Problem 53 - Algebra for College Students

Question

Carlos's first computer had a processing speed of (1.6) \(\left(10^{6}\right)\) hertz. He recently purchased a laptop computer with a processing speed of \((1.33)\left(10^{9}\right)\) hertz. Approximately how many times faster is the processing speed of his laptop than that of his first computer? Express the result in decimal form.

Step-by-Step Solution

Answer

The laptop is approximately 831.25 times faster than the first computer.

Step 1: Understand the Problem

We are given two different processing speeds for two computers. The processing speed of Carlos's first computer is \(1.6 \times 10^{6}\) hertz, and his new laptop has a processing speed of \(1.33 \times 10^{9}\) hertz. We need to find how many times faster the laptop is compared to the first computer.

Step 2: Write the Expression for Comparison

To find how many times faster the laptop is than Carlos's first computer, we'll use the formula: \[ \text{Factor} = \frac{\text{Laptop speed}}{\text{First computer speed}} \] Substitute the given values into this formula to get: \[ \text{Factor} = \frac{1.33 \times 10^{9}}{1.6 \times 10^{6}} \]

Step 3: Simplify the Expression

Divide the coefficients (1.33 by 1.6) and then divide the powers of ten. This involves simplifying the expression: \[ \text{Factor} = \frac{1.33}{1.6} \times \frac{10^{9}}{10^{6}} \] \[ \text{Factor} = 0.83125 \times 10^{3} \] Convert \(10^{3}\) into decimal form: \(10^{3} = 1000\). Thus, the expression becomes: \[ \text{Factor} = 0.83125 \times 1000 \]

Step 4: Calculate the Result

Multiply the simplified expression to find how many times faster the laptop is: \[ \text{Factor} = 0.83125 \times 1000 = 831.25 \] This says that the processing speed of the laptop is approximately 831.25 times faster than Carlos's first computer.

Key Concepts

Processing Speed

Processing speed in computers refers to how quickly a computer can perform tasks. We measure it in hertz (Hz), which indicates the number of cycles per second the processor completes. Higher hertz mean faster processing capabilities. In this context, understanding the role of processing speed helps us compare different computers. For Carlos's computers, the speeds were given in scientific notation: his first computer had a speed of \(1.6 \times 10^{6}\) Hz and his new laptop, \(1.33 \times 10^{9}\) Hz.
This tells us two important things:

• The higher the number next to the exponential, the faster the processing speed.
• Each increase in the power of ten significantly increases speed, owing to the mathematical power of exponents.

Understanding processing speed is crucial for evaluating the overall performance of a computer, especially when multitasking or running heavy applications.

Comparison of Values

To determine which computer is faster, we must compare processing speeds using a mathematical expression. The goal is to find how many times faster Carlos's new laptop is compared to his old computer. We achieve this by dividing the laptop's processing speed by the first computer's speed:
\[\text{Factor} = \frac{1.33 \times 10^{9}}{1.6 \times 10^{6}}\]This expression involves simplifying both the coefficients and the exponents. Simplifying is crucial because it lets us focus on the size and direction of each value. The quotient of the coefficients (\(\frac{1.33}{1.6}\)) simplifies to 0.83125, while the quotient of the exponents gives us \(10^3\), due to subtraction of powers:

• \(10^9 / 10^6 = 10^{9-6} = 10^3\)

Combining these gives us a full expression that we can further simplify and evaluate for a clearer comparison.

Scientific Notation

Scientific notation is a method to write very large or very small numbers in a more concise and readable form. It is particularly useful in fields like science and engineering, where such numbers are common. In scientific notation, numbers are expressed as a product of a coefficient (a number between 1 and 10) and a power of ten.
With Carlos's computers, the processing speeds were:

• First computer: \(1.6 \times 10^{6}\)
• Laptop: \(1.33 \times 10^{9}\)

The benefits of scientific notation in comparisons include:

• Facilitating easy computation by simplifying multiplication and division.
• Clear visualization of scale differences, especially when comparing vastly different sizes.

In our comparison, knowing scientific notation allowed us to quickly assess and compute the factor by simplifying powers of ten. This significantly eased the complexity of the operation, resulting in a tangible number that reflects the performance difference between Carlos’s two computers.