Problem 111 - Fundamentals of Mathematics

Question

There are 24 hours in a day. What fraction of a day is 14 hours?

Step-by-Step Solution

Answer

14 hours is \( \frac{7}{12} \) of a day.

Step 1: Identify Total Parts of the Day

First, recognize that a whole day consists of 24 hours. This will be the total number of parts when considering fractions.

Step 2: Identify the Fractional Part of the Day

We need to find what fraction 14 hours are out of the total 24 hours in a day. This is done by placing 14 hours over 24 hours, resulting in the fraction \( \frac{14}{24} \).

Step 3: Simplify the Fraction

To express the fraction in its simplest form, determine the greatest common divisor (GCD) of 14 and 24. The GCD is 2. Divide both the numerator and the denominator of the fraction by 2: \( \frac{14}{24} = \frac{7}{12} \).

Step 4: Verify the Simplified Fraction

Multiply the simplified fraction's numerator and denominator by 2 to check if it returns back to the original fraction: \( \frac{7}{12} \times \frac{2}{2} = \frac{14}{24} \). This confirms that \( \frac{7}{12} \) is indeed the simplified fraction.

Key Concepts

Simplification

Simplifying fractions is like cleaning up a messy room—everything becomes neater and easier to understand. When we have a fraction like \( \frac{14}{24} \), the numbers on top and bottom can sometimes be reduced to smaller numbers while still representing the same quantity. This process is called simplification.

A fraction is simplified by dividing both the numerator (top number) and the denominator (bottom number) by the largest number that they both share as a factor, which is known as the greatest common divisor (GCD). By finding this common factor, both parts of the fraction shrink down, making them simpler to read and use. This doesn’t change the fraction's actual value—just its appearance.

For example, with the fraction \( \frac{14}{24} \), we can simplify it by dividing both 14 and 24 by their greatest common divisor. In this case, it’s 2, so we divide both numbers by 2 and get \( \frac{7}{12} \). Now, this fraction is in its simplest form.

Greatest Common Divisor

The greatest common divisor (GCD) is like a magic key that helps us unlock simpler versions of fractions. It's the largest number that divides both the numerator and the denominator without leaving a remainder.

To find the GCD, we list out the factors of each number, which are numbers that can be multiplied together to reach the original one. For the numbers in our fraction, 14 and 24, here are the factors:

• Factors of 14: 1, 2, 7, 14
• Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

By comparing these lists, we see that 2 is the biggest number they both share. Therefore, 2 is the GCD of 14 and 24.

Using the GCD to simplify fractions ensures that the fraction is expressed in the smallest possible whole numbers, like \( \frac{7}{12} \) from our example, without changing its actual value.

Numerator and Denominator

Fractions are composed of two main parts: the numerator and the denominator. Understanding these terms is crucial for working with fractions.

The numerator is the top part of the fraction, representing how many parts of a whole we have. In our fraction \( \frac{14}{24} \), 14 is the numerator. It tells us we are considering 14 parts out of a whole.

The denominator, on the other hand, is the bottom part. This tells us into how many equal parts the whole is divided. In the example, 24 is the denominator, indicating that there are 24 equal parts in the whole.

When simplifying fractions, both the numerator and the denominator need to be divided by their greatest common divisor to make the fraction easier to use, but it's important to remember that each part plays a distinct role in expressing the fraction's value.