Problem 47 - Introductory and Intermediate Algebra for College Students 4th

Question

Perform the indicated operation. Where possible, reduce the answer to its lowest terms. $$\frac{1}{10} \cdot \frac{5}{6}$$

Step-by-Step Solution

Answer

The simplified result of \(\frac{1}{10} \cdot \frac{5}{6}\) is \(\frac{1}{12}\).

Step 1: Apply the multiplication rule

To multiply two fractions, multiply the numerators together to form the new numerator and multiply the denominators together to form the new denominator. This gives: \(\frac{1}{10} \cdot \frac{5}{6} = \frac{1 \cdot 5}{10 \cdot 6}\)

Step 2: Perform the multiplications

Perform the multiplications in the numerator and denominator: \(\frac{1 \cdot 5}{10 \cdot 6} = \frac{5}{60}\)

Step 3: Simplify the fraction

Divide the numerator and denominator by their greatest common divisor (gcd) to reduce the fraction to its lowest terms. The gcd of 5 and 60 is 5. Therefore, \(\frac{5}{60}\) can be simplified as \(\frac{5 ÷ 5}{60 ÷ 5} = \frac{1}{12}\)

Key Concepts

Fraction Multiplication

When multiplying fractions, the process is straightforward: you multiply the top numbers (numerators) together and the bottom numbers (denominators) together. Taking the exercise \(\frac{1}{10} \cdot \frac{5}{6}\), you multiply the numerators 1 and 5 to get 5, and the denominators 10 and 6 to get 60, resulting in the fraction \(\frac{5}{60}\).

Remember, you don't need to find a common denominator when multiplying fractions, as you would with addition or subtraction. This makes multiplication of fractions one of the simpler operations involving fractions.

Simplifying Fractions

Simplifying fractions means reducing them to their lowest terms. To do this, you find the greatest common divisor (gcd) of the numerator and the denominator, and then divide both by this number. For example, in the fraction \(\frac{5}{60}\), the gcd is 5. By dividing both the numerator and the denominator by 5, the fraction simplifies to \(\frac{1}{12}\).

It's important to simplify fractions to make them easier to understand and work with in subsequent calculations. The simplified form is considered the most elegant and is generally preferred in mathematical expression.

Greatest Common Divisor

The greatest common divisor (gcd), also known as the greatest common factor (gcf), is the largest number that can evenly divide two or more integers. To find the gcd, you can list out the factors of each number and find the highest number that appears in all lists, or use more advanced methods like Euclid's algorithm.

For the fraction \(\frac{5}{60}\), we list the factors of 5 (1, 5) and 60 (1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60). The highest number that is a factor of both is 5. This gcd is used to simplify the fraction to its lowest terms.

Algebraic Operation

Multiplying fractions is an example of an algebraic operation, which is a step in solving algebra problems that involves arithmetic with letters and numbers. In algebra, operations on fractions follow the same rules as those with numbers. Performing these operations correctly is crucial for solving equations and simplifying expressions in algebra.

Understanding fraction multiplication as an algebraic operation helps students to later tackle more complex problems involving variables in fractions. Remember, when fractions include variables, you multiply them exactly the same way: multiply numerators with numerators and denominators with denominators, then simplify if possible.