Problem 31 - Algebra 1: Concepts and Skills

Question

Choose the expression that completes the following equation: 720 seconds \(\cdot \quad ?=12\) minutes. A. \(\frac{1 \text { minute }}{60 \text { seconds }}\) B. \(\frac{60 \text { seconds }}{1 \text { minute }}\)

Step-by-Step Solution

Answer

The expression that completes the equation is \(\frac{1 \, \text{minute}}{60 \, \text{seconds}}\) (option A).

Step 1: Analyze the question

Consider the equation \(720 \, \text{seconds} \cdot ? = 12 \, \text{minutes}\). This equation is asking what we need to multiply with 720 seconds in order to obtain 12 minutes.

Step 2: Convert minutes to seconds

Convert the minutes of the right hand side of the equation to seconds so we are working in a consistent unit. Multiplying \(12 \, \text{minutes} \times 60 \, \text{seconds/minute}\) gives 720 seconds.

Step 3: Identify the correct conversion rate

Finally, to get from 720 seconds to 720 seconds, we need to multiply by 1. The factor that represents this 1 in the context of seconds and minutes is \(\frac{1\, \text{minute}}{60 \, \text{seconds}}\) which is option A.

Key Concepts

Converting Seconds to Minutes

When converting seconds to minutes, it's essential to understand the relationship between these two units of time. There are 60 seconds in a minute. Therefore, converting seconds into minutes involves dividing the number of seconds by 60. This conversion helps in simplifying calculations and understanding time duration better.
For example, if you have 120 seconds, you can convert it to minutes by doing the following:

• Divide 120 by 60.
• The result is 2 minutes.

Understanding this basic conversion process is critical and provides a foundation for more complex time-related calculations.

Understanding Equivalent Expressions

Equivalent expressions might look different but have the same value when solved. In the context of time conversion, for instance, converting 720 seconds to 12 minutes involves using different expressions that represent the same thing. Each expression provides a way to describe these conversions.
Here's an example:

• The expression for converting 720 seconds to minutes could be \( \frac{720}{60} \), which equals 12 minutes.
• Alternatively, using 1 as a conversion factor can be represented within this context by the expression \( 720 \times \frac{1 \text{ minute}}{60 \text{ seconds}} \).

Recognizing equivalent expressions allows you to choose the one that best fits your problem and helps ensure accurate conversions and calculations.

The Role of Conversion Factors

Conversion factors are crucial in translating between units, such as seconds and minutes. These factors are essentially fractions or ratios that are always equal to 1. They represent the relationship between the units being converted. By multiplying by a conversion factor, we change units without altering the actual value.

In our exercise, the conversion factor is \( \frac{1 \text{ minute}}{60 \text{ seconds}} \). This factor is derived from recognizing that 1 minute equals 60 seconds, making the ratio equal to 1.

• This means multiplying by the conversion factor does not change the original quantity.
• Instead, it shifts the units from seconds to minutes.

This principle holds for any unit conversion, making conversion factors a powerful tool in various mathematical and real-world applications.