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Direct and Inverse Proportions

Direct and Inverse Proportions

Sometimes a change in the proportions of one quantity means a change in the proportions of the other! For example, when you buy more apples, you’ll have to pay more money. Similarly, if we increase the speed of a vehicle, the time that it takes to cover some distance goes down. The first is an example of direct proportionality and the second is an example of quantities that are inversely proportional. Let’s read further.

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Direct Proportion

Suppose you increase the number of books in your bag. What will happen to its weight? It will also increase. This is direct proportionality. Let’s take two quantities, x and y which are believed to be in direct proportion. Now, what would that mean? It means, that these two quantities, x and y, will increase and decrease in such a manner that the ratio of their corresponding values always remains constant.

inversely proportional

Let’s understand this with the help of an equation. If, x/y = k, where k is a positive number or a constant, then x and y are said to vary directly. Let’s say if y1 and y2 are the values of y corresponding to the values x1 and x2 of x, respectively, then:

How to solve problems with Direct Proportion?

There are two methods to solve a problem with Direct Proportion.

Method 1

We know, that in a direct proportion:

So, in the case of a direct proportion, one ratio would be given. Then, we’ll have to use the equation above and find all the unknown quantities. Let’s understand this with the help of an example:

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Q: The cost of 5 kg of a particular quality of sugar is Rs 200. Tabulate the cost of 1,2, 4, 10 and 14 kg of sugar of the same type?
Solution: Let x kg of sugar cost y rupees:

x 1 2 4 5 10 14
y ? ? ? 200 ? ?

We already know that with the increase in the quantity of sugar, the cost of the sugar will also increase in the same ratio. This is the general rule of direct proportion. Now, to solve the problem, we’ll use the equation above:

Now, we have:

1/y = 5/200 or y = Rs 40

2/y = 5/200 or y=Rs 80

Using, the above two results, we can find the other values as well. So, the complete table would be as below:

x 1 2 4 5 10 14
y 40 80 160 200 400 560

Method 2

We already know that in the case of a direct proportion: x / y = k i.e. x = k × y. So, now, we can find the value of the k from the values that are already known to us and then use the formula above to calculate all the unknown values. Hope, with that you have understood both the methods of solving a problem of direct proportion.

Inversely Proportional

Now, what is an inverse proportion? So, two quantities, x and y would be inversely proportional, if an increase in the value of x leads to a proportional decrease in the value of y and vice versa. The speed and time of a journey is one example. The increase or decrease in the proportion would be in such a manner that the corresponding values happen to be constant.

inversely proportional

This means that if, x y = k, then here k is always constant. That is when both x and y are said to vary inversely. Let’s now understand the concept of inverse proportion with the help of an equation. In inverse proportion: x1 y1 = x2 y2. What does this mean? This means, that if y1 and y2 are the values of y corresponding to the values of x1 and x2 of x, respectively, then x1 y1 = x2 y2

How to solve problems with Inversely Proportional variables?

Again, there are two methods to solve a problem with inversely proportional variables.

Method 1

We know that in the inverse proportion,

x1 y1 = x2 y2 = x2 y2 = x2 y2

So, when you are told to solve this problem, one pair would always be given. Then, we can use the equation above, to find the terms that are unknown to use.

Method 2

We know that in the inverse proportion, x × y= k. This means that x = k/y. So, to find the value of the k, you can use the known values and then use the formula above to calculate all the unknown values. Now, let’s understand this situation with the help of an example.

Example: If 20 workers can build a wall in 48 hours, how many workers will be required to do the same work in 30 hours?
Solution: Let the number of workers employed to build the wall in 30 hours be y. We have the following table.

Number of Hours 48 30
Number of workers 20 y

Obviously more the number of workers, faster will they build the wall.
So, the number of hours and number of workers vary in inverse proportion.
So 48 × 20 = 30 × y
y=32 workers
So  to finish the work in 30 hours, 32 workers are required

Solved Examples for You

Question 1: Following are the car parking charges near an Airport up to

  1. 2 hours Rs 60
  2. 6 hours Rs 100
  3. 12 hours Rs 14
  4. 24 hours Rs 180

Check if the parking charges are in direct proportion to the parking time.

Answer : We know that two quantities are in direct proportion if whenever the values of one quantity increase, then the value of another quantity increase in such a way that ratio of the quantities remains same. Here, the charges are not increasing in direct proportion to the parking time because of 2/60 ≠ 6/100 ≠ 12/140 ≠ 24/180

Question 2: What does inversely proportional mean?

Answer:It refers to the relationship between two variables in which the product is a constant.Furthermore, in it when one variables increases the other decreases in proportion so that the product is unchanged. E.g. if b is inversely proportional to a then the equation is in the form b = k/a, where k is a constant.

Question 3: What is the difference between direct and inverse proportion?

Answer: In a direct proportion the ratio between matching quantities remain the same if they we divide them. On the other hand, in an inverse or indirect proportion as one-quantity increases, the other automatically decreases.

Question 4: State the equation for inverse proportion?

Answer:The equation for inverse proportion is x  y = k or x = k/ y. Therefore, for finding the value of the constant k, you can use the known values and then use this formula to calculate all the unknown values.

Question 5: Does proportional mean equal?

Answer:When anything is proportional to anything else, it means that they change in respect to each other and it does not mean that the values are equal. However, the constant of proportionality serves as a multiplier.

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