Problem 67 - Fundamentals of Mathematics

Question

For the following problems, find the products. Be sure to reduce. $$\frac{14}{15} \cdot \frac{21}{28} \cdot \frac{45}{7}$$

Step-by-Step Solution

Answer

The product is \(\frac{9}{2}\).

Step 1: Multiply the Numerators

First, multiply all the numerators of the fractions: \( 14 \times 21 \times 45 \). Calculate this product step-by-step: - First, \( 14 \times 21 = 294 \).- Then, multiply the result by 45: \( 294 \times 45 = 13230 \). So, the product of the numerators is 13230.

Step 2: Multiply the Denominators

Next, multiply all the denominators of the fractions: \( 15 \times 28 \times 7 \). Calculate this product step-by-step: - First, \( 28 \times 7 = 196 \).- Then, multiply the result by 15: \( 15 \times 196 = 2940 \). So, the product of the denominators is 2940.

Step 3: Form the New Fraction

Combine the results from Steps 1 and 2 to form a new fraction by placing the product of the numerators over the product of the denominators: \[\frac{13230}{2940}\].

Step 4: Simplify the Fraction

To simplify \(\frac{13230}{2940}\), find the greatest common divisor (GCD) of 13230 and 2940. Using prime factorization or Euclidean algorithm, we find that the GCD is 210. - Divide both the numerator and the denominator by 210: \(\frac{13230 \div 210}{2940 \div 210} = \frac{63}{14}\).

Step 5: Further Simplify the Fraction

Continue simplifying if possible. Divide the numerator and the denominator by their GCD (in this case, 7): \(\frac{63 \div 7}{14 \div 7} = \frac{9}{2}\).So, the simplified product is \(\frac{9}{2}\).

Key Concepts

Simplifying Fractions

Simplifying fractions is an essential process that involves reducing a fraction to its simplest form, making it easier to understand and use. This process entails finding a common factor of the numerator and the denominator. You divide both by this common factor, ensuring that the value of the fraction remains unchanged.

For example, if we have the fraction \( \frac{63}{14} \), we look for the largest factor that divides both the numerator (63) and the denominator (14) equally. In this case, it's 7. Dividing both terms by 7, we get \( \frac{9}{2} \).

Simplifying fractions improves the ease of interpretation and calculation. Always check if a fraction can be further simplified to avoid overly complex numbers.

Greatest Common Divisor

The Greatest Common Divisor (GCD), sometimes called the greatest common factor (GCF), is the largest number that divides two or more numbers without leaving a remainder. Finding the GCD is a crucial step in simplifying fractions as it helps reduce the fraction to its simplest form.

• Identify the numbers you need to find the GCD for. In this scenario, 13230 and 2940 are your targets.
• Use methods like prime factorization or Euclidean algorithm to determine the GCD.
• In the example given, the GCD of 13230 and 2940 is 210.

After identifying the GCD, divide the numerator and the denominator by this number to simplify the fraction into its simplest form. Understanding the concept of GCD is instrumental in manipulating fractions efficiently.

Numerator and Denominator

In a fraction, the numerator and the denominator are crucial terms distinguishing its two parts. The numerator is the top number, indicating how many parts of a whole we are dealing with, while the denominator is the bottom number, showing the whole is divided into how many parts.

• In the calculation \( \frac{13230}{2940} \), the numerator is 13230, and the denominator is 2940.
• Multiplying the numerators and denominators of fractions during an operation is a fundamental process.
• The order matters; multiply all numerators together to find the new numerator, and the denominators together to find the new denominator.

These parts of a fraction allow us to accurately represent parts of a whole, particularly important in rational arithmetic. Understanding how numerators and denominators interact is essential for proper fraction operations.

Prime Factorization

Prime factorization is the process of expressing a number as a product of its prime factors. This technique is incredibly useful for finding the GCD when simplifying fractions.

• Break down each number into its prime components. For instance, 13230 is reduced into prime factors.
• Identify common prime factors between the numerator and the denominator.
• Multiply these common primes to find the GCD.

Prime factorization simplifies computations and provides solutions for more complex numerical problems, such as identifying the GCD. It allows you to see clearly which numbers are shared between the numerator and the denominator, aiding in simplification.