Problem 46 - Elementary Algebra

Question

The International Association for Wireless Communication predicted \(196.9\) billion text messages in 2011 , compared to \(33.5\) million text messages sent in 1996 . Fin the increase in the number of text messages, writing the answer in place value notation. (Source: www.ctia.org, Oct. 11, 2011)

Step-by-Step Solution

Answer

The increase is 196,866,500,000 text messages.

Step 1 - Identify given values

First, identify the number of text messages in both years. In 1996, there were 33.5 million text messages, and in 2011, there were 196.9 billion text messages.

Step 2 - Convert units to match

Since we need to compare these numbers directly, convert the 1996 value to the same unit as the 2011 value. 33.5 million can be written as 0.0335 billion.

Step 3 - Calculate the increase

Subtract the 1996 value from the 2011 value to find the increase: difference = 196.9 billion - 0.0335 billion

Step 4 - Perform the subtraction

Now, perform the subtraction: \[ 196.9 - 0.0335 = 196.8665 \text{ billion} \]

Step 5 - Write the answer in place value notation

Express the final result in standard place value notation: 196,866,500,000 text messages.

Key Concepts

Place Value Notation

Place value notation is a way of writing numbers using digits, where each digit has a place and a value depending on its position in the number. For instance, in the number 196,866,500,000, the digit 1 represents one hundred billion, the digit 9 represents ninety billion, and so on. This system makes it easier to understand large numbers and perform arithmetic operations. Writing our final increase in text messages as 196,866,500,000 helps in visualizing the magnitude of the number clearly.

Unit Conversion

Unit conversion is the process of changing a quantity from one unit to another. In this exercise, we had to convert text messages from 'millions' to 'billions' to make sure the units matched for subtraction. To convert 33.5 million to billions, you need to remember that a million is \( 10^6 \) and a billion is \( 10^9 \). Hence, \( 33.5 \text{ million} = 33.5 \times 10^6 \), which equals \( 0.0335 \text{ billion} \). This conversion step is crucial for accurate comparisons and calculations.

Subtraction in Algebra

Subtraction in algebra involves finding the difference between numbers. In this problem, we are subtracting to find the increase in text messages. After converting both numbers to the same unit, the next step is straightforward subtraction. We subtract 0.0335 billion from 196.9 billion: \[ 196.9 - 0.0335 = 196.8665 \text{ billion} \]. This technique is essential in solving problems that involve analyzing growth, comparing data, or calculating changes over time.