Problem 2 - Technical Mathematics with Calculus

Question

Multiply and reduce. Do some by calculator. $$\frac{3}{7} \times \frac{21}{24}$$

Step-by-Step Solution

Answer

\frac{3}{8}

Step 1: Set up the multiplication

Write the multiplication of the two fractions as \(\frac{3}{7} \times \frac{21}{24}\)

Step 2: Multiply numerators and denominators

Multiply the numerators (top numbers) together and the denominators (bottom numbers) together. This results in \(\frac{3 \times 21}{7 \times 24} = \frac{63}{168}\)

Step 3: Reduce the fraction

Find the greatest common divisor (GCD) of the numerator and denominator and divide both by this number to simplify the fraction. The GCD of 63 and 168 is 21. Thus, \(\frac{63 \div 21}{168 \div 21} = \frac{3}{8}\).

Key Concepts

Reduce Fractions

Reducing fractions, also known as simplifying fractions, means to convert a fraction into its simplest form where no greater common divisor other than 1 exists between the numerator and the denominator. This process often makes the numbers smaller and the fraction easier to work with or understand. To reduce a fraction, you'll need to find the greatest common divisor (GCD) for the numerator and the denominator, and then divide both by this number.

Let's consider our initial fraction from the exercise, \(\frac{63}{168}\). To reduce this, we first need to find the GCD of 63 and 168. Once identified, we can then divide both the numerator (63) and the denominator (168) by the GCD to find the reduced form of the fraction.

Why Reduce Fractions?

Reducing fractions is not just a mathematical exercise; it makes subsequent calculations simpler and results easier to interpret. For instance, \(\frac{3}{8}\) is much more accessible and easily compared to other fractions or whole numbers than \(\frac{63}{168}\).

Greatest Common Divisor

The greatest common divisor (GCD), also known as the greatest common factor (GCF), is the largest number that divides both the numerator and the denominator without leaving a remainder. Finding the GCD is a critical step in the process of reducing fractions. The GCD of two numbers can be found using various methods, including prime factorization, using the Euclidean algorithm, or simply by listing out the factors of each number and choosing the largest common one.

For example, to find the GCD of the numbers 63 and 168, we might list the factors of each:

• Factors of 63: 1, 3, 7, 9, 21, 63
• Factors of 168: 1, 2, 3, 4, 6, 7, 8, 12, 14, 21, 24, 42, 56, 84, 168

From the list above, the largest number that appears in both lists is 21, making it the GCD of 63 and 168. By dividing both the numerator and denominator by the GCD, we significantly reduce the fraction to its simplest form.

Numerators and Denominators

Understanding the roles of numerators and denominators is key to working with fractions. A fraction consists of two parts: the numerator, which is the top number, represents how many parts we have; while the denominator, the bottom number, signifies the total number of equal parts that make up a whole. When multiplying fractions, as in the given exercise, we multiply the numerators together to get the new numerator, and the denominators together to get the new denominator.

The product of \(\frac{3}{7} \times \frac{21}{24}\) gives us a new fraction, \(\frac{63}{168}\), where 63 is the numerator obtained from multiplying 3 and 21, and 168 is the denominator from multiplying 7 and 24. This multiplication doesn't change the real value of the fraction; it just expands it to its non-reduced form which we then simplify by finding and using the GCD.