Problem 80 - Algebra 1: Concepts and Skills

Question

Write the answer as a fraction or as a mixed number in simplest form. $$ \frac{6}{7} \cdot \frac{3}{5} $$

Step-by-Step Solution

Answer

The product of the fractions \( \frac{6}{7} \cdot \frac{3}{5}\) is \( \frac{18}{35}\)

Step 1: Multiply Numerators

First, multiply the numerators of the two fractions. Here, the numerators are 6 and 3. Therefore, the product of the numerators is \(6 \times 3 = 18\).

Step 2: Multiply Denominators

Next, multiply the denominators of the two fractions, which are 7 and 5. As a result, the product of the denominators is \(7 \times 5 = 35\).

Step 3: Write as a New Fraction

Combine the product of the numerators and the product of the denominators to form a new fraction. That is \( \frac{18}{35} \).

Step 4: Simplify the Fraction

Check if the fraction can be simplified further. However, there is no number other than 1 that divides both 18 and 35. Therefore, the fraction is already in its simplest form, \( \frac{18}{35} \).

Key Concepts

Understanding Numerators

In the world of fractions, the numerator plays a crucial role. It is the number located at the top of the fraction. The numerator tells us how many parts of the whole or set we are considering. For example, in the fraction \(\frac{6}{7}\), the number 6 is the numerator. Similarly, in \(\frac{3}{5}\), 3 is the numerator.
Numerators are the numbers we focus on first when multiplying fractions. When you multiply fractions, you multiply the numerators together to find the new numerator for your product. It's just like counting apples: if you have groups of apples, you'll multiply the number of apples in each group to see how many apples you have in total. Remember, it's the top numbers that we multiply first during fraction multiplication.

Delving into Denominators

Denominators are the numbers below the line in fractions. They represent the total number of equal parts the whole is divided into. For instance, in the fraction \(\frac{6}{7}\), the denominator is 7, indicating that the whole is divided into 7 parts. In \(\frac{3}{5}\), the denominator is 5.
When multiplying fractions, you multiply the denominators to get the denominator of the new fraction. This process ensures that the fractions are combined correctly to reflect the division of the whole. Think of it like stacking floors of a building; each level represents a chunk of space that affects the entire structure.
Remember: the denominator tells us the size of the pieces, while the numerator counts them.

Simplifying to the Simplest Form

Once you multiply the numerators and denominators, your work isn't over just yet. The next step in understanding fraction multiplication is simplifying the fraction, if possible. Simplifying means reducing the fraction to its smallest terms without changing its value. To simplify \(\frac{18}{35}\), look for the greatest common divisor (GCD) of both numbers.

• If there's a number larger than 1 that divides both the numerator and the denominator, divide by that number to simplify.
• In the given solution, no such number exists for 18 and 35 other than 1, so \(\frac{18}{35}\) is already in its simplest form.

Achieving the simplest form not only makes your answer more elegant and easier to compare with other fractions, but it also represents the fraction in the most concise way.

Exploring Improper Fractions

Improper fractions are a key concept in understanding fractions fully. An improper fraction is one where the numerator is equal to or larger than the denominator. This means that the fraction represents a number that is equal to or greater than one.

• For example, \(\frac{9}{7}\) is an improper fraction since 9 is greater than 7.
• This concept is the opposite of a proper fraction, where the numerator is less than the denominator, representing a number less than one.

Knowing how to work with improper fractions is essential, especially when converting between mixed numbers and improper fractions. Although \(\frac{18}{35}\) from the original exercise is a proper fraction, having an understanding of improper fractions will aid in dealing with a wider variety of mathematical problems.