It Took Ivan $7 \frac{1}{2}$ Hours To Drive 412.5 Miles At A Constant Speed. How Fast Was He Driving? Show How You Know.

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Understanding the Problem

In this problem, we are given the time it took Ivan to drive a certain distance and asked to find his speed. To solve this problem, we need to use the formula for speed, which is distance divided by time.

The Formula for Speed

The formula for speed is:

Speed = Distance / Time

Given Information

  • Distance: 412.5 miles
  • Time: 7 \frac{1}{2} hours

Converting Mixed Numbers to Improper Fractions

To make calculations easier, we need to convert the mixed number 7 \frac{1}{2} to an improper fraction. To do this, we multiply the whole number part (7) by the denominator (2) and add the numerator (1).

7 \frac{1}{2} = 7 \times 2 + 1 = 14 + 1 = 15/2

So, the time is 15/2 hours.

Converting the Time to a Decimal

To make calculations easier, we can convert the improper fraction 15/2 to a decimal. To do this, we divide the numerator (15) by the denominator (2).

15/2 = 7.5

So, the time is 7.5 hours.

Calculating Speed

Now that we have the distance and time in decimal form, we can plug these values into the formula for speed.

Speed = Distance / Time = 412.5 miles / 7.5 hours = 55 miles per hour

Conclusion

Therefore, Ivan was driving at a speed of 55 miles per hour.

Why This Matters

Understanding how to calculate speed is an important skill in mathematics, particularly in physics and engineering. It can be used to determine the time it takes to travel a certain distance, the distance traveled in a certain time, or the speed of an object.

Real-World Applications

Calculating speed has many real-world applications, such as:

  • Determining the time it takes to travel to work or school
  • Calculating the distance traveled in a road trip
  • Determining the speed of a vehicle in a traffic accident
  • Calculating the speed of a projectile in physics

Tips and Tricks

  • Always make sure to convert mixed numbers to improper fractions before making calculations.
  • Use decimals instead of fractions when possible to make calculations easier.
  • Make sure to plug in the correct values into the formula for speed.

Common Mistakes

  • Forgetting to convert mixed numbers to improper fractions
  • Forgetting to convert improper fractions to decimals
  • Plugging in the wrong values into the formula for speed

Conclusion

Q: What is the formula for speed?

A: The formula for speed is:

Speed = Distance / Time

Q: How do I convert a mixed number to an improper fraction?

A: To convert a mixed number to an improper fraction, you need to multiply the whole number part by the denominator and add the numerator. For example, to convert 7 \frac{1}{2} to an improper fraction, you would multiply 7 by 2 and add 1:

7 \frac{1}{2} = 7 \times 2 + 1 = 14 + 1 = 15/2

Q: How do I convert an improper fraction to a decimal?

A: To convert an improper fraction to a decimal, you need to divide the numerator by the denominator. For example, to convert 15/2 to a decimal, you would divide 15 by 2:

15/2 = 7.5

Q: What if I have a fraction with a denominator that is not a power of 10?

A: If you have a fraction with a denominator that is not a power of 10, you can convert it to a decimal by dividing the numerator by the denominator. For example, to convert 3/4 to a decimal, you would divide 3 by 4:

3/4 = 0.75

Q: How do I calculate speed if I have a distance in miles and a time in hours?

A: To calculate speed, you need to divide the distance by the time. For example, if you have a distance of 412.5 miles and a time of 7.5 hours, you would divide 412.5 by 7.5:

Speed = 412.5 miles / 7.5 hours = 55 miles per hour

Q: What if I have a distance in kilometers and a time in hours?

A: To calculate speed, you need to divide the distance by the time. For example, if you have a distance of 660 kilometers and a time of 5 hours, you would divide 660 by 5:

Speed = 660 kilometers / 5 hours = 132 kilometers per hour

Q: How do I calculate speed if I have a distance in feet and a time in minutes?

A: To calculate speed, you need to convert the distance from feet to miles and the time from minutes to hours. Then, you can divide the distance by the time. For example, if you have a distance of 5280 feet and a time of 30 minutes, you would convert 5280 feet to miles (5280/5280 = 1 mile) and 30 minutes to hours (30/60 = 0.5 hours):

Speed = 1 mile / 0.5 hours = 2 miles per hour

Q: What if I have a distance in meters and a time in seconds?

A: To calculate speed, you need to convert the distance from meters to kilometers and the time from seconds to hours. Then, you can divide the distance by the time. For example, if you have a distance of 1000 meters and a time of 10 seconds, you would convert 1000 meters to kilometers (1000/1000 = 1 kilometer) and 10 seconds to hours (10/3600 = 0.00278 hours):

Speed = 1 kilometer / 0.00278 hours = 360 kilometers per hour

Q: How do I calculate speed if I have a distance in yards and a time in minutes?

A: To calculate speed, you need to convert the distance from yards to miles and the time from minutes to hours. Then, you can divide the distance by the time. For example, if you have a distance of 1080 yards and a time of 20 minutes, you would convert 1080 yards to miles (1080/1760 = 0.612 miles) and 20 minutes to hours (20/60 = 0.333 hours):

Speed = 0.612 miles / 0.333 hours = 1.83 miles per hour

Conclusion

Calculating speed is an important skill that has many real-world applications. By understanding how to calculate speed, you can determine the time it takes to travel a certain distance, the distance traveled in a certain time, or the speed of an object.