Problem 30 - Beginning and Intermediate Algebra

Question

Simplify each expression. \(\frac{3}{4} \cdot \frac{1}{2}+\frac{2}{3}\)

Step-by-Step Solution

Answer

The simplified expression is \( \frac{25}{24} \).

Step 1: Multiply the Fractions

The expression starts with a multiplication of two fractions: \( \frac{3}{4} \cdot \frac{1}{2} \). Multiply the numerators: \( 3 \times 1 = 3 \), and the denominators: \( 4 \times 2 = 8 \). So, \( \frac{3}{4} \cdot \frac{1}{2} = \frac{3}{8} \).

Step 2: Find a Common Denominator

To add \( \frac{3}{8} \) and \( \frac{2}{3} \), we need a common denominator. The least common denominator of 8 and 3 is 24.

Step 3: Convert to Common Denominator

Convert \( \frac{3}{8} \) to a fraction with denominator 24. Multiply both the numerator and denominator by 3 to get: \( \frac{3 \times 3}{8 \times 3} = \frac{9}{24} \). Convert \( \frac{2}{3} \) similarly by multiplying by 8: \( \frac{2 \times 8}{3 \times 8} = \frac{16}{24} \).

Step 4: Add the Fractions

Now, add the fractions with the common denominator: \( \frac{9}{24} + \frac{16}{24} = \frac{9 + 16}{24} = \frac{25}{24} \).

Step 5: Simplify the Result

The sum \( \frac{25}{24} \) is an improper fraction. Since there are no common factors between 25 and 24 other than 1, it is already in its simplest form. If needed, it can also be expressed as a mixed number: 1 \(\frac{1}{24}\).

Key Concepts

Fractions

Fractions are a way of representing a part of a whole. Each fraction consists of a numerator and a denominator. The numerator tells us how many parts we have, and the denominator tells us into how many equal parts the whole is divided.

• If the numerator is smaller than the denominator, the fraction is "proper." Examples include \(\frac{3}{4}\) or \(\frac{1}{2}\).
• If the numerator is larger, the fraction is "improper," such as \(\frac{5}{3}\).

Understanding how to manipulate fractions is essential. You need to know how to multiply, divide, add, and subtract them to simplify expressions correctly. For instance, when you multiply fractions, as in \(\frac{3}{4} \cdot \frac{1}{2}\), you multiply the numerators together and the denominators together to get a new fraction. This process is straightforward but very essential.

Common Denominator

To perform addition or subtraction with fractions, they must share a common denominator. This means converting each fraction so they have the same denominator while retaining the same overall value.
Finding a common denominator is akin to finding a common language between fractions. To find one, you can often use the least common multiple (LCM) of the denominators.

• The LCM of 8 and 3 is 24. This makes 24 the smallest number divisible by both 8 and 3.

Transforming fractions like \(\frac{3}{8}\) and \(\frac{2}{3}\) into equivalent fractions with a common denominator of 24 goes like this:

• Multiply \(\frac{3}{8}\) by \(3/3\) to get \(\frac{9}{24}\).
• Multiply \(\frac{2}{3}\) by \(8/8\) to get \(\frac{16}{24}\).

This conversion is crucial, as it allows the fractions to maintain their proportions and be added together.

Improper Fractions

An improper fraction is when the numerator is greater than or equal to the denominator. Such fractions represent a value greater than or equal to one. They're often encountered in calculations involving the addition or multiplication of mixed numbers or fractions.
In the step-by-step solution, we get \(\frac{25}{24}\) as the result, which is an improper fraction. While this fraction is already in its simplest form, you might change it into a mixed number for easier interpretation. Here's how:

• Divide 25 by 24. The quotient (1) is the whole number part.
• The remainder (1) becomes the numerator of the fractional part after division.
• This gives us the mixed number 1 \(\frac{1}{24}\).

Improper fractions are not incorrect; they just need an understanding of what they signify. They often convey more intuitive results when expressed as mixed numbers. Nevertheless, keeping the fraction improper can be useful, especially when performing operations requiring the use of fractions.