Problem 73 - Elementary Algebra

Question

\(\frac{1}{4}+\frac{5}{8} \cdot \frac{2}{3}\)

Step-by-Step Solution

Answer

\(\frac{2}{3}\)

Step 1 - Multiply the fractions

First, multiply the fractions \(\frac{5}{8} \cdot \frac{2}{3}\). Multiply the numerators together and the denominators together: \(\frac{5 \cdot 2}{8 \cdot 3} = \frac{10}{24}\).

Step 2 - Simplify the fraction

Simplify the fraction \(\frac{10}{24}\) by finding the greatest common divisor of 10 and 24, which is 2. Divide both the numerator and the denominator by 2: \(\frac{10 \div 2}{24 \div 2} = \frac{5}{12}\).

Step 3 - Add the fractions

Add \(\frac{1}{4}\) and \(\frac{5}{12}\) by finding a common denominator. The least common multiple of 4 and 12 is 12. Convert \(\frac{1}{4}\) to \(\frac{3}{12}\). Now add the fractions: \(\frac{3}{12} + \frac{5}{12} = \frac{8}{12}\).

Step 4 - Simplify the final result

Simplify \(\frac{8}{12}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 4: \(\frac{8 \div 4}{12 \div 4} = \frac{2}{3}\).

Key Concepts

Fraction Multiplication

Multiplying fractions is a straightforward process! You multiply the numerators together and the denominators together. Let's take the fractions \(\frac{5}{8}\) and \(\frac{2}{3}\). To multiply them, we perform the following steps: \(\frac{5}{8} \times \frac{2}{3} = \frac{5 \times 2}{8 \times 3} = \frac{10}{24}\).

Easy, right? The result, \(\frac{10}{24}\), might look a little messy, which leads us to the next concept: simplifying fractions.

Simplifying Fractions

After multiplying fractions, the result might not be in the simplest form. Simplifying a fraction means making it as simple as possible. To simplify \(\frac{10}{24}\), find the greatest common divisor (GCD) of 10 and 24. The GCD here is 2.

Now, divide both the numerator and the denominator by their GCD: \(\frac{10 \text{ ÷ } 2}{24 \text{ ÷ } 2} = \frac{5}{12}\).

Voila! Your fraction is now easier to read and more elegant: \(\frac{5}{12}\). Understanding how to simplify makes future calculations less overwhelming and more accurate.

Fraction Addition

Adding fractions requires a bit more care. You must have a common denominator before you can add the numerators. In our example, we need to add \(\frac{1}{4}\) and \(\frac{5}{12}\).

First, we'll find a common denominator. The least common multiple (LCM) of 4 and 12 is 12. So, \(\frac{1}{4}\) becomes \(\frac{3}{12}\).

Now, add: \(\frac{3}{12} + \frac{5}{12} = \frac{8}{12}\).

We have our sum, but it's not yet in its simplest form.

Finding Common Denominators

Finding a common denominator is essential for adding and subtracting fractions. You need a common denominator to combine the fractions correctly. Let’s break down the steps using our example: adding \(\frac{1}{4}\) and \(\frac{5}{12}\).

1. Identify the denominators of the fractions (4 and 12).
2. Calculate the LCM of these denominators. In this case, the LCM of 4 and 12 is 12.
3. Convert each fraction to have this common denominator: Converting \(\frac{1}{4}\) to \(\frac{3}{12}\) because \(\frac{1 \times 3}{4 \times 3} = \frac{3}{12}\).

With common denominators, addition becomes simple: just add the numerators \(\frac{3}{12} + \frac{5}{12} = \frac{8}{12}\).

Finally, don't forget to simplify \(\frac{8}{12} = \frac{2}{3}\).