Question

Subtract the following fractions and mixed numbers. Reduce to lowest terms. \(\frac{4}{5}-\frac{1}{6}=\) ______

Short answer

\(\frac{19}{30}\)

Step-by-step solution

Find the Least Common Denominator (LCD)

To subtract the fractions \(\frac{4}{5}\) and \(\frac{1}{6}\), we first need to find a common denominator. The denominators are 5 and 6. The least common multiple of 5 and 6 is 30, so the LCD is 30.

Convert Fractions to Equivalent Fractions

Convert each fraction to an equivalent fraction with the denominator of 30. For \(\frac{4}{5}\), multiply both the numerator and the denominator by 6 to get \(\frac{4 \times 6}{5 \times 6} = \frac{24}{30}\). For \(\frac{1}{6}\), multiply both the numerator and the denominator by 5 to get \(\frac{1 \times 5}{6 \times 5} = \frac{5}{30}\).

Subtract the Fractions

Subtract the fractions with the common denominator: \(\frac{24}{30} - \frac{5}{30} = \frac{24 - 5}{30} = \frac{19}{30}\).

Simplify the Fraction

Check if \(\frac{19}{30}\) can be simplified further. The greatest common divisor (GCD) of 19 and 30 is 1, so \(\frac{19}{30}\) is already in its simplest form.

Key concepts

Understanding the Least Common Denominator

When working with fraction subtraction, finding the least common denominator (LCD) is crucial. It allows you to combine fractions with different denominators easily. Think of the LCD as the smallest number that both denominators can divide into without leaving a remainder.
For example, in the problem \(\frac{4}{5} - \frac{1}{6}\), the denominators are 5 and 6. To find their LCD, you look for the smallest multiple that both numbers share.
A quick way to find the LCD is to start listing the multiples of each number:

• Multiples of 5: 5, 10, 15, 20, 25, 30, ...
• Multiples of 6: 6, 12, 18, 24, 30, ...

As you can see, 30 is the first common multiple, hence making it the LCD. Once you know the LCD, you can convert each fraction to have this shared denominator, smoothing the way for subtraction.

Creating Equivalent Fractions

Fractions can be easily manipulated to become equivalent fractions by altering their numerators and denominators in such a way that the value of the fraction does not change.
For subtraction, especially, creating equivalent fractions is a key step. When you have determined the least common denominator, the next step is to modify each fraction to this common denominator without changing their mathematical value.
Take \(\frac{4}{5}\) as an example: By multiplying both its numerator and denominator by 6, it becomes \(\frac{24}{30}\). This maintains its value, only adjusting the denominator to match the LCD. Similarly, change \(\frac{1}{6}\) by multiplying its numerator and denominator by 5, resulting in \(\frac{5}{30}\). Now, the fractions are equivalent in terms of having the same denominator, allowing them to be subtracted easily.

Simplifying Fractions to Their Lowest Terms

After subtracting fractions that share a common denominator, the last piece of the puzzle is simplifying the result. Simplification means reducing a fraction to its lowest terms, where the numerator and denominator share no common factors other than 1.
In the exercise, after subtracting \(\frac{24}{30} - \frac{5}{30}\), you get \(\frac{19}{30}\). Checking for any possible simplification, determine if 19 and 30 share any factors other than 1.
Since the greatest common divisor (GCD) of 19 and 30 is 1, the fraction is already as simple as it gets. Thus, \(\frac{19}{30}\) is the final simplified answer. Remember, simplifying fractions is an important skill as it makes the fractions easier to interpret and helps keep your answers neat and accurate.