Problem 3 - Prealgebra

Question

A use the tables of conversion factors given in this section and other sections in this chapter to make the following conversions. (Round your answers to the nearest hundredth.) [Example 1] 5 hours 20 minutes to a. Minutes b. Hours

Step-by-Step Solution

Answer

5 hours 20 minutes equals 320 minutes or 5.33 hours.

Step 1: Convert Hours to Minutes

The first part of conversion involves changing hours into minutes. Since 1 hour is equal to 60 minutes, multiply the number of hours by 60.\[ 5 \text{ hours} \times 60 = 300 \text{ minutes} \]

Step 2: Add Minutes from the Given Time

After converting the hours into minutes, add the additional 20 minutes to the total from Step 1.\[ 300 \text{ minutes} + 20 \text{ minutes} = 320 \text{ minutes} \]

Step 3: Calculate Total Hours in Decimal Form

To find the total hours in decimal form, first convert the 20 minutes to hours by dividing by 60, since 1 hour is 60 minutes.\[ 20 \div 60 = \frac{1}{3} \approx 0.33 \]

Step 4: Combine Hours and Decimal Part

Combine the whole hours with the decimal part calculated in Step 3 to find the total hours.\[ 5 + 0.33 = 5.33 \text{ hours} \]

Key Concepts

Understanding Conversion Factors

Conversion factors are tools that help us switch between different units of measurement. They are particularly useful when dealing with time, as they let us translate hours into minutes and vice versa. This is possible because conversion factors express the relationship between different units mathematically.

In our example, the conversion factor between hours and minutes is 60. This means:

• 1 hour = 60 minutes

Whenever you have a quantity in hours and want to convert it to minutes, you multiply by 60. Conversely, when converting from minutes to hours, you divide by 60. These simple calculations help ensure accuracy and consistency in measuring time.

Converting Hours to Minutes

To convert hours to minutes, you need to use the conversion factor we discussed. Let's take the number of hours and multiply it by the conversion factor, which is 60. This will give you the equivalent number of minutes.

For example, if you have 5 hours, you calculate the minutes by performing the following multiplication:

• \( 5 \text{ hours} \times 60 = 300 \text{ minutes} \)

After finding the minutes from the hours, add any additional minutes you have from the original time. For instance, adding 20 minutes to your 300 minutes gives:

• \( 300 \text{ minutes} + 20 \text{ minutes} = 320 \text{ minutes} \)

These steps give you a straightforward way to convert time from hours to minutes.

The Concept of Decimal Hours

Decimal hours refer to hours expressed as a decimal number, akin to how we represent fractions in everyday calculations. When you have minutes not totaling a full hour, you need to convert these into decimal form by dividing by 60 because 60 minutes make a whole hour.

In this exercise, we take the leftover time of 20 minutes and convert it:

• \( 20 \div 60 = \frac{1}{3} \approx 0.33 \)

Including this in our total, we shift the representation from hours and minutes to just decimal hours, like so:

• \( 5 + 0.33 = 5.33 \text{ hours} \)

Decimal hours provide a convenient and precise way to express time in many mathematical and professional contexts.

Making Mathematical Calculations

When dealing with time conversions, mathematical calculations are key. Knowing how to properly multiply and divide using conversion factors allows for accurate transformations between units.

Remember the basic operations:

• Multiplication to convert larger units to smaller units (e.g., hours to minutes).
• Division to convert smaller units into larger units (e.g., minutes to hours).

These calculations require precision, and it's important to carefully carry out each step to avoid errors. For example, multiplying 5 hours by 60 gives 300 minutes, and then including an additional 20 minutes results in 320 minutes total. Meanwhile, transforming 20 minutes into hours is done via division: 20 divided by 60 equals approximately 0.33 hours.

With these simple steps, we can effectively manage and manipulate data related to time.