Calculating Speed in km/h from Distance and Time

This problem requires us to calculate the speed of a person who crosses a certain distance in a given amount of time. The distance is provided in meters and the time in minutes, but the required speed unit is kilometers per hour (km/h). Therefore, we need to convert the given units before calculating the speed.

Given Information

• Distance covered = 700 meters
• Time taken = \(3\frac{1}{2}\) minutes

Unit Conversion for Speed Calculation

To find the speed in km/h, we must convert the distance from meters to kilometers and the time from minutes to hours.

Converting Distance to Kilometers

We know that 1 kilometer = 1000 meters.

So, 700 meters = \(\frac{700}{1000}\) kilometers = 0.7 kilometers.

Distance in km = 0.7 km.

Converting Time to Hours

The time taken is \(3\frac{1}{2}\) minutes.

First, convert the mixed fraction to an improper fraction: \(3\frac{1}{2} = \frac{(3 \times 2) + 1}{2} = \frac{6 + 1}{2} = \frac{7}{2}\) minutes = 3.5 minutes.

We know that 1 hour = 60 minutes.

So, 3.5 minutes = \(\frac{3.5}{60}\) hours.

Time in hours = \(\frac{3.5}{60}\) hours.

Calculating Speed in km/h

The formula for speed is:

Speed = \(\frac{\text{Distance}}{\text{Time}}\)

Using the converted units:

Speed = \(\frac{0.7 \text{ km}}{\frac{3.5}{60} \text{ hours}}\)

Speed = \(0.7 \times \frac{60}{3.5}\) km/h

To simplify the calculation, we can write 0.7 as \(\frac{7}{10}\) and 3.5 as \(\frac{35}{10}\).

Speed = \(\frac{7}{10} \times \frac{60}{\frac{35}{10}}\) km/h

Speed = \(\frac{7}{10} \times \frac{60 \times 10}{35}\) km/h

Speed = \(\frac{7}{10} \times \frac{600}{35}\) km/h

Now, we can cancel common factors. 7 divides 35 five times, and 10 divides 600 sixty times.

Speed = \(\frac{1}{1} \times \frac{60}{5}\) km/h

Speed = \(\frac{60}{5}\) km/h

Speed = 12 km/h

Alternatively, using decimals directly:

Speed = \(0.7 \times \frac{60}{3.5}\) km/h

Speed = \(0.7 \times \frac{600}{35}\) km/h (Multiply numerator and denominator by 10 to remove decimal)

Speed = \(7 \times \frac{600}{350}\) km/h (Multiply 0.7 by 10 to get 7)

Speed = \(7 \times \frac{60}{35}\) km/h (Cancel 10 from 600 and 350)

Speed = \(1 \times \frac{60}{5}\) km/h (Cancel 7 from 7 and 35)

Speed = 12 km/h

The speed of the person is 12 km/h.

Revision Table: Key Concepts

Additional Information: Understanding Speed, Distance, and Time

Speed, distance, and time are fundamental concepts in physics and mathematics. They are related by simple formulas that are essential for solving many problems.

• Distance: This is the total length covered by a moving object. It is typically measured in meters (m), kilometers (km), miles, etc.
• Time: This is the duration for which the motion occurs. It is typically measured in seconds (s), minutes, hours (h), etc.
• Speed: This is the rate at which an object covers distance. It tells us how fast an object is moving. It is calculated as distance divided by time. Common units are meters per second (m/s) and kilometers per hour (km/h).
• Consistency of Units: When using the speed formula, it is crucial that the units for distance and time are consistent with the units of speed. If speed is required in km/h, distance must be in km and time in hours.
• Conversion Factors: Knowing standard conversion factors (like 1 km = 1000 m, 1 hour = 60 minutes, 1 minute = 60 seconds) is vital for solving problems involving mixed units.

This problem demonstrates a typical application of these concepts, requiring careful unit conversion before applying the speed formula.