Calculating Train Speeds and Their Sum

This problem involves using the relationship between speed, distance, and time, and applying a given ratio to find the speeds of two trains and their sum.

Understanding the Problem

We are given:

• The ratio of the speeds of two trains is 2 : 7.
• The first train covers a distance of 250 km in 5 hours.

We need to find the sum of the speeds of both trains in km/h.

Step-by-Step Solution to Find the Sum of Speeds

Step 1: Calculate the speed of the first train

The formula relating speed, distance, and time is:

$ \text{Speed} = \frac{\text{Distance}}{\text{Time}} $

For the first train:

• Distance = 250 km
• Time = 5 hours

Speed of the first train $= \frac{250 \text{ km}}{5 \text{ hours}}$

Speed of the first train $= 50 \text{ km/h}$

Step 2: Use the speed ratio to find the speeds

The ratio of the speeds of the two trains is given as 2 : 7.

Let the speeds of the two trains be $2x$ and $7x$ km/h, where $x$ is a common factor.

We know the speed of the first train is 50 km/h. So, we can set up the equation:

$ 2x = 50 \text{ km/h} $

Now, solve for $x$:

$ x = \frac{50}{2} $

$ x = 25 $

Now we can find the speed of the second train using $7x$:

Speed of the second train $= 7x = 7 \times 25 \text{ km/h}$

Speed of the second train $= 175 \text{ km/h}$

Step 3: Calculate the sum of the speeds

The sum of the speeds of both trains is the speed of the first train plus the speed of the second train.

Sum of speeds $= (\text{Speed of first train}) + (\text{Speed of second train})$

Sum of speeds $= 50 \text{ km/h} + 175 \text{ km/h}$

Sum of speeds $= 225 \text{ km/h}$

The sum of the speeds of both the trains is 225 km/h.

Summary of Speed Calculations

Revision Table: Key Concepts

Additional Information: Speed, Distance, and Time

The relationship between speed, distance, and time is fundamental in physics and many quantitative problems. It can be rearranged to find any of the three quantities if the other two are known.

• To find Distance: $ \text{Distance} = \text{Speed} \times \text{Time} $
• To find Time: $ \text{Time} = \frac{\text{Distance}}{\text{Speed}} $

Understanding ratios is also crucial. When a ratio is given, it often represents parts of a total or relationships between quantities. Introducing a variable like $x$ helps in solving for the actual values when one value corresponding to a part of the ratio is known.

In this problem, the speed of the first train corresponded to the '2 parts' of the 2:7 ratio, allowing us to find the value of one 'part' ($x=25$), and subsequently the speed corresponding to the '7 parts' for the second train.