# Direct proportion

Direct proportion is the ratio between two quantities expressed in a definite value. Direct proportion is also known as direct variation and is represented by the symbol of proportionality ∝. The same symbol also denotes inverse proportionality between anything, but the right-hand value is reversed. For example, x ∝ y means x is directly proportional to y. Therefore, x ∝ 1/y denotes x is inversely proportional to y.

Direct proportion plays an essential role in everyday life. Our age is directly proportional to the food we eat or the exercise we do. Getting good marks in exams is directly proportional to the amount of study we do before exams. It is helpful for businesses to grow their revenue. If a customer has enjoyed a service provided by the industry, the profit will increase because they will visit the company more and more. Therefore, the growth of the company or collecting revenue is directly proportional to customer satisfaction.

## Symbol of proportionality

∝ is the symbol used for denoting proportionality between two or more situations. It is different from the alpha and infinity symbols. When this symbol denotes two quantities,

X ∝ Y, this means X is directly dependent upon Y. Whatever change will happen in X, Y will also change its value.

We can remove the proportionality symbol by inculcating a constant of proportionality. If x ∝ y, then x = Ky, where K is the constant of proportionality and has some specific value depending upon the relation derived for whatsoever situation.

If two or more values are expressed in equal proportions, they can be represented in the form of ratios, like a1 and a2 are two ratios ∝ to b1 and b2. Then equal proportions can be represented as-

## Examples for representing the direct proportion

If a machine manufactures 20 units per hour, and these units are directly proportional to how many hours it has worked, then according to the direct proportion phenomenon, the more work the machine does, the more units are manufactured. This means working and production are in direct proportion.

This can be written as: Units ∝ Hours Worked.

Therefore, if it works for 2 hours, we get 40 Units. If it works 4 hours, we get 80 Units, and so on.

Example

Find the height of a tree that casts a shadow of 10 meters, if another tree of 7 metres casts a shadow of 5 metres under similar conditions.

Solution

We know, the length of the shadow increases with the increase in height of the tree. Therefore, the height of the tree and the height of the shadow are in direct proportion.

We can represent the condition as-

Therefore, we get b1 = 14 metres, which is the height of the tree.

Example

A train travels 200 km in 5 hours. How much time will it take to cover 600 km?

Solution:

We know, the distance covered by a train is directly proportional to its time.

Therefore, 200/5 = 600/time taken

We get: time taken = 15 hours.

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