Problem 8 - Technical Mathematics with Calculus

Question

Multiply and reduce. Do some by calculator. $$\frac{3}{5} \times \frac{2}{7} \times \frac{5}{9}$$

Step-by-Step Solution

Answer

The simplified result of multiplying the fractions \(\frac{3}{5} \times \frac{2}{7} \times \frac{5}{9}\) is \(\frac{2}{21}\).

Step 1: Multiply the numerators

Multiply the numerators (the top numbers) of all the fractions together. In this case, multiply 3, 2, and 5: $$3 \times 2 \times 5 = 30$$

Step 2: Multiply the denominators

Multiply the denominators (the bottom numbers) of all the fractions together. Multiply 5, 7, and 9: $$5 \times 7 \times 9 = 315$$

Step 3: Form the new fraction

Combine the results from Step 1 and Step 2 to form a new fraction: $$\frac{30}{315}$$

Step 4: Reduce the fraction

To simplify the fraction, find the greatest common divisor (GCD) of the numerator and the denominator. Use a calculator to find the GCD of 30 and 315, which is 15. Divide both the numerator and the denominator by the GCD: $$\frac{30 \div 15}{315 \div 15} = \frac{2}{21}$$

Key Concepts

Numerators and Denominators

Understanding the roles of numerators and denominators is crucial when working with fractions. In any fraction, the numerator is the number above the fraction bar, indicating how many parts we have. The denominator, below the fraction bar, tells us into how many equal parts the whole is divided.

For example, in the fraction \(\frac{3}{5}\), 3 is the numerator and 5 is the denominator. This means we have 3 parts out of a whole that is divided into 5 equal parts. When multiplying fractions like \(\frac{3}{5} \times \frac{2}{7} \times \frac{5}{9}\), we follow this principle for each fraction, multiplying all numerators together to find the new numerator, and all denominators together for the new denominator.

Simplifying Fractions

Simplifying fractions involves rewriting them in the simplest form where the numerator and denominator have no common factors other than 1. This process makes fractions easier to understand and compare.

It is accomplished by dividing both the numerator and the denominator by their greatest common divisor (GCD). For instance, in the fraction \(\frac{30}{315}\), we identify that 30 and 315 are both divisible by 15. By performing the division \(\frac{30 \/ 15}{315 \/ 15}\), we simplify the fraction to its lowest terms, resulting in a more comprehensible fraction: \(\frac{2}{21}\).

Greatest Common Divisor (GCD)

The Greatest Common Divisor (GCD), also known as the greatest common factor, is the largest positive integer that divides two or more integers without a remainder. Finding the GCD is an important step in the process of reducing fractions.

To determine the GCD, various methods can be used such as factoring, using the Euclidean algorithm, or simply by using a calculator as seen with the GCD of 30 and 315. The GCD of these numbers is 15, enabling us to simplify the fraction \(\frac{30}{315}\) to \(\frac{2}{21}\) as noted in the previous steps.

Reducing Fractions

Reducing fractions means transforming them into their simplest form without changing their value. It is done by dividing both the numerator and denominator by their GCD. Reduced fractions are easier to work with, especially in addition, subtraction, or comparison of fractions.

Using our previous example, \(\frac{30}{315}\) is reduced by dividing both terms by the GCD, who is 15. The end result is \(\frac{2}{21}\), which is the fraction in its reduced form. This cannot be simplified further since 2 and 21 do not have any common divisors other than 1.