Problem 63 - Fundamentals of Mathematics

Question

For the following problems, find the products. Be sure to reduce. $$\frac{21}{25} \cdot \frac{15}{14}$$

Step-by-Step Solution

Answer

The product is \( \frac{9}{10} \).

Step 1: Identify the Fractions

We are given two fractions: \(\frac{21}{25}\) and \(\frac{15}{14}\). Our task is to find their product and ensure it is in its simplest form.

Step 2: Multiply the Numerators

The numerators are 21 and 15. Multiply them together. \[ 21 \times 15 = 315 \]

Step 3: Multiply the Denominators

The denominators are 25 and 14. Multiply them together. \[ 25 \times 14 = 350 \]

Step 4: Combine the Product

Combine the results from Steps 2 and 3 to form a new fraction: \[ \frac{315}{350} \]

Step 5: Simplify the Fraction

To simplify \(\frac{315}{350}\), find the greatest common divisor (GCD) of 315 and 350. Start with their prime factorizations: - 315: \( 3^2 \times 5 \times 7 \)- 350: \( 2 \times 5^2 \times 7 \).The GCD is 35. Divide both the numerator and the denominator by 35. \[ \frac{315 \div 35}{350 \div 35} = \frac{9}{10} \]

Step 6: Verify Simplification

Ensure the fraction \( \frac{9}{10} \) is in its simplest form. Check if there are any common factors other than 1. Since 9 and 10 have no common divisors other than 1, \( \frac{9}{10} \) is the simplest form.

Key Concepts

Simplifying Fractions

Simplifying fractions is an essential skill in mathematics. It makes fractions easier to understand and work with. To simplify a fraction, our goal is to reduce it to its smallest possible form.

A fraction is simplified when the numerator and the denominator have no common factors other than 1. To simplify a fraction like \( \frac{315}{350} \), we can follow a simple process:

• Identify any common factors that both numbers share.
• Divide both the numerator and the denominator by their greatest common factor.
• Verify the result to ensure that it is fully simplified.

Use these steps to effectively reduce fractions in your homework or any math problem.

Multiplying Fractions

Multiplying fractions might sound tricky, but it's simpler than you think. The process involves a straightforward two-step multiplication of the numerators and denominators.

Let's break it down:

• First, multiply the numerators (the top numbers) of the fractions together.
• Next, multiply the denominators (the bottom numbers) of the fractions together.
• Finally, combine your results into a new fraction.

For example, multiplying \( \frac{21}{25} \) by \( \frac{15}{14} \) gives a new fraction \( \frac{315}{350} \) from the products of \( 21 \times 15 \) and \( 25 \times 14 \). This fraction can then be simplified using the steps for simplification. With practice, multiplying fractions can be an easy and quick task.

Greatest Common Divisor

The greatest common divisor (GCD) helps us reduce fractions to their simplest form. The GCD of two numbers is the largest integer that divides them both without leaving a remainder.

How do we find the GCD of two numbers? Here's a step-by-step method:

• List the factors of each number. Factors are numbers that divide the original number exactly.
• Identify the common factors shared by both numbers.
• Choose the largest factor that both numbers share. This is the GCD.

For instance, when simplifying \( \frac{315}{350} \), we notice both numbers share a common factor of 35. Thus, 35 is the GCD. By dividing both the numerator and the denominator by the GCD, we simplify the fraction to \( \frac{9}{10} \).Understanding the concept of the GCD greatly aids in simplifying fractions and other math operations.