Solving Speed and Distance Problems Using Algebraic Methods

In the field of mathematics, understanding the relationship between speed, distance, and time can help us solve complex problems. Today, we'll explore a specific problem that involves a person's travel distance and speed reduction over a certain period. Let's break down this problem and solve it step by step using algebraic methods.

The Problem

One person travels a certain distance in 30 hours. When the person's speed is reduced by ( frac{1}{15} ) part, he travels less than 10 km in the same time. The question is: What is the speed of the person in km/h?

Step-by-Step Solution

Let's start by defining the variables and setting up the equations based on the given problem.

Step 1: Define the original speed

Let the original speed of the person be ( v ) km/h.

The distance traveled in 30 hours at this speed is:

Distance Speed × Time v × 30

When the speed is reduced by ( frac{1}{15} ) of the original speed, the new speed becomes:

v - frac{1}{15}v frac{14}{15}v

At this reduced speed, the person travels less than 10 km in the same 30 hours. Therefore, we can express this as:

frac{14}{15}v × 30 - 10

Substituting the value:

frac{14}{15}v × 30 - 10 28v - 10

Simplifying this inequality:

28v - 10 0

Adding 10 to both sides:

28v 10

Dividing both sides by 28:

v frac{10}{28}

Further simplifying:

v frac{5}{14}

Since the speed must be a positive value, we find that the speed of the person must be less than approximately 0.357 km/h.

The distance covered in the original speed must also be checked to ensure it is a meaningful distance. Therefore, the original speed must be:

v × 30 30v

At the reduced speed, the distance covered is:

frac{14}{15}v × 30 28v

According to the problem, 28v - 10 10, which gives us:

28v 20

Dividing both sides by 28:

v frac{20}{28}

Simplifying:

v frac{5}{14}

Thus, the speed of the person is less than (frac{5}{14}) km/h, confirming that the speed is very low.

In summary, the speed of the person is:

v 0.357 , text{km/h}

This indicates that the person travels at a very slow speed and can only maintain this speed to fulfill both conditions of distance and time.

Another Approach

Another method to solve this problem involves directly using the given conditions and setting up an equation.

Let's assume the person's speed is ( x ) km/h. The distance covered at this speed in 30 hours would be:

3 , text{km}

When the speed is reduced by ( frac{1}{10} ), the new speed becomes:

x - frac{1}{10}x frac{9}{10}x

The distance covered at the reduced speed in 30 hours would be:

30 times frac{9}{10}x 27x , text{km}

According to the problem, the difference in distance is 9 km:

3 - 27x 9

Simplifying:

3x 9

Dividing both sides by 3:

x 3 , text{km/h}

This confirms that the speed of the person is 3 km/h.

I hope this helps!