Problem 74 - Elementary Algebra

Question

\(\frac{1}{6}+\frac{2}{3} \cdot \frac{1}{8}\)

Step-by-Step Solution

Answer

The answer is \ \ \( \frac{1}{4} \).

Step 1: Identify the Operations

Observe the given expression: \ \ \(\frac{1}{6} + \frac{2}{3} \times \frac{1}{8}\)\ There are two operations to perform: multiplication and addition.

Step 2: Perform the Multiplication

Start with the multiplication \ \ \(\frac{2}{3} \times \frac{1}{8}\).\ Multiply the numerators together and the denominators together: \ \ \(\frac{2 \times 1}{3 \times 8} = \frac{2}{24}\).\ Simplify the fraction by dividing the numerator and denominator by their greatest common divisor (2): \ \ \(\frac{2 \text{ (numerator) }}{24 \text{ (denominator) }} = \frac{1}{12}\)

Step 3: Add the Fractions

Now, add \ \ \(\frac{1}{6} + \frac{1}{12}\).\ To add these fractions, find a common denominator.\ The least common multiple (LCM) of 6 and 12 is 12.\ Convert the fraction for the common denominator: \ \ \(\frac{1}{6} = \frac{2}{12}\).\ Now add the fractions: \ \ \(\frac{2}{12} + \frac{1}{12} = \frac{3}{12}\).\ Simplify the fraction by dividing the numerator and denominator by their greatest common divisor (3): \ \ \(\frac{3}{12} = \frac{1}{4}\)

Key Concepts

Multiplying Fractions

Multiplying fractions is straightforward. You multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. For example, if you have \(\frac{2}{3} \times \frac{1}{8}\), you multiply 2 by 1 to get the new numerator and 3 by 8 to get the new denominator. So, \(\frac{2}{3} \times \frac{1}{8} = \frac{2 \times 1}{3 \times 8} = \frac{2}{24}\). Always remember to simplify your fraction if possible. In this case, both the numerator and the denominator can be divided by 2. Therefore, \(\frac{2}{24} = \frac{1}{12}\). Simplifying fractions helps ensure that your answer is in its simplest form.

Adding Fractions

To add fractions, they must have a common denominator. This means the bottom numbers of both fractions should be the same. If the fractions don't have a common denominator, you'll need to find one. Let's look at the example \(\frac{1}{6} + \frac{1}{12}\). Here, the common denominator is 12, because it's the least common multiple of both 6 and 12. So, you convert \(\frac{1}{6}\) to \(\frac{2}{12}\) since \(\frac{1}{6} = \frac{2}{12}\). Now that the denominators are the same, you can add the numerators directly: \(\frac{2}{12} + \frac{1}{12} = \frac{3}{12}\). Next, simplify the fraction if possible. In this case, \(\frac{3}{12}\) simplifies to \(\frac{1}{4}\) because both 3 and 12 can be divided by 3.

Simplifying Fractions

Simplifying fractions makes them easier to understand and work with. A fraction is in its simplest form when both the numerator and denominator are as small as possible (other than being 1). To simplify fractions, divide both the numerator and the denominator by their greatest common divisor (GCD). For example, in the fraction \(\frac{6}{8}\), both 6 and 8 can be divided by 2. So, \(\frac{6 \text{(numerator)}}{8 \text{(denominator)}} = \frac{6 \div 2}{8 \div 2} = \frac{3}{4}\). Always check if both the numerator and denominator share a common divisor and keep dividing until no further simplification is possible.