Problem 30 - Prealgebra

Question

Perform the indicated operation. Again, remember to use the appropriate conversion factor. [Examples \(2,3]\) Subtract 2 hours 54 minutes from 3 hours 7 minutes.

Step-by-Step Solution

Answer

The result is 13 minutes.

Step 1: Convert hours to minutes

To make the subtraction straightforward, we convert both quantities entirely into minutes. Note that 1 hour is equivalent to 60 minutes. Convert 3 hours 7 minutes to minutes:\[3 \text{ hours} \times 60 + 7 = 180 + 7 = 187 \text{ minutes}\]

Step 2: Convert the second duration to minutes

Now, convert 2 hours 54 minutes into minutes:\[2 \text{ hours} \times 60 + 54 = 120 + 54 = 174 \text{ minutes}\]

Step 3: Subtract the minutes

Subtract the total minutes of the second duration from the first:\[187 \text{ minutes} - 174 \text{ minutes} = 13 \text{ minutes}\]

Step 4: Express the result as hours and minutes

Since the result is entirely in minutes and is less than 60, you can express the result purely in minutes. It does not convert into full hours:\[13 \text{ minutes}\]

Key Concepts

Addition and Subtraction of Time

Time calculations often require a careful approach, because the units we deal with are not in the usual base 10 system. When adding or subtracting time, we deal with two main units: hours and minutes.
For addition and subtraction of time, it is important to ensure that the units match and are aligned in a way that makes arithmetic operations straightforward. Typically, we convert everything into the smallest unit involved - in this case, minutes - before performing the addition or subtraction.
Subtracting or adding time accurately requires the following steps:

• Ensure all time units are converted into minutes. This ensures consistency and accuracy in calculations.
• Perform the addition or subtraction using these converted values.
• Convert the result back into a larger unit, like hours and minutes, if necessary.

Remember, this conversion is crucial when the time units mix. Mistakes usually happen when this conversion is overlooked.

Unit Conversion

Unit conversion is essential in countless mathematical problems, especially time calculations. Since time measures hours and minutes, converting these units accurately is a vital skill.
In time conversion, the key is remembering that

• 1 hour equals 60 minutes.
• Ensure all time is expressed in the same unit for calculations.

For example, 3 hours 7 minutes combines two units. To convert this entirely into minutes: multiply the hours by 60 and then add the remaining minutes. It looks like this: \[3 \times 60 + 7 = 180 + 7 = 187 \text{ minutes}\] This method changes mixed units into a single unit, simplifying the calculation. Similarly, for 2 hours 54 minutes, convert as follows: \[2 \times 60 + 54 = 120 + 54 = 174 \text{ minutes}\] Perform these conversions before any addition or subtraction. This eliminates the complexity that varying units introduce.

Problem-solving Steps

Solving problems involving time subtraction starts with a structured approach. Breaking tasks into smaller steps simplifies complex operations.
The problem-solving process for time subtraction includes:

• First, list all components in their initial state, noting the hours and minutes separately.
• Convert each component entirely into minutes to simplify the arithmetic operation. This eliminates conversion errors mid-calculation.
• Subtract the respective minute volumes from one another.
• Assess the final result. If it exceeds 60 minutes, consider converting it back to an hour and minute format for clarity.

For the given problem: after converting both times, subtract as follows:\[187 \text{ minutes} - 174 \text{ minutes} = 13 \text{ minutes}\]By maintaining consistent units throughout, math becomes precise, minimizing the chance of making errors. Finally, review your result to ensure it matches the problem's requirements, potentially involving writing in the preferred units.