A computer can perform 1,000,000 calculations per second. At this rate, how many hours will it take this computer to...

1. Translate the problem requirements: We need to find how long it takes to complete \(3.6 \times 10^{11}\) calculations when the computer performs \(1,000,000\) calculations per second, with the final answer in hours.
2. Set up the basic rate relationship: Use the fundamental relationship that Time = Total Work ÷ Rate to find the time in seconds.
3. Convert scientific notation for easier calculation: Express both the rate and total calculations in scientific notation to simplify the division.
4. Convert from seconds to hours: Transform the time result from seconds to hours using the conversion factor of \(3,600\) seconds per hour.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're being asked to find. We have a computer that can do calculations very quickly, and we want to know how long it will take to complete a specific job.

Think of it like this: if you can eat 10 cookies per minute, and you have 600 cookies to eat, how long will it take? You'd divide 600 by 10 to get 60 minutes.

Here's what we know:

• The computer performs \(1,000,000\) calculations per second
• We need to complete \(3.6 \times 10^{11}\) total calculations
• We want our final answer in hours (not seconds)

Process Skill: TRANSLATE - Converting the word problem into mathematical quantities we can work with

2. Set up the basic rate relationship

We'll use the fundamental relationship that most people understand intuitively:

Time needed = Total amount of work ÷ Rate of work

Just like in our cookie example, if we know how much work needs to be done and how fast we can do it, we can find the time.

So: Time in seconds = Total calculations ÷ Calculations per second
Time in seconds = \(3.6 \times 10^{11} \div 1,000,000\)

3. Convert scientific notation for easier calculation

To make our division easier, let's express both numbers in scientific notation:

• \(1,000,000 = 1 \times 10^6\)
• \(3.6 \times 10^{11}\) stays as is

Now our division becomes:
Time in seconds = \((3.6 \times 10^{11}) \div (1 \times 10^6)\)

When we divide numbers in scientific notation, we divide the regular numbers and subtract the exponents:
= \(3.6 \div 1 \times 10^{11-6}\)
= \(3.6 \times 10^5\)
= \(360,000\) seconds

4. Convert from seconds to hours

Now we need to convert \(360,000\) seconds into hours. We know that:

• 1 minute = 60 seconds
• 1 hour = 60 minutes = \(3,600\) seconds

So: Time in hours = Time in seconds ÷ \(3,600\)
Time in hours = \(360,000 \div 3,600\)

To make this division easier, notice that:
\(360,000 = 36 \times 10,000 = 36 \times 10^4\)
\(3,600 = 36 \times 100 = 36 \times 10^2\)

So: \(360,000 \div 3,600 = (36 \times 10^4) \div (36 \times 10^2) = 10^4 \div 10^2 = 10^2 = 100\)

Final Answer

The computer will take 100 hours to perform all the required calculations.

This matches answer choice B. 100.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding the unit conversion requirement
Students often overlook that the question asks for the answer in hours, not seconds. They may set up the calculation correctly but forget that they need to convert their final result from seconds to hours, leading them to select an answer that's \(3,600\) times too large.

2. Confusing rate vs. time relationships
Some students might incorrectly set up the relationship as Time = Rate × Work instead of Time = Work ÷ Rate. This fundamental misunderstanding of the rate formula would lead them to multiply \(3.6 \times 10^{11}\) by \(1,000,000\) instead of dividing, resulting in a drastically incorrect answer.

Errors while executing the approach

1. Scientific notation calculation errors
When dividing \((3.6 \times 10^{11}) \div (1 \times 10^6)\), students frequently make mistakes with exponent subtraction. They might incorrectly calculate \(10^{11-6}\) as \(10^5\) but then misinterpret \(3.6 \times 10^5\) as \(36,000\) instead of \(360,000\), or make errors in the exponent arithmetic itself.

2. Unit conversion arithmetic mistakes
Even when students correctly identify that 1 hour = \(3,600\) seconds, they often make computational errors when dividing \(360,000\) by \(3,600\). Common mistakes include incorrectly simplifying this division or losing track of zeros during the calculation process.

3. Errors while selecting the answer
No likely faltering points - once students correctly execute the calculations, the final answer of 100 hours directly matches one of the given answer choices without requiring further interpretation or rounding.