Problem 66 - Prealgebra

Question

According to the U.S. Census Bureau, it is estimated that the average person watches 4 hours of TV each day. Represent the number of hours of TV watched each day as a fraction in lowest terms.

Step-by-Step Solution

Answer

The number of hours of TV watched each day, as a fraction in lowest terms, is \( \frac{1}{6} \).

Step 1: Understanding the Problem

The problem asks us to convert the average time watched (4 hours) into a fraction. Here, the total time considered in one day is 24 hours since there are 24 hours in a day.

Step 2: Setting Up the Fraction

We are asked to find out the number of hours watched as a fraction of a day, so we set up the fraction as \( \frac{4}{24} \) where 4 is the number of hours watched and 24 is the total number of hours in a day.

Step 3: Simplifying the Fraction

To simplify \( \frac{4}{24} \), we find the greatest common divisor (GCD) of 4 and 24, which is 4. We then divide both the numerator and the denominator by their GCD: \( \frac{4 \div 4}{24 \div 4} = \frac{1}{6} \).

Key Concepts

Simplifying Fractions

When you are dealing with fractions, sometimes you will come across ones that can be simplified into smaller, more manageable terms. Simplifying a fraction means making the numbers smaller, but keeping the value of the fraction the same.

To simplify a fraction, you need to find a common number that both the numerator and the denominator can be divided by evenly. Once you find this number, which is also known as the greatest common divisor, you divide both elements of the fraction by it.

• Start with identifying the fraction that needs simplifying.
• Find a number that both the numerator and denominator can be divided by without leaving a remainder.
• Divide both parts of the fraction by this number to simplify it.

For example, the fraction \( \frac{4}{24} \) can be simplified by first finding their GCD, which is 4, and then dividing both the 4 and 24 by this number resulting in \( \frac{1}{6} \).

Numerator

In any fraction, the numerator is the number located above the line. It represents how many parts of the whole we are dealing with.

For instance, in the fraction \( \frac{4}{24} \), the numerator is 4. This shows that out of the total parts - represented by the denominator - the numerator counts how many are considered. When we simplify, we are essentially changing the number of parts counted (numerator) without changing the proportion it forms with the denominator.

• The numerator is crucial in understanding the fraction size.
• By simplifying, we adjust both the numerator and the denominator proportionally.

This helps show the same proportion with smaller, simpler numbers.

Denominator

The denominator is the number at the bottom of the fraction. It tells us into how many pieces the whole is divided.

Taking our fraction \( \frac{4}{24} \) again as an example, the denominator is 24. This means the whole day is divided into 24 equal parts (hours in this context).

• The denominator stays the lower part of the fraction in both unsimplified and simplified forms.
• Simplifying the fraction adjusts it to a more manageable form.

Understanding the denominator helps in simplifying the fraction properly, as any adjustment to the numerator when simplifying must be equally considered in the denominator.

Greatest Common Divisor

The Greatest Common Divisor (GCD) is an important concept when working with fractions, especially when simplifying them. It is the largest number that divides two numbers without leaving a remainder.

Finding the GCD is crucial because it allows you to reduce fractions to their simplest form, ensuring no common factors remain between the numerator and the denominator.

• To find the GCD, list the factors of both numbers.
• Identify the largest factor common to both lists.
• Use this factor to divide both numerator and denominator.

For instance, for the numbers 4 and 24, the greatest common divisor is 4. Using the GCD, the fraction \( \frac{4}{24} \) simplifies to \( \frac{1}{6} \). By mastering the GCD, you can easily simplify any fraction.