Question

A bus is traveling at a constant rate of 50 miles per hour. At this rate, how far will the bus travel in \(3 \frac{4}{6}\) hours? (A) 150 miles (B) 160 miles (C) \(162.5\) miles (D) \(175.5\) miles

Short answer

The bus will travel approximately 183.33 miles.

Step-by-step solution

Convert Mixed Number to Improper Fraction

First, convert the mixed number \(3 \frac{4}{6}\) into an improper fraction. The mixed number \(3 \frac{4}{6}\) can be converted as follows: \(3 \frac{4}{6} = 3 + \frac{4}{6}\). To simplify \(\frac{4}{6}\), divide both the numerator and the denominator by their greatest common divisor, which is 2. Thus, \(\frac{4}{6} = \frac{2}{3}\). Now, convert \(3 \frac{2}{3}\) into an improper fraction: \(3 = \frac{9}{3}\) so \(3 + \frac{2}{3} = \frac{11}{3}\).

Calculate the Distance

The problem states the rate is 50 miles per hour, and we have found the time to be \(\frac{11}{3}\) hours. To find the distance, use the formula: \(\text{Distance} = \text{Rate} \times \text{Time}\). Hence, \(\text{Distance} = 50 \times \frac{11}{3}\). This can be calculated as: \(50 \times \frac{11}{3} = \frac{550}{3}\).

Simplify the Distance

Convert the improper fraction to a decimal or mixed number for easier interpretation. Divide 550 by 3: \(550 \div 3 = 183.333\ldots\) or approximately 183.33 when rounded to two decimal places. This means the bus travels approximately 183.33 miles in \(3 \frac{4}{6}\) hours.

Key concepts

Rate and Distance Calculations

Rate and distance problems are a common type of exercise in mathematics, often involving a formula that is simple yet critical: \( \text{Distance} = \text{Rate} \times \text{Time} \). This formula is useful in scenarios where an object travels at a constant speed over a period of time.

To solve such problems, you need to:

• Determine the constant rate of travel (how fast the object moves).
• Identify the time duration for which the object is traveling.
• Apply the formula by multiplying the rate by the time to get the distance.

In the given exercise, the bus travels at a rate of 50 miles per hour over a period of \(3 \frac{4}{6}\) hours. By plugging these values into the distance formula, you can compute how far the bus will travel. This approach is straightforward if you follow the steps accurately. Understanding and applying this formula can help you tackle many practical problems related to travel and delivery systems effectively.

Improper Fractions

Improper fractions are fractions where the numerator is larger than the denominator. They are particularly useful in calculations as they simplify arithmetic operations involving mixed numbers.

In converting mixed numbers to improper fractions:

• Multiply the whole number by the denominator of the fractional part.
• Add the result to the numerator of the fractional part.
• Write the sum over the original denominator.

For instance, with the mixed number \(3 \frac{4}{6}\), you convert it by first simplifying the fractional part, \(\frac{4}{6} = \frac{2}{3}\), then calculate the improper fraction by \(3 \times 3 + 2 = 11\), giving \(\frac{11}{3}\). Improper fractions are beneficial in multiplication or division problems like rate and distance calculations, as they remove the need to separately handle whole numbers and fractions.

Mixed Numbers

Mixed numbers combine whole numbers and fractions, making them intuitive to understand in real-world measurements, like time or weight. However, for mathematical operations, converting them to improper fractions usually simplifies the process.

Here's how to work with mixed numbers:

• Recognize the mixed number's components: a whole part and a fractional part.
• Convert it to an improper fraction for computation ease.
• After performing the calculation, interpret results by converting back to a mixed number if necessary.

In the given exercise, converting the mixed number \(3 \frac{4}{6}\) to an improper fraction simplifies multiplication. After determining the distance as an improper fraction or decimal, you can interpret it as needed for clarity or application, demonstrating how mixed numbers can smoothly transition into improper fractions for calculations.