Problem 70 - Prealgebra

Question

Based on the most recent data available, there are approximately \(1,320,000,000\) Internet users in the world. North America makes up about \(240,000,000\) of this total. Represent the number of Internet users in North America as a fraction of the total expressed in lowest terms.

Step-by-Step Solution

Answer

The number of Internet users in North America is \( \frac{2}{11} \) of the total.

Step 1: Setup the Fraction

First, represent the number of Internet users in North America as a fraction of the total number of Internet users in the world. This means placing the number of North American users (240,000,000) in the numerator and the total number of users (1,320,000,000) in the denominator. This gives us the fraction: \[ \frac{240,000,000}{1,320,000,000} \]

Step 2: Simplify the Fraction

Next, simplify the fraction to its lowest terms. To do this, we need to find the greatest common divisor (GCD) of 240,000,000 and 1,320,000,000. Both numbers end in multiple zeros, so let's start by dividing both the numerator and the denominator by 10 until we can't anymore. This simplifies to: \[ \frac{240,000}{1,320,000} \] Keep simplifying by recognizing that 240,000 and 1,320,000 can both be divided by the same greatest common factor, which is 120,000. Divide both by 120,000: \[ \frac{240,000 \div 120,000}{1,320,000 \div 120,000} = \frac{2}{11} \]

Step 3: Verify the Simplified Fraction

We confirm the fraction \( \frac{2}{11} \) is in its simplest form because 2 and 11 share no common divisors other than 1, ensuring it cannot be simplified further.

Key Concepts

Greatest Common Divisor

When simplifying a fraction, identifying the greatest common divisor (GCD) is an essential step. The GCD is the largest number that can perfectly divide both the numerator and the denominator of a fraction without leaving any remainder.
To find it:

• List out the factors of each number in the fraction.
• Identify the largest factor that these numbers share.

In our example of Internet users, finding the GCD of 240,000,000 and 1,320,000,000 helps us to simplify:

• Initially, we reduce the numbers by dividing both by 10 multiple times.
• Finally, after several reductions, 120,000 emerges as the GCD, allowing for further simplification.

Using the greatest common divisor ensures fractions are reduced to their most basic form efficiently.

Lowest Terms

Expressing a fraction in its lowest terms means it's simplified so that the numerator and denominator have no common factors other than 1. This is the simplest possible form a fraction can take.
A simplified fraction is easier to interpret and compare with other fractions. To achieve lowest terms:

• Identify and divide by the greatest common divisor.
• Check that no further division can evenly reduce both numbers.

In the Internet user example, simplifying 240,000,000 over 1,320,000,000 by the GCD of 120,000 gives us 2/11. This fraction cannot be reduced further, indicating it's at its lowest terms.

Numerator and Denominator

Understanding the roles of the numerator and denominator is fundamental in fraction manipulation.

• The numerator, located above the line in a fraction, signifies the part or count. In our case, it represents North American Internet users: 240,000,000.
• The denominator is the total or whole, positioned below the line. For our example, it is the global Internet user count: 1,320,000,000.

This relationship tells us what proportion the numerator is of the denominator. Initially, this proportion is 240,000,000 over 1,320,000,000. By simplifying, we better understand that for every 11 Internet users globally, 2 are in North America. Knowing how to correctly determine the numerator and denominator allows for accurate problem solving in fractions.