Problem 26 - Fundamentals of Mathematics

Question

(Section 4.6) Find the quotient: \(\frac{14}{15} \div \frac{4}{45}\).

Step-by-Step Solution

Answer

\(\frac{21}{2}\)

Step 1: Understand the Problem

The problem asks us to find the quotient when dividing two fractions: \(\frac{14}{15}\) and \(\frac{4}{45}\). In mathematical terms, this is expressed as \(\frac{14}{15} \div \frac{4}{45}\).

Step 2: Rewrite Division as Multiplication

When dividing by a fraction, it's equivalent to multiplying by its reciprocal. Therefore, we rewrite the problem as \(\frac{14}{15} \times \frac{45}{4}\) by taking the reciprocal of \(\frac{4}{45}\) which is \(\frac{45}{4}\).

Step 3: Multiply the Fractions

Multiply the numerators and denominators: \(\frac{14 \times 45}{15 \times 4}\). This simplifies to \(\frac{630}{60}\).

Step 4: Simplify the Fraction

Simplify \(\frac{630}{60}\). Divide both the numerator and the denominator by their greatest common divisor, which is 30: \(\frac{630 \div 30}{60 \div 30} = \frac{21}{2}\).

Key Concepts

Reciprocal

When you are dividing fractions, one crucial step is using the reciprocal of the divisor. The reciprocal of a number is simply another number that, when multiplied by it, equals one. For a fraction like \( \frac{4}{45} \), the reciprocal is found by swapping the numerator and the denominator, giving us \( \frac{45}{4} \).
This concept is key because multiplying by a reciprocal turns a division problem into a more straightforward multiplication problem, which is usually easier to solve.

• To find a reciprocal, flip the fraction; the numerator becomes the denominator and vice versa.
• The product of a number and its reciprocal always equals 1.

Using the reciprocal is a powerful method, especially when handling fraction division in mathematics, as it simplifies calculations immensely.

Simplifying Fractions

Simplifying fractions involves reducing them to their lowest terms. This means the numerator and the denominator have no other common factor besides one. Simplifying makes fractions easier to understand and work with.
In the solution, the expression \( \frac{630}{60} \) was simplified. You need to find the greatest common divisor (GCD) of the numerator and denominator. For our example:

• The GCD of 630 and 60 is 30.
• Divide both the numerator (630) and the denominator (60) by 30 to get \( \frac{21}{2} \).

By simplifying, you make the fraction more visually manageable and easier to interpret, especially in subsequent calculations or interpretations.

Numerator and Denominator

Understanding the components of a fraction is fundamental. A fraction is made up of two parts: the numerator and the denominator. The numerator, positioned above the line, indicates how many parts out of the whole are being considered. The denominator, below the line, illustrates the total number of equal parts in the whole.
For example, in \( \frac{14}{15} \), 14 is the numerator and 15 is the denominator. In division or multiplication of fractions, these components are crucial:

• When multiplying fractions, multiply the numerators together and the denominators together.
• Keep track of both during operations; after multiplying, always consider simplifying the resulting fraction.

Knowing the role of each part helps in performing operations accurately and understanding the outcome of these operations in any context.