# DIRECT AND INVERSE PROPORTION SHORT NOTES AND IMPORTANT QUESTIONS

Hi students, welcome to **Amans Maths Blogs (AMB)**. On this page, you will get the short notes and important question of **CBSE Class 8 Chapter 13 Direct and Inverse Proportion**.

Read : Square & Square Roots

These short notes and important questions are taken from the NCERT text books and other books. If you want to get the **NCERT Solution for class 8 Maths Direct and Inverse Proportion Chapter**, then **click here**.

If you need sample questions based on direct and inverse proportion for practice, then you are at right place.

Read : Cubes & Cube Roots

## Direct and Inverse Proportions Short Notes

If the values of two quantities depend on each other in such a way that a change in one result in a corresponding change in other, then the two quantities are called in variations. There are two types of variations as below.

### Direct Variations

If two quantities are related in such a way that if an **increase in one quantity** produces a proportional **increase in other quantity** or if **decrease in one quantity** produces a proportional **decrease in other quantity**, then the two quantities are known as **DIRECT VARIATIONS**.

For example:

If we need to buy **more** pens, then we require **more** money to buy the pens. Here, the number of pens and cost of money are in direct variation.

If a school has **more** number of students, then the school is required **more** number of teachers to teach the students. Here, the number of students and the number of teachers are in direct variations.

If we need to travel **less** distance, then we require **less** time taken to complete the journey. Here, the the distance and the time taken are in direct variation.

If we need to **less** number of books, then we require **less** money to buy the books. Here, the number of books and the money are in direct variation.

Thus, if two quantities x and y are in direct variations, then x/y is constant. It means, x is proportional to y. x = ky x/y = k (constant).

In such a case, if y_{1} and y_{2} are two values of y corresponding to the values of x_{1} and x_{2} of x respectively, then . Similarly, .

### Indirect Variations

If two quantities are related in such a way that if an **increase in one quantity** produces a proportional **decrease in other quantity** or if **decrease in one quantity** produces a proportional **increase in other quantity**, then the two quantities are known as **INDIRECT VARIATIONS**.

For example:

If we have **more** number of workers, then **less** time to complete the work. Here, the number of workers and time taken are in inverse variation.

If a car has **more** speed, then it takes **less** time to travel Here, the speed of the car and the time taken are in inverse variations.

If we need to complete a work in **less** time, then we require **more** number of employee to complete the task. Here, the the time taken to complete the work and the number of employees are in inverse variation.

Thus, if two quantities x and y are in inverse variations, then xy is constant. It means, x is proportional to 1/y. x = k/y xy = k (constant).

In such a case, if y_{1} and y_{2} are two values of y corresponding to the values of x_{1} and x_{2} of x respectively, then x_{1}y_{1} = x_{2}y_{2}. Similarly, x_{1}y_{1} = x_{2}y_{2} = x_{3}y_{3}…

## Direct and Inverse Proportions Important Questions

**Direct and Inverse Proportion Important Question No 1:**

A worker is paid Rs 2000 for 8 days work. If he works for 20 days, how much will he paid?

**Answer: Rs 5000**

**Direct and Inverse Proportion Important Question No 2:**

Sanjay bought 12 registers for Rs 156. Find the cost of 8 registers.

Read : Linear Equations in One Variable

**Answer: Rs 104**

**Direct and Inverse Proportion Important Question No 3:**

If 36 men can do a piece of work in 25 days, then in how many days will 15 men complete it?

**Answer: 60 Days**

**Direct and Inverse Proportion Important Question No 4:**

A jeep finishes a journey in 9 hours at a speed of 60 km/h. By how much should its speed be increased so that it may take only 6 hours to finish the same journey?

**Answer: 30 km/h**

**Direct and Inverse Proportion Important Question No 5:**

An overhead tank has two taps. One is working as inlet and other as an outlet. If the tank can be filled in 3 hours by the inlet tap and it can be finished by the outlet tap in 4 hours, then find the time required to fill the tank if both the taps are opened simultaneously.

**Answer: 12 hours**

**Direct and Inverse Proportion Important Question No 6:**

45 cows graze a field in 13 days. how many cows will graze the same filed in 9 days?

**Answer: 65 days**

**Direct and Inverse Proportion Important Question No 7:**

A contractor employed 210 men to build a house in 60 days. After 12 days, he was joined by 70 more man. In how many days will the remaining work be finished?

**Answer: 36 days**

**Direct and Inverse Proportion Important Question No 8:**

Sudha cycles to her school at an average speed of 12 km/h. It takes her 20 minutes to reach the school. If she wants to reach her school in 15 minutes, what should be her average speed?

**Answer: 16 km/h**

**Direct and Inverse Proportion Important Question No 9:**

Two pipes A and B can fill a tank in 36 hours and 45 hours respectively. If both the pipes are opened simultaneously, how much time will be taken to fill the tank?

**Answer: 20 hours**

**Direct and Inverse Proportion Important Question No 10:**

A and B undertake to due to do a piece of work for Rs 600. A alone can do it in 6 days while B alone can do it in 8 days with the help of C, they finish it in 3 days. Find the share of each.

**Answer: Rs. 300, Rs. 225, Rs. 75**

**Direct and Inverse Proportion Important Question No 11:**

What number must be added to each of the numbers 6, 15, 20 and 43 to make four numbers proportional?

**Answer: 3**

**Direct and Inverse Proportion Important Question No 12:**

If A : B = 3 : 4, B : C = 7 : 9, find A : B : C.

**Answer: 21 : 28 : 36**

**Direct and Inverse Proportion Important Question No 13:**

If x varies directly as y and inversely as z and if x = a, y = b and z = c. Find x when y = b^{2} and z = c^{2}.

**Answer: ab/c**

**Direct and Inverse Proportion Important Question No 14:**

If z varies as px + y and if z = 3 when x = 1 and y = 2 and z = 5, when x = 2 and y = 3, find p.

**Answer: 1**

**Direct and Inverse Proportion Important Question No 15:**

P and Q together can complete a job in 8 days and 16 days respectively. They work on alternate days with Q starting the job. In how many days will the job be completed?

**Answer: 11 days**

**Direct and Inverse Proportion Important Question No 16:**

A mixture containing alcohol and water in the ratio 3 : 2. If 4 lt of water is added to the mixture the ratio becomes 3 : 4. Find the quantity of water in the given mixture.

Answer: 4 litres

**Direct and Inverse Proportion Important Question No 17:**

If 30 men working 7 hours a day can do a piece of work in 18 days. In how many days will 21 men working 8 hours a day, do the same work?

**Answer: 22.5 days**

**Direct and Inverse Proportion Important Question No 18:**

10 students can do a job in 8 days, but on the starting day, 2 of them informed that they are not coming. By what fraction the number of days required for whole work will change?

**Answer: 1/4**

**Direct and Inverse Proportion Important Question No 19:**

If the work done by (x – 1) men in (x + 1) days is to the work done by (x + 2) men in (x – 1) days is in the ratio of 9 : 10. Find x.

**Answer: 8**

**Direct and Inverse Proportion Important Question No 20:**

What number must be added to each of the numbers 10, 18, 22, 38 to get the numbers which are in proportion?

**Answer: 2**

**Direct and Inverse Proportion Important Question No 21:**

Sharad covers two-third of a certain distance at 4 km/h and remaining at 5 km/h. If he takes 42 minutes in all, then what is the total distance?

**Answer: 3 km**

**Direct and Inverse Proportion Important Question No 22:**

Sheetal has enough money to buy 75 machines worth Rs 200 each. How many machine can she buy if she gets a discount of Rs. 50 on each machine?

**Answer: 100 machine**

**Direct and Inverse Proportion Important Question No 23:**

A job can be completed by P in 15 days and by Q in 10 days. If they work on alternate days starting with P. In how many days will the job be completed?

**Answer: 12 days**

**Direct and Inverse Proportion Important Question No 24:**

If 12 boys earn Rs 840 in 7 days, what will 15 days earn in 6 days?

**Answer: Rs 900**

**Direct and Inverse Proportion Important Question No 25:**

In a hostel of 50 girls, there are food provision for 40 days. If 30 more girls join the hostel, how will these provision last?

**Answer: 25 Days**

**Direct and Inverse Proportion Important Question No 26.**

A and B together can do a task in t hours. A and B individually can complete the task in (t + 2) hours and (t + 18) hours respectively. Find t.

**Answer: 6 hours**

**Direct and Inverse Proportion Important Question No 27.**

A can do a piece of work in 25 days and b can finish it in 20 days. They work together for 5 days and then A goes away. In how many days will B finish the remaining work?

**Answer: 11 days**

**Direct and Inverse Proportion Important Question No 28.**

An employer reduces the number of employees in the ratio of 15 : 8 and increases their wages in the ratio 14 : 25. In what ratio, the wages bill is increased or decreased?

**Answer: 21 : 20**

**Direct and Inverse Proportion Important Question No 29.**

A and B can do a piece of work in 10 days, B and C in 15 days, C and A in 12 days. How long would A and B take separately to do the same work?

**Answer: 17(1/7) days, 24 days**

**Direct and Inverse Proportion Important Question No 30.**

A contractor undertakes to construct a road in 20 days and engages 12 workers. After 16 days, he find that 2/3 part of the work has been done. How many more workers should be now engage in order to finish the job in time?

**Answer: 12 workers**

**Direct and Inverse Proportion Important Question No 31.**

Three taps A, B, C can fill a tank in 10 minutes, 15 minutes and 20 minutes respectively. If the taps were shut off after 5 minutes, what is the volume of water that overflowed (in %).

**Answer: 8(1/3) %**

**Direct and Inverse Proportion Important Question No 32.**

8 men or 12 boys can do a piece of work in 25 days. In how many days can 6 men and 11 boys complete the work?

**Answer: 15**

**Direct and Inverse Proportion Important Question No 33.**

6 men and 9 women can do a piece of work in 4 days. 4 men and 5 women can do it in 8 days. In how many days can 20 men and 6 women do the same work?

Answer: 3

**Direct and Inverse Proportion Important Question No 34.**

If Px/(b – c) = Qy/(c – a) = Rz/(a – b), then find the value of Pax + Qby + Rcz?

**Answer: 0**

**Direct and Inverse Proportion Important Question No 35.**

If x + y + z = 0 and (x^{3} + y^{3})/xyz = (y^{3} + z^{3})/xyz = (z^{3} + y^{3})/xyz = a, then find the value of a

**Answer: 2**

**Direct and Inverse Proportion Important Question No 36.**

90 men can dig a well in 2 days after they have worked for 5 days. How many more men should be employed so that to complete the work in another 10 days?

**Answer: 45**

**Direct and Inverse Proportions Important Question No 37.**

In a school, the maximum strength of each class is 60. In class, then total number of students is a multiple of 5. On a certain day, 4 boys were absent and the ratio of the numbers of the remaining boys to that of the girls was 5 : 4. Find the ratio of total boys to that of girls.

**Answer: 3 : 2**

**Direct and Inverse Proportions Important Question No 38.**

In a test, Rakesh got 50% marks and scored 10 marks more than the pass mark. In the same test, Suresh got 55% marks and scored 20 marks more than pass mark. Find the pass mark.

**Answer: 90**

**Direct and Inverse Proportions Important Question No 39.**

A person leaves his house and travel at 4 km/h reaches his office in 10 minutes later. Had he traveled at 6 km/h, he would have reached 20 minutes early. Find the distance from his house to the office.

**Answer: 6 m**

**Direct and Inverse Proportions Important Question No 40.**

How long will a train 100 m long and traveling at a speed of 45 km/h take to cross a platform of length 150 m?

**Answer: 20 sec**