1 - Find a Common Denominator for the Numerator

The numerator consists of two fractions: \( \frac{1}{6} \) and \( \frac{3}{10} \). To add these fractions, find a common denominator. The smallest common multiple of 6 and 10 is 30.

Step 2 - Convert Fractions to Have a Common Denominator

Convert \( \frac{1}{6} \) and \( \frac{3}{10} \) to fractions with a denominator of 30. \( \frac{1}{6} = \frac{1 \times 5}{6 \times 5} = \frac{5}{30} \) and \( \frac{3}{10} = \frac{3 \times 3}{10 \times 3} = \frac{9}{30} \)

Step 3 - Add the Fractions in the Numerator

Now that the fractions have a common denominator, add them together. \( \frac{5}{30} + \frac{9}{30} = \frac{5 + 9}{30} = \frac{14}{30} \)

Step 4 - Simplify the Original Expression

The original expression \( \frac{\frac{1}{6}+\frac{3}{10}}{\frac{14}{30}} \) can now be written as \( \frac{\frac{14}{30}}{\frac{14}{30}} \).

Step 5 - Simplify

Any fraction where the numerator and denominator are the same equals 1. Thus, \( \frac{\frac{14}{30}}{\frac{14}{30}} = 1 \).

Key Concepts

Common Denominator

When adding or comparing fractions, they must share the same denominator. This makes the fractions 'compatible' and easier to work with. The denominator is the bottom number of a fraction, which shows how many equal parts the whole is divided into. To find a common denominator, you can use the least common multiple (LCM) of the denominators. For example, for the fractions \( \frac{1}{6} \) and \( \frac{3}{10} \), the denominators are 6 and 10. The LCM of 6 and 10 is 30.
Now you should convert each fraction so that they have this common denominator. This step is essential because it allows you to add or subtract the fractions by merely working with the numerators, which is the next key concept.

Adding Fractions

Once the fractions have a common denominator, you can add them easily. Only the numerators (the top numbers) get added together, while the denominator remains the same. For example, if you convert \( \frac{1}{6} \) to \( \frac{5}{30} \) and \( \frac{3}{10} \) to \( \frac{9}{30} \), you then add the numerators: 5 + 9 = 14. So, you get \( \frac{14}{30} \).
This step simplifies working with fractions and helps ensure accurate results. Understanding this process is crucial for solving more complex fraction problems, like the one in the exercise.

Simplify Fractions

After adding fractions, it's often necessary to simplify the result. Simplifying a fraction means reducing it to its simplest form. This is done by dividing both the numerator and the denominator by their greatest common divisor (GCD). In the exercise, after adding the fractions, the result was \( \frac{14}{30} \). To simplify, you would find the GCD of 14 and 30, which is 2, and then divide both by 2, though in this specific case, it just leads back to 1 because the fraction is divided by itself. Simplifying ensures that the fraction is as simple as possible and easily understandable.

Numerator and Denominator

In fractions, the numerator is the top number, and the denominator is the bottom number. The numerator indicates how many parts of the whole you have, while the denominator tells you into how many equal parts the whole is divided. For any fraction to be properly simplified or compared, understanding what the numerator and denominator represent is key. In our exercise, we had to convert fractions to have a common denominator and then add the numerators. By understanding these roles, you can manipulate and simplify fractions correctly.