Problem 3 - Technical Mathematics with Calculus

Question

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

Step-by-Step Solution

Answer

\(\frac{2}{3} \times \frac{9}{7} = \frac{6}{7}\)

Step 1: Multiplying Fractions

To multiply fractions, multiply the numerators (top numbers) together and the denominators (bottom numbers) together separately. So, multiply 2 by 9 for the numerator, and multiply 3 by 7 for the denominator.

Step 2: Calculating Numerator and Denominator

Multiplying the numerators: 2 \times 9 = 18. Multiplying the denominators: 3 \times 7 = 21. The product of the fractions is now \(\frac{18}{21}\).

Step 3: Reducing the Fraction

To reduce the fraction, find the greatest common divisor (GCD) of the numerator and the denominator and divide both by that number. The GCD of 18 and 21 is 3. Divide both numerator and denominator by 3 to reduce the fraction to its simplest form.

Step 4: Simplify the Fraction

Dividing the numerator and denominator by 3: \frac{18 \div 3}{21 \div 3} = \frac{6}{7}. The final answer is \(\frac{6}{7}\).

Key Concepts

Reducing Fractions

Reducing fractions is an essential skill in mathematics, ensuring that answers are presented in their simplest form. It involves dividing both the numerator (the top number) and the denominator (the bottom number) by their greatest common divisor (GCD). For example, if you have the fraction \( \frac{18}{21} \) after multiplying two fractions, you look for the largest number that divides evenly into both 18 and 21.

To find this number, list the factors of each:

• Factors of 18: 1, 2, 3, 6, 9, 18
• Factors of 21: 1, 3, 7, 21

The largest number found in both lists is 3, which means the GCD of 18 and 21 is 3. Dividing both the numerator and the denominator by 3 gives you \( \frac{6}{7} \), which is the reduced form of the fraction.

Greatest Common Divisor (GCD)

The greatest common divisor (GCD), also known as the greatest common factor (GCF), is the largest number that can evenly divide two or more integers. Finding the GCD is crucial when simplifying fractions.

To calculate the GCD of two numbers, such as 18 and 21, you can use various methods, including the Euclidean algorithm, prime factorization, or simply listing out the factors and choosing the highest common one. Once the GCD is determined, it serves as the divisor to reduce fractions to their simplest form. In our example, the GCD of 18 and 21 is 3, and using it to reduce the fraction ensures a minimal and more comprehensible result.

Simplifying Fractions

Simplifying fractions is a process that makes a fraction as simple as possible, meaning that the numerator and denominator share no common divisors other than 1. It's a fundamental step in presenting your answer neatly and understandably.

In our earlier example, \( \frac{18}{21} \) is simplified by dividing both the numerator and the denominator by their GCD, which is 3. After doing this, the simplified fraction is \( \frac{6}{7} \). Not only is the fraction now in its most reduced form, but it also becomes much easier to interpret and use in subsequent calculations or comparisons with other fractions.

Arithmetic Operations with Fractions

Arithmetic operations with fractions involve addition, subtraction, multiplication, and division. Multiplication is perhaps the most straightforward operation: simply multiply the numerators to find the new numerator, and multiply the denominators to find the new denominator.

For instance, to multiply \( \frac{2}{3} \) by \( \frac{9}{7} \), you calculate \(2 \times 9 \) for the new numerator and \(3 \times 7 \) for the new denominator, which results in \( \frac{18}{21} \). This initial product often requires further simplification, as in our case, where we reduce it to \( \frac{6}{7} \). Other arithmetic operations with fractions, like addition or subtraction, typically require finding a common denominator before carrying out the operation to ensure accuracy.