Problem 61 - Algebra A Combined Function

Question

Perform the indicated operations. See Examples 11 through \(23 .\) $$ \frac{7}{10} \cdot \frac{5}{21} $$

Step-by-Step Solution

Answer

The result is \(\frac{1}{6}\).

Step 1: Understand Multiplying Fractions

To multiply two fractions, multiply the numerators together and multiply the denominators together. Therefore, for this problem, we will multiply the numerators 7 and 5, and the denominators 10 and 21.

Step 2: Multiply the Numerators

First, calculate the product of the numerators: \(7 \times 5 = 35\).

Step 3: Multiply the Denominators

Next, calculate the product of the denominators: \(10 \times 21 = 210\).

Step 4: Form the New Fraction

Combine the products of the numerators and the denominators to form the new fraction: \(\frac{35}{210}\).

Step 5: Simplify the Fraction

To simplify \(\frac{35}{210}\), find the greatest common divisor (GCD) of 35 and 210. In this case, the GCD is 35.Divide both the numerator and the denominator by their GCD:\(\frac{35}{35} = 1\)\(\frac{210}{35} = 6\)Thus, \(\frac{35}{210}\) simplifies to \(\frac{1}{6}\).

Key Concepts

Simplifying Fractions

Simplifying fractions is an important step in dealing with fractions effectively. It means reducing the fraction to its simplest form where the numerator and the denominator have no common divisors other than 1. This results in a fraction that is easier to understand and compare with other fractions.

To simplify a fraction like \(\frac{35}{210}\), you need to find the greatest common divisor (GCD) of the numerator and denominator. Divide both the top (numerator) and the bottom (denominator) by this GCD. In our example, dividing 35 and 210 by their GCD, 35, simplifies the fraction to \(\frac{1}{6}\). A well-simplified fraction is always useful for easier calculation and interpretation.

Numerator and Denominator

In the process of multiplying fractions, understanding the roles of the numerator and the denominator is key. The **numerator** is the top part of the fraction, and it represents how many parts of a whole we are considering. The **denominator** is the bottom part, indicating into how many parts the whole is divided.

When multiplying fractions, as in the given problem \(\frac{7}{10} \cdot \frac{5}{21}\), you multiply the numerators together \((7 \times 5 = 35)\) and then multiply the denominators together \((10 \times 21 = 210)\). These products form a new fraction, \(\frac{35}{210}\), where 35 is the new numerator and 210 is the new denominator.

• The numerator shows the accumulated parts counted by multiplying.
• The denominator shows how the total parts are defined by multiplying.

Understanding these components helps in both performing operations on fractions and ensuring accuracy in calculations.

Greatest Common Divisor (GCD)

The Greatest Common Divisor (GCD) is crucial when it comes to simplifying fractions. It refers to the largest number that divides both the numerator and the denominator without leaving a remainder. Finding the GCD is essential for reducing fractions to their simplest form.

There are multiple methods to find the GCD:

• **Prime Factorization**: Break down both numbers into their prime factors and multiply the common factors.
• **Euclidean Algorithm**: An efficient method for larger numbers, repeatedly subtracting the smaller number from the larger until zero is reached.

For our example, the GCD of 35 and 210 is 35. We divide both the numerator and the denominator by this number to simplify the fraction to \(\frac{1}{6}\). Understanding and finding the GCD ensures that fractions are reduced correctly, making computations easier.