Problem 4 - Prealgebra

Question

Express each of the following rates as a ratio with the given units. Kilometers/Hour In 6 hours an airplane travels \(4,200\) kilometers. What is the rate of the airplane in kilometers per hour?

Step-by-Step Solution

Answer

The airplane travels at 700 kilometers per hour.

Step 1: Identify Given Values

You are given that in 6 hours, the airplane travels 4,200 kilometers. The first task is to identify the given values: total distance is 4,200 kilometers and total time is 6 hours.

Step 2: Define the Rate Formula

We need to find the rate using the formula for speed, which is \[\text{Speed} = \frac{\text{Distance}}{\text{Time}}.\]

Step 3: Substitute Known Values

Substitute the known values into the formula:\[\text{Speed} = \frac{4,200\text{ kilometers}}{6\text{ hours}}.\]

Step 4: Perform the Division

Divide 4,200 by 6 to find the rate of speed:\[\text{Speed} = \frac{4,200}{6} = 700.\]

Step 5: State the Rate with Units

The calculated rate is 700. Therefore, the speed of the airplane is 700 kilometers per hour.

Key Concepts

Rates and Ratios

Rates and ratios are fundamental concepts in mathematics that help us compare two quantities. A rate is a type of ratio where the two quantities have different units. For example, the airplane travels at a rate of 700 kilometers per hour. This shows the relationship between distance and time. It tells us how many kilometers the airplane covers in one hour.
Understanding rates allows us to simplify complex quantities into more manageable parts. Here are a few key points to remember:

• Rates often compare different units, such as kilometers per hour or miles per gallon.
• Ratios typically compare quantities of the same kind, like the number of apples to oranges.

You can calculate a rate like speed using the formula \( ext{Speed} = \frac{\text{Distance}}{\text{Time}}\). Substituting the distance and time into this formula gives you the rate at which the airplane travels.

Distance and Speed Calculations

Distance and speed are related through time in what's called the "speed formula." Knowing any two of these quantities allows you to find the third by rearranging the formula. To determine speed, you divide the distance by the time. In the given example, the airplane's speed is calculated as \[\text{Speed} = \frac{4,200 \text{ kilometers}}{6 \text{ hours}} = 700 \text{ kilometers per hour}.\]
Here are some important steps for distance and speed calculations:

• Ensure the units for distance and time are compatible, such as kilometers and hours.
• Use division to calculate speed: \(\text{Speed} = \frac{\text{Distance}}{\text{Time}}\).

If you have the speed and the time, you can find the distance by multiplying these values: \(\text{Distance} = \text{Speed} \times \text{Time}\). This understanding helps in situations ranging from calculating travel times to planning trips.

Unit Conversion

Unit conversion is essential when comparing quantities measured in different systems or units. It ensures consistency and correctness in calculations. For instance, if the distance were provided in miles and the speed in kilometers per hour, you would need to convert units to make them compatible.
When doing unit conversion, remember these tips:

• Always check the units you're working with to ensure consistency.
• Use conversion factors, like \(1 \text{ mile} = 1.60934 \text{ kilometers}\).
• Practicing converting between units helps avoid mistakes.

Whether converting miles to kilometers, or even metric units to imperial, understanding unit conversion allows you to handle mixed-unit math problems confidently. This skill proves invaluable in real life, where sources often provide information in varied units.