Problem 27 - Prealgebra

Question

Miles/Hour A car travels 675.4 miles in \(12 \frac{1}{2}\) hours. Give the rate in miles per hour to the nearest hundredth.

Step-by-Step Solution

Answer

54.03 miles per hour.

Step 1: Converting Mixed Number to Improper Fraction

First, convert the mixed number \(12 \frac{1}{2}\) hours into an improper fraction. Multiply the whole number 12 by the denominator 2 and add it to the numerator 1 to get \(\frac{25}{2}\) hours.

Step 2: Calculating the Rate

To find the rate in miles per hour, divide the total distance by the total time in hours. Use the formula: \[\text{Rate} = \frac{\text{Distance}}{\text{Time}}\]Substitute the values: \[\text{Rate} = \frac{675.4}{\frac{25}{2}}\]

Step 3: Simplifying the Equation

Dividing by a fraction is the same as multiplying by its reciprocal. Thus, multiply 675.4 by \(\frac{2}{25}\):\[\text{Rate} = 675.4 \times \frac{2}{25}\] This simplifies to \(\frac{1350.8}{25}\).

Step 4: Performing the Division

Divide 1350.8 by 25 to find the rate in miles per hour. You will get approximately 54.032.

Step 5: Rounding to the Nearest Hundredth

Round 54.032 to the nearest hundredth. Check the thousandth's place, 2, which is less than 5, so you round down, resulting in 54.03.

Key Concepts

Improper Fractions

An improper fraction is a type of fraction where the numerator is greater than or equal to the denominator. This is in contrast to a proper fraction where the numerator is less than the denominator. Improper fractions are useful in mathematical calculations because they can simplify arithmetic operations, such as multiplying and dividing fractions.

Converting a mixed number into an improper fraction is simple. Consider the mixed number, which has both a whole number and a fractional part, like \(12 \frac{1}{2}\). To convert it, multiply the whole number (12) by the denominator of the fractional part (2) and add the numerator (1). This results in the equation: \(12 \times 2 + 1 = 25\). So, \(12 \frac{1}{2}\) converts to \(\frac{25}{2}\).

This step makes it easier to perform operations like multiplication or division, which are commonly encountered in mathematical problems.

Rounding Numbers

Rounding numbers is the process of adjusting a number to make it simpler yet very close to the original value. This process is particularly useful in making numbers easier to work with, especially when dealing with decimals.

Imagine you have a number like 54.032 and you need to round it to the nearest hundredth. First, identify the hundredth place; in this case, it’s the third digit after the decimal point (3). Next, look at the digit immediately after the hundredth place (2). If this digit is 5 or higher, you would round the number up. If it is less than 5, as it is here, you round down, resulting in 54.03.

Rounding makes numbers friendlier for estimation and is especially handy in real-world calculations where extreme precision is unnecessary.

Distance and Time Relationship

The relationship between distance and time is fundamental to understanding how objects move. Speed or rate is the measure that quantifies how much distance is covered over a specific period of time. The formula used is:

• \(\text{Rate} = \frac{\text{Distance}}{\text{Time}}\)

This formula is integral in finding the speed, such as in our example of calculating miles per hour (mph).

By dividing the total distance by the total time, you can easily determine how fast an object, like a car, is traveling. In our exercise, the car travels 675.4 miles in \(12\frac{1}{2}\) hours, and by employing the formula, you determine the car's rate of travel. Understanding this relationship helps in various real-world contexts, from scheduling travel times to optimizing routes.

Division Operations

Division is one of the four basic arithmetic operations, pivotal in dividing quantities into equal parts. When working with fractions, division can seem tricky, but it follows a straightforward rule: dividing by a fraction is equivalent to multiplying by its reciprocal.

Let's consider the problem of dividing 675.4 by the improper fraction \(\frac{25}{2}\). Dividing by \(\frac{25}{2}\) is the same as multiplying 675.4 by \(\frac{2}{25}\).

This step is crucial in simplifying the calculation. After transforming the division into a multiplication operation, the resulting multiplication \(675.4 \times \frac{2}{25}\) simplifies to \(\frac{1350.8}{25}\). This clarity in approach allows for a neat transition to correctly dividing 1350.8 by 25, ultimately leading to the conclusion of approximately 54.03 miles per hour after rounding.