Problem 34 - Algebra and Trigonometry

Question

Perform the indicated operations. 0.25\(\left(\frac{8}{9}+\frac{1}{2}\right)\)

Step-by-Step Solution

Answer

The result is \( \frac{25}{72} \).

Step 1: Addition of fractions

Add the fractions \( \frac{8}{9} \) and \( \frac{1}{2} \). To do this, find a common denominator, which is 18 in this case. Convert \( \frac{8}{9} \) to \( \frac{16}{18} \) and \( \frac{1}{2} \) to \( \frac{9}{18} \). Now add them: \( \frac{16}{18} + \frac{9}{18} = \frac{25}{18} \).

Step 2: Multiplication by 0.25

Multiply the sum of the fractions \( \frac{25}{18} \) by 0.25. To do this, convert 0.25 to a fraction, which is \( \frac{1}{4} \). Then multiply: \( \frac{25}{18} \times \frac{1}{4} = \frac{25}{72} \).

Step 3: Simplifying the fraction

The fraction \( \frac{25}{72} \) is already in its simplest form since 25 and 72 have no common factors other than 1.

Key Concepts

Common Denominator

When working with fractions, finding a common denominator is crucial for addition or subtraction. A common denominator is a shared multiple of the denominators of the fractions involved. By finding this shared multiple, you can rewrite each fraction so that all have the same denominator, making calculations straightforward. For instance, consider adding the fractions \(\frac{8}{9}\) and \(\frac{1}{2}\). Here, the denominators are 9 and 2. The smallest common multiple of 9 and 2 is 18. This becomes the common denominator. Thus, you transform \(\frac{8}{9}\) to \(\frac{16}{18}\) and \(\frac{1}{2}\) to \(\frac{9}{18}\) by multiplying both the numerator and the denominator by appropriate factors. This step ensures that both fractions are expressed with the same terms, enabling seamless fraction addition or subtraction.

Fraction Addition

Adding fractions is simple once they have a common denominator. With fractions like \(\frac{16}{18}\) and \(\frac{9}{18}\), the process becomes a matter of adding the numerators while keeping the denominator unchanged.

• Add the numerators: 16 and 9.
• The sum is 25, so the resultant fraction is \(\frac{25}{18}\).

In fraction addition, ensure the result is in its simplest form by dividing the numerator and the denominator by their greatest common factor. For \(\frac{25}{18}\), it is already simplified since 25 and 18 share no common factors other than 1. This efficient approach keeps your calculations accurate and your results simplified.

Fraction Multiplication

Multiplying fractions is a straightforward process. Multiply the numerators to get the new numerator, and multiply the denominators to find the new denominator. In this exercise, you multiply the fraction \(\frac{25}{18}\) by \(\frac{1}{4}\), which involves the following steps:

• Multiply the numerators: 25 \(\times\) 1 = 25.
• Multiply the denominators: 18 \(\times\) 4 = 72.

So, \(\frac{25}{18} \times \frac{1}{4} = \frac{25}{72}\). It's important to always check if the resulting fraction is in its simplest form. In this scenario, \(\frac{25}{72}\) is already simplified. Fraction multiplication, unlike addition or subtraction, doesn't require a common denominator but focuses directly on multiplying across numerators and denominators.