Problem 49 - Prealgebra

Question

Set up the following conversions as you have been doing. Then perform the calculations on a calculator. Change 751 miles to feet.

Step-by-Step Solution

Answer

751 miles is equal to 3,963,280 feet.

Step 1: Identify the Conversion Factors

To convert miles to feet, you need a conversion factor. There are 5280 feet in one mile. This is the conversion factor we will use.

Step 2: Set Up the Conversion

We start by writing the given quantity, which is 751 miles, and multiply it by the conversion factor to change miles to feet. The setup is as follows: \[751\, \text{miles} \times \frac{5280\, \text{feet}}{1\, \text{mile}}\]

Step 3: Perform the Calculation

Now we multiply 751 by 5280 to find the number of feet in 751 miles. Use a calculator to perform this multiplication:\[751 \times 5280 = 3,963,280\]So, 751 miles equals 3,963,280 feet.

Key Concepts

Conversion Factor

In the world of conversion, a crucial role is played by what we call a "conversion factor." Essentially, a conversion factor is a number used to change one set of units into another, keeping the quantity the same. Imagine you're in a supermarket and you have a voucher for a discount—just for a different brand. Here, you would exchange your voucher for a similar value, perhaps even the same amount in a different brand. Similarly, when converting units, we're not changing the value of what we have; we're merely shifting its format or unit.
In our example of converting miles to feet, the conversion factor is the number of feet in a mile. Think of this as the necessary ingredient to switch from one unit to another. In this scenario, the conversion factor is 5280, because there are 5280 feet in a single mile. Using conversion factors like this ensures that our results are consistent and accurate.

Miles to Feet

Converting miles to feet might seem tricky initially, but it's a straightforward process once you understand the method. Imagine a mile as a long path; along this path, there are exactly 5280 steps, or feet.
To convert the measurement from miles to feet, we need to multiply the number of miles by the conversion factor (5280 feet per mile). This conversion doesn’t change the "distance," only the unit we express it in. For example, converting 751 miles into feet involves expressing that same distance with a different unit.

• Start by recognizing that 1 mile is equal to 5280 feet.
• Then, for 751 miles, multiply by 5280.
• This conversion approach allows us to express 751 miles in terms of feet, which might be more convenient for different applications, like construction or mapping, where smaller units are more practical.

Multiplication

When performing unit conversions, multiplication is the most common mathematical operation used. It's the same process you'd use if you were trying to figure out how much money you'd have if you tripled your savings. Multiplication simply tells us how many times to add a number to itself.
In our conversion scenario, we multiply 751 (miles) by 5280 (the number of feet in a mile). Multiplying these provides the total number of feet in those miles. This is an essential technique in unit conversion, as it allows the dimensionality (here, distance) to remain constant while the units change.
Effectively, multiplication here serves as our tool to transform our units:

• The original value (751 miles) is scaled by a factor (5280), adjusting the unit from larger (miles) to smaller (feet) without altering the actual physical distance.
• This fundamental approach underscores the importance of accuracy in multiplication— especially with large numbers—since we want an exact conversion.