Problem 70 - Algebra A Combined Function

Question

Identify each statement as true or false. If false, correct the multiplication. See the Concept Check in this section. $$ \frac{2}{3} \cdot \frac{2}{4}=\frac{2}{7} $$

Step-by-Step Solution

Answer

The statement is false; the correct multiplication is \( \frac{2}{3} \cdot \frac{2}{4} = \frac{1}{3} \).

Step 1: Review the Given Problem

The statement given is \( \frac{2}{3} \cdot \frac{2}{4} = \frac{2}{7} \). We need to verify if this statement is correct or incorrect.

Step 2: Multiply the Fractions

To check the accuracy of the given statement, perform the multiplication of the fractions. Multiply the numerators: \( 2 \times 2 = 4 \) and multiply the denominators: \( 3 \times 4 = 12 \). This results in: \( \frac{4}{12} \).

Step 3: Simplify the Result

Simplify the fraction \( \frac{4}{12} \) by finding the greatest common divisor (GCD) of 4 and 12, which is 4. Divide the numerator and the denominator by the GCD: \( \frac{4}{4} = 1 \) and \( \frac{12}{4} = 3 \). Thus, \( \frac{4}{12} \) simplifies to \( \frac{1}{3} \).

Step 4: Compare Results

Compare the simplified result \( \frac{1}{3} \) with the statement result \( \frac{2}{7} \). Since \( \frac{1}{3} eq \frac{2}{7} \), the original statement is false.

Step 5: Correct the Statement

The corrected multiplication should be \( \frac{2}{3} \cdot \frac{2}{4} = \frac{1}{3} \).

Key Concepts

Simplifying Fractions

Simplifying fractions is the process of reducing them to their simplest form. It means making the fraction as simple as possible, which usually involves making the numerator and denominator smaller. This is achieved by dividing both the numerator and the denominator by their greatest common divisor (GCD). For instance, consider the fraction \(\frac{4}{12}\). To simplify it:

• Find the GCD of 4 and 12, which is 4.
• Divide both the numerator and the denominator by 4.
• This results in \(\frac{1}{3}\).

Simplifying fractions is important in math because it makes calculations easier and the results more understandable. A simplified fraction is usually preferred when comparing fractions, adding, subtracting, multiplying, or dividing them.

Comparing Fractions

Comparing fractions is an essential skill that helps in determining which fraction is larger or smaller, or if they are equal. One way to compare fractions is by simplifying them first. In our original example, simplify \(\frac{4}{12}\) to \(\frac{1}{3}\). Now, compare it to another fraction, say \(\frac{2}{7}\).To compare, consider:

• Convert both fractions to have a common denominator.
• You can use cross-multiplication: \(1 \cdot 7 = 7\) and \(2 \cdot 3 = 6\).
• Since 7 is greater than 6, \(\frac{1}{3}\) is greater than \(\frac{2}{7}\).

Comparing fractions can also be done visually or by converting them to decimal form, but understanding these techniques helps to solidify the concept in mathematical terms.

Greatest Common Divisor

The greatest common divisor (GCD) of two numbers is the largest number that divides both numbers without leaving a remainder. Finding the GCD is crucial in simplifying fractions. Here's how you can find the GCD:1. **List the factors**: Write down all factors of each number. - For 4: 1, 2, 4 - For 12: 1, 2, 3, 4, 6, 122. **Identify the common factors**: Identify the numbers that appear in both lists. - Common factors: 1, 2, 43. **Choose the greatest**: The largest number in the list of common factors is the GCD.In this example, the GCD of 4 and 12 is 4. With this knowledge, you can simplify fractions efficiently, like turning \(\frac{4}{12}\) into \(\frac{1}{3}\). Understanding the GCD concept helps in dealing with more complex mathematical problems where fraction simplification is necessary.