Problem 47 - Prealgebra

Question

Add and subtract as indicated. $$\frac{11}{12}-\frac{9}{10}$$

Step-by-Step Solution

Answer

\(\frac{1}{60}\).

Step 1: Determine the Least Common Denominator (LCD)

To add or subtract fractions, we need a common denominator. The denominators here are 12 and 10. The least common multiple of these numbers is 60, so that will be our least common denominator (LCD).

Step 2: Rewrite Each Fraction with the LCD

Convert each fraction so that they have the same denominator of 60. For \(\frac{11}{12}\), multiply both numerator and denominator by 5 to get \(\frac{55}{60}\). For \(\frac{9}{10}\), multiply both numerator and denominator by 6 to get \(\frac{54}{60}\).

Step 3: Perform the Subtraction

Now that the fractions have been converted to have the same denominator, subtract the numerators while keeping the denominator the same: \(\frac{55}{60} - \frac{54}{60} = \frac{1}{60}\).

Step 4: Simplification Confirmation

Check if \(\frac{1}{60}\) can be simplified further. Since 1 is a prime number and 60 does not share any common factors with 1 other than 1 itself, \(\frac{1}{60}\) is in its simplest form.

Key Concepts

Least Common Denominator

When working with fraction subtraction, the first key step is finding a common denominator so that the fractions can be combined or subtracted. The least common denominator (LCD) is essentially the smallest number that both denominators can divide into evenly. In the problem we solved, the denominators of the fractions were 12 and 10. To find the LCD, you can use the least common multiple (LCM) of the two denominators. We determined that the LCM of 12 and 10 is 60, which serves as the LCD in this subtraction problem. Finding the LCD ensures that each fraction references the same whole, allowing us to perform the subtraction effectively. Identifying the least common denominator allows you to rewrite each fraction as an equivalent fraction with this common denominator, which is crucial for accurate subtraction or addition of fractions.

Simplifying Fractions

Whenever you work with fractions, it's important to simplify the result whenever possible. Simplifying a fraction means reducing it to its lowest terms. This involves ensuring there's no common factor between the numerator and the denominator other than 1.In our example, after performing the subtraction, we obtained the fraction \(\frac{1}{60}\). The next step is to check if this fraction can be simplified further. However, since 1 is a prime number, and its only divisor is 1, \(\frac{1}{60}\) is already in its simplest form. Simplifying fractions is not only about neatness. It makes it easier to interpret and compare different fractions because you are working with the most reduced representation of an amount.

Common Denominators

Adding or subtracting fractions requires common denominators because, without them, the fractions are referencing different sized parts of a whole. It's a bit like trying to subtract squares from circles; without a common shape (or number), the subtraction doesn't make sense.To bring fractions to a common denominator, you essentially find a shared base to which both fractions can conform. In our case, we needed the denominators 12 and 10 to both convert into 60, the LCD. - Multiply each fraction's numerator and denominator by the number that allows their denominators to transform into the common denominator. - For \(\frac{11}{12}\), multiply by 5 to get \(\frac{55}{60}\).- For \(\frac{9}{10}\), multiply by 6 to get \(\frac{54}{60}\).With common denominators, you can easily subtract one fraction from another, as they now measure the same parts of the same whole. After subtraction, ensure to simplify the resulting fraction for the final answer.