Question

Evaluate each expression. $$\frac{6}{11}-\frac{1}{2}$$

Short answer

The answer is \(\frac{1}{22}\).

Step-by-step solution

Find the least common denominator

Identify the least common denominator (LCD) for the fractions \(\frac{6}{11}\) and \(\frac{1}{2}\). The LCD is the smallest number that both 11 and 2 divide into evenly. Calculate the LCD: \( 11 \times 2 = 22 \). So, the LCD is 22.

Convert fractions to have the same denominator

Rewrite each fraction with the least common denominator of 22. Convert \(\frac{6}{11}\) to \(\frac{6 \times 2}{11 \times 2} = \frac{12}{22}\). Convert \(\frac{1}{2}\) to \(\frac{1 \times 11}{2 \times 11} = \frac{11}{22}\).

Subtract the fractions

Now that the fractions have the same denominator, subtract the numerators and place the result over the common denominator. Compute \(\frac{12}{22} - \frac{11}{22} = \frac{12 - 11}{22} = \frac{1}{22}\).

Key concepts

least common denominator

When subtracting or adding fractions, the least common denominator (LCD) is crucial. The LCD helps ensure that both fractions share a common groundwork, making them easier to work with. To find the LCD, look for the smallest number both denominators divide into evenly. For instance, with \(\frac{6}{11}\) and \(\frac{1}{2}\), we need to determine a number both 11 and 2 can divide into. Here, multiplying 11 by 2 gives us 22, making 22 the LCD. This step simplifies the entire fraction operation!

rewriting fractions

Rewriting fractions is about changing them to have that shared common denominator we just discussed. This doesn't alter their value; it just makes them compatible for subtraction or addition. To rewrite \(\frac{6}{11}\) with a denominator of 22, we multiply both the numerator and the denominator by 2, giving \(\frac{12}{22}\). Likewise, to rewrite \(\frac{1}{2}\), we multiply both the numerator and the denominator by 11, resulting in \(\frac{11}{22}\). Now both fractions have the same denominator, they're ready to be subtracted!

subtracting fractions

Now comes the actual subtraction! With the fractions sharing the same denominator (22 in this case), we focus on the numerators. For \(\frac{12}{22} - \frac{11}{22}\), we simply subtract 11 from 12, ending up with 1. This difference goes over the common denominator. Hence, \(\frac{12}{22} - \frac{11}{22} = \frac{1}{22}\). The process becomes straightforward once the fractions are prepared with a unified denominator.

fraction simplification

Fraction simplification follows the subtraction (or addition) step. It involves reducing the fraction to its simplest form. A fraction is simplified when there are no common factors (other than 1) between the numerator and denominator. For example, \(\frac{1}{22}\) is already simplified because there's no number other than 1 that divides both 1 and 22 evenly. Simplification helps in making fractions easier to understand and work with.