Question

Add the following fractions and mixed numbers. Reduce to lowest terms. \(\frac{1}{4}+\frac{1}{6}+\frac{1}{8}=\) ______

Short answer

The sum is \(\frac{13}{24}\).

Step-by-step solution

Find the Least Common Denominator (LCD)

To add these fractions, we first need a common denominator. The denominators are 4, 6, and 8. The least common multiple of these numbers is 24. Thus, the least common denominator is 24.

Convert Fractions to Equivalent Fractions

Convert each fraction to an equivalent fraction with the denominator of 24.- For \(\frac{1}{4}\), multiply both the numerator and the denominator by 6: \(\frac{1 \times 6}{4 \times 6} = \frac{6}{24}\).- For \(\frac{1}{6}\), multiply both the numerator and the denominator by 4: \(\frac{1 \times 4}{6 \times 4} = \frac{4}{24}\).- For \(\frac{1}{8}\), multiply both the numerator and the denominator by 3: \(\frac{1 \times 3}{8 \times 3} = \frac{3}{24}\).

Add the Fractions

Now, add the equivalent fractions: \(\frac{6}{24} + \frac{4}{24} + \frac{3}{24}\).Combine the numerators: \(6 + 4 + 3 = 13\).So, \(\frac{6}{24} + \frac{4}{24} + \frac{3}{24} = \frac{13}{24}\).

Simplify the Resulting Fraction

The fraction \(\frac{13}{24}\) is already in its lowest terms. The numerator 13 is a prime number and does not have any common factors with 24 other than 1.

Key concepts

Least Common Denominator

Adding fractions often involves finding a shared bottom number, known as the denominator. But not just any denominator will do; we aim for the Least Common Denominator (LCD). This is the smallest number that can be divided evenly by each of the original fractions' denominators. For example, in the fractions \( \frac{1}{4} \), \( \frac{1}{6} \), and \( \frac{1}{8} \), the denominators are 4, 6, and 8, respectively. To find the LCD, we need the least common multiple of these numbers. We calculate this by listing the multiples of each number and finding the smallest multiple they all share. Multiples of 4 include 4, 8, 12, 16, 20, 24. Multiples of 6 are 6, 12, 18, 24, and multiples of 8 are 8, 16, 24. The smallest common multiple is 24. Therefore, 24 is used as the least common denominator for our addition task.

Equivalent Fractions

Once we have the least common denominator, the next step is to convert each fraction to what’s referred to as an equivalent fraction. An equivalent fraction is a different-looking fraction that has the same value as the original. To achieve this, you adjust the numerator to fit the new denominator. Using our earlier examples, we aim to transform each fraction so its denominator is 24. For \( \frac{1}{4} \), we multiply the numerator and denominator by 6, resulting in: \( \frac{1 \times 6}{4 \times 6} = \frac{6}{24} \). Similarly, \( \frac{1}{6} \) becomes \( \frac{4}{24} \) after multiplying each by 4, and \( \frac{1}{8} \) becomes \( \frac{3}{24} \) with a multiplier of 3. These new fractions: \( \frac{6}{24} \), \( \frac{4}{24} \), and \( \frac{3}{24} \) are thus equivalent to their originals, yet now they share a common denominator.

Simplifying Fractions

After converting and adding fractions, simplifying them is the final step. Simplifying means reducing a fraction to its lowest terms. This occurs when the numerator and the denominator no longer share any common divisors other than 1. In our example, after adding the equivalent fractions \( \frac{6}{24} + \frac{4}{24} + \frac{3}{24} \), we end up with \( \frac{13}{24} \). To check if this is in its simplest form, we look at the numerator, 13. Since 13 is a prime number (only divisible by 1 and itself), it doesn't share any common factors with 24 except for 1. Hence, \( \frac{13}{24} \) is already simplified. In fractions, simplicity usually means clarity, making it easier to understand or use in further calculations or comparisons.