Problem 40 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. Write the answer as a fraction or as a mixed number in simplest form. (Skills Review pp. 764-765) $$ \frac{2}{3} \cdot \frac{2}{5}+\frac{1}{5} $$

Step-by-Step Solution

Answer

The answer to the given expression simplified is \( \frac{7}{15} \)

Step 1: Multiplying Fractions

We begin by multiplying the two fractions \( \frac{2}{3} \) and \( \frac{2}{5} \). To multiply fractions, we simply multiply the numerators together and the denominators together. \( \frac{2}{3} \cdot \frac{2}{5} = \frac{2 \cdot 2}{3 \cdot 5} = \frac{4}{15} \)

Step 2: Finding a Common Denominator

Now we have the fraction \( \frac{4}{15} \) and we need to add it to \( \frac{1}{5} \). To add fractions, they must have the same denominator. In this case, the least common denominator of 15 and 5 is 15. So we will convert \( \frac{1}{5} \) to have 15 as the denominator. Doing so, \( \frac{1}{5} = \frac{3}{15} \)

Step 3: Adding Fractions

Now that we have two fractions with the same denominator, we can add them together. To add fractions, we just add the numerators, keeping the same denominator. \( \frac{4}{15} + \frac{3}{15} = \frac{7}{15} \)

Key Concepts

Multiplying Fractions

When you multiply fractions, you encounter a very straightforward process. Each fraction has a numerator and a denominator, and to multiply them, you handle these parts separately. Specifically, you multiply the numerators with each other and the denominators with each other.
For example, if you multiply \( \frac{2}{3} \) by \( \frac{2}{5} \), you proceed as follows:

• Multiply the numerators: \( 2 \times 2 = 4 \)
• Multiply the denominators: \( 3 \times 5 = 15 \)

Therefore, the product of \( \frac{2}{3} \) and \( \frac{2}{5} \) is \( \frac{4}{15} \). It’s a simple multiplication strategy that doesn’t require the fractions to have any specific properties such as having the same denominator. The result will remain in its simplest form unless indicated otherwise.

Common Denominator

Finding a common denominator is a crucial step when you want to add or subtract fractions. First, understand what a denominator is: it’s the bottom part of the fraction, indicating how many equal parts the whole is divided into.
To find a common denominator, look for the least common multiple (LCM) of the denominators involved. In this exercise, you need to add \( \frac{4}{15} \) and \( \frac{1}{5} \).

• The denominators here are 15 and 5.
• Recognize that 15 is already a common denominator of both fractions because 15 is a multiple of 5.

Therefore, the least common denominator here is 15. You convert \( \frac{1}{5} \) to \( \frac{3}{15} \) to match the denominator of \( \frac{4}{15} \), allowing these two fractions to be added together.

Adding Fractions

Once fractions have the same denominator, you can add them by simply adding the numerators. The denominator stays unchanged since it represents the common unit they share.
Let's add \( \frac{4}{15} \) and \( \frac{3}{15} \):

• Add the numerators: \( 4 + 3 = 7 \)
• Keep the denominator the same: 15

The result of adding \( \frac{4}{15} \) and \( \frac{3}{15} \) is \( \frac{7}{15} \). It's already in its simplest form as the numerator and denominator share no common factors. Adding fractions is a simple operation once they share a common denominator, turning potentially complex calculations into straightforward arithmetic.