Problem 10 - Prealgebra

Question

Use the rule for order of operations to simplify each of the following. [Examples 1–3] $$\left(\frac{2}{9}+\frac{1}{3}\right)\left(\frac{1}{5}+\frac{1}{10}\right)$$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{1}{6}\).

Step 1: Apply the Rule for Parentheses

First, simplify the expressions inside the parentheses separately. Calculate \(\frac{2}{9} + \frac{1}{3}\). In order to add fractions, they must have a common denominator. Convert \(\frac{1}{3}\) to \(\frac{3}{9}\) so both fractions have a denominator of 9.\[\frac{2}{9} + \frac{3}{9} = \frac{5}{9}\]

Step 2: Simplify the Second Parenthesis

Next, simplify \(\frac{1}{5} + \frac{1}{10}\). Change \(\frac{1}{5}\) to \(\frac{2}{10}\) so both fractions have a denominator of 10.\[\frac{2}{10} + \frac{1}{10} = \frac{3}{10}\]

Step 3: Multiply the Results of the Parentheses

Now, multiply the simplified results from each parenthesis: \(\frac{5}{9}\) and \(\frac{3}{10}\).\[\frac{5}{9} \times \frac{3}{10} = \frac{5 \times 3}{9 \times 10} = \frac{15}{90}\]

Step 4: Simplify the Fraction

Simplify \(\frac{15}{90}\) by dividing both the numerator and denominator by their greatest common divisor, which is 15.\[\frac{15}{90} = \frac{15 \div 15}{90 \div 15} = \frac{1}{6}\]

Key Concepts

Understanding Fractions

Fractions are a way to express numbers that are not whole. They are written as a ratio of two numbers, with a top number called the numerator and a bottom number called the denominator. A fraction like \( \frac{3}{4} \) tells you there are 3 parts out of a total of 4 parts. This concept is essential in understanding parts of a whole and is widely used in mathematics.

You can think of the denominator as "how many equal parts the whole is divided into," and the numerator as "how many of those parts you have." For example, in \( \frac{1}{3} \), the whole is divided into 3 equal parts, and you have 1 part.

• The numerator (top number) indicates how many parts you have.
• The denominator (bottom number) indicates the number of equal parts the whole is divided into.

This basic understanding of fractions will help you in adding, subtracting, multiplying, and dividing them.

Finding a Common Denominator

When adding or subtracting fractions, they must have a common denominator. This means that the bottom numbers (denominators) must be the same. The common denominator is a shared multiple of the denominators of the fractions involved.

For instance, to add \( \frac{2}{9} \) and \( \frac{1}{3} \), you should rewrite \( \frac{1}{3} \) as \( \frac{3}{9} \). This gives them the same denominator, allowing you to easily add them: \( \frac{2}{9} + \frac{3}{9} = \frac{5}{9} \)

Here’s how you find a common denominator:

• Identify the denominators of the fractions you want to add or subtract.
• Determine the least common multiple (LCM) of these denominators.
• Adjust the fractions to have this common denominator by finding equivalent fractions.

This helps ensure that the fractions are easy to work with, making addition or subtraction possible.

Multiplying Fractions

Multiplying fractions is straightforward and does not require a common denominator. You simply multiply the numerators together and the denominators together. For example, when you multiply \( \frac{5}{9} \) and \( \frac{3}{10} \):

\[ \frac{5}{9} \times \frac{3}{10} = \frac{5 \times 3}{9 \times 10} = \frac{15}{90} \]

Multiplication of fractions could be summarized into these steps:

• Multiply the numerators to get the new numerator.
• Multiply the denominators to get the new denominator.
• Write the result as a new fraction.

After multiplying, you may need to simplify the fraction to its simplest form, making it easier to understand or use in calculations.

Simplifying Fractions

Simplifying a fraction means to make it as simple as possible while keeping it equal in value. The goal is to find the smallest equivalent fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD).

For example, after multiplying to get \( \frac{15}{90} \), you can simplify this fraction. First, find the GCD of 15 and 90, which is 15. Divide both the numerator and the denominator by this GCD:

\[ \frac{15}{90} = \frac{15 \div 15}{90 \div 15} = \frac{1}{6} \]

To simplify a fraction:

• Identify the GCD of the numerator and the denominator.
• Divide both the numerator and the denominator by the GCD.
• The result is the fraction in its simplest form.

Simplifying makes fractions easier to interpret and work with, especially in solving algebraic equations or real-world problems.