Inverse Proportion

Inverse proportion is an important concept in mathematics, especially in the field of ratio and proportion. When we say that two quantities are in inverse proportion, we mean that as one quantity increases, the other decreases such that their product remains constant. It may seem a bit complicated at first, but this detailed explanation will give you a clear idea about inverse proportion.

Defining inverse proportion

In basic terms, inverse proportion occurs when two variables, let's call them x and y, are such that the product of x * y is always a single number, which we will refer to as k, where k is not zero. Mathematically, it is expressed as:

x * y = k

This relation implies that if one variable increases, the other must decrease proportionately to keep the product k constant.

Simple examples of inverse proportions

Example 1: Speed and travel time

Let's consider a situation where you travel a certain distance. The time taken to cover this distance is inversely proportional to your speed of travel.

Suppose it takes you 4 hours to cover a distance of 200 km at a speed of 50 km per hour. If you increase your speed to 100 km per hour, how much time will it take you?

Using inverse proportion, if:

Speed (x) * Time (y) = Distance

Here speed and time are inversely proportional. So:

50 km/h * 4 hours = 200 km

Now, if you increase the speed to 100 km / h, you should find the new time y as:

100 km/h * y = 200 km

Solving for y, we get:

y = 200 km / 100 km/h = 2 hours

Hence, it will take 2 hours to travel at 100 km/hr, which is half the original time.

Example 2: Tasks and workers

Imagine a situation involving work and the number of workers. The time taken to complete a job is inversely proportional to the number of workers. More workers means the job will be completed faster.

For example, if 5 workers can complete a job in 10 days, how many days will it take for 10 workers to complete the same job?

Here, using inverse proportion:

Workers (x) * Time (y) = Constant Work Output (k)

Substitute the values:

5 workers * 10 days = 50 worker-days (constant)

Now, if 10 workers are used:

10 workers * y = 50 worker-days

Solving for y, we get:

y = 50 worker-days / 10 workers = 5 days

Thus, 10 workers will complete the work in 5 days.

Visual representation

Imagine that we plot the relationship between two variables on an inverse proportionality graph. This forms a hyperbolic curve, showing that as one value increases, the other decreases.

In this graph, the curve shows the inverse relationship, which shows that when one axis increases, the other decreases.

Real-world applications of inverse proportions

1. Economics

Inverse proportions can be found in economic scenarios, such as the relationship between supply and demand. Generally, if the price of a product decreases (and other factors remain constant), demand increases.

2. Science

Another example is the relationship between pressure and volume in gases. According to Boyle's law, for a fixed amount of gas at a constant temperature, pressure is inversely proportional to volume.

3. Engineering

In electrical engineering, when the voltage is held constant, the current flowing through a resistor is inversely proportional to the resistance, which demonstrates Ohm's law.

Practice problems

Problem 1: Construction work

If 8 workers take 20 days to build a wall, how many days will 16 workers take?

Using x1 * y1 = x2 * y2:

8 workers * 20 days = 16 workers * y

y = (8 workers * 20 days) / 16 workers = 10 days

Problem 2: Car travel

If a car takes 6 hours to travel between two cities at a speed of 40 km/h, how much time will it take to cover the same distance at a speed of 80 km/h?

40 km/h * 6 hours = 80 km/h * y

y = (40 km/h * 6 hours) / 80 km/h = 3 hours

Conclusion

Inverse proportion is a fascinating and useful concept that provides information about the relationship between variables that are dependent on one another in an inverse way. A clear understanding of inverse proportion can help solve many practical problems in fields ranging from mathematics and physics to economics and engineering. Through visualization and real-life applications, the idea of inverse proportion becomes a tangible and more intuitive concept.

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