Solving Train and Car Speed, Distance, Time Problem

This problem involves calculating speeds and times for a train and a car based on their given relationship and individual journeys. We will use the fundamental relationship between speed, distance, and time: Speed = Distance / Time.

Step-by-Step Calculation

Calculating the Car's Average Speed

We are given that the car covers a distance of 990 km in 15 hours. Using the formula, we can find the car's average speed.

Distance covered by car = 990 km

Time taken by car = 15 hours

Average speed of car = \(\frac{\text{Distance}}{\text{Time}}\)

Average speed of car = \(\frac{990 \text{ km}}{15 \text{ hours}}\)

Average speed of car = 66 km/hour

Calculating the Train's Average Speed

The question states that the average speed of the train is 180% of the average speed of the car. To find the train's speed, we calculate 180% of 66 km/hour.

Average speed of train = 180% of Average speed of car

Average speed of train = \(\frac{180}{100} \times 66 \text{ km/hour}\)

Average speed of train = \(1.8 \times 66 \text{ km/hour}\)

Average speed of train = 118.8 km/hour

Calculating the Time Taken by the Train

We need to find the time taken by the train to cover a distance of 891 km. We will use the formula Time = Distance / Speed.

Distance to be covered by train = 891 km

Average speed of train = 118.8 km/hour

Time taken by train = \(\frac{\text{Distance}}{\text{Speed}}\)

Time taken by train = \(\frac{891 \text{ km}}{118.8 \text{ km/hour}}\)

To calculate this value:

Time taken by train = \(\frac{891}{118.8} = \frac{8910}{1188}\) hours

Now, simplify the fraction:

• Divide both numerator and denominator by 9: \(\frac{8910 \div 9}{1188 \div 9} = \frac{990}{132}\)
• Divide both numerator and denominator by 6: \(\frac{990 \div 6}{132 \div 6} = \frac{165}{22}\)
• Divide both numerator and denominator by 11: \(\frac{165 \div 11}{22 \div 11} = \frac{15}{2}\)

So, the time taken by the train is \(\frac{15}{2}\) hours.

Converting this improper fraction to a mixed number:

\(\frac{15}{2} = 7 \text{ with a remainder of } 1\)

Thus, \(\frac{15}{2} = 7 \frac{1}{2}\) hours.

The time taken by the train to cover the distance of 891 km is \(7 \frac{1}{2}\) hours.

Time taken by Train = \(\frac{891 \text{ km}}{118.8 \text{ km/hour}} = 7.5 \text{ hours} = 7 \frac{1}{2} \text{ hours}\)

Revision Table: Key Concepts for Speed, Distance, Time

Additional Information on Percentage Calculations

Understanding percentages is crucial for solving problems involving comparisons like the train's speed being a percentage of the car's speed. 180% means \(\frac{180}{100}\) or 1.8 times the original value.

To calculate 'X% of a value Y', you can use the formula:

X% of Y = \(\left(\frac{X}{100}\right) \times Y\)

In this problem, X = 180 and Y = 66 km/hour. So, 180% of 66 km/hour is \(\left(\frac{180}{100}\right) \times 66 = 1.8 \times 66 = 118.8\) km/hour.

Problems involving speed, distance, and time often require converting between units or interpreting percentages correctly. Always ensure that the units for distance and time are consistent when using the formulas.