Problem 13 - Fundamentals of Mathematics

Question

\(\frac{14}{15}\) of \(\frac{20}{21}\) is what number?

Step-by-Step Solution

Answer

The number is \(\frac{8}{9}\).

Step 1: Understand the Problem

We are asked to find what number is equivalent to multiplying two fractions: \(\frac{14}{15}\) and \(\frac{20}{21}\). This is determined by performing the multiplication of these fractions.

Step 2: Multiply the Numerators

Multiply the numerators of the fractions. Here, the numerators are 14 and 20. Thus, the calculation is:\[14 \times 20 = 280\]

Step 3: Multiply the Denominators

Multiply the denominators of the fractions. The denominators are 15 and 21. Thus, the calculation is:\[15 \times 21 = 315\]

Step 4: Form the Resulting Fraction

The results from Step 2 and Step 3 give us the fraction:\[\frac{280}{315}\]

Step 5: Simplify the Fraction

To simplify \(\frac{280}{315}\), we find the greatest common divisor (GCD) of 280 and 315 and divide both the numerator and the denominator by this number. The GCD of 280 and 315 is 35. Therefore, \[\frac{280}{315} = \frac{280 \div 35}{315 \div 35} = \frac{8}{9}\]

Key Concepts

Fraction Simplification

Fraction simplification is a fundamental concept in mathematics that makes working with fractions much easier. When a fraction is 'simplified,' it means it is reduced to its smallest form. This simpler form is easier to interpret and often more useful in further calculations.

The process of simplification involves dividing both the numerator and the denominator by their greatest common divisor (GCD). This results in a fraction that still represents the same value but is expressed with smaller, co-prime numbers. Here’s how it typically goes:

• Identify the GCD of the numerator and the denominator.
• Divide both the numerator and denominator by the GCD.

These steps ensure the fraction is in its simplest form, which is often easier to understand or use in subsequent mathematical operations.

Greatest Common Divisor (GCD)

The Greatest Common Divisor, or GCD, is essential when simplifying fractions. It is the largest number that accurately divides both the numerator and the denominator without leaving a remainder. Finding the GCD helps reduce fractions to their simplest form, preserving the same value in a more concise expression.

There are multiple ways to determine the GCD:

• Listing Factors: Write down all factors of each number and identify the greatest factor common to both lists.
• Prime Factorization: Break down both numbers into their prime factors. Multiply all prime factors they have in common.

By using the GCD, you can easily simplify the fraction, making it more manageable and easier to understand. This technique not only simplifies computation but also helps in comparing fractions, finding common denominators, and more.

Numerator and Denominator

The terms 'numerator' and 'denominator' are critical in understanding fractions. A fraction consists of two numbers separated by a line. The top number, known as the numerator, indicates how many parts we have. The bottom number, the denominator, tells us how many parts make up a whole.

To multiply fractions like \[\frac{14}{15} \text{ and } \frac{20}{21},\]we first multiply the numerators together and the denominators together:

• Numerator: Multiply 14 and 20 to get 280.
• Denominator: Multiply 15 and 21 to get 315.

Understanding these roles helps in performing operations such as addition, subtraction, multiplication, and division of fractions efficiently. The concept of maintaining proper distinction between numerator and denominator is essential for correct mathematical procedures and simplifications.