NCERT Solutions for Class 8 Maths Chapter 13 Direct and Inverse Proportions: You must have come across many situations where changes in one quantity result in changes in other quantities. In mathematics, two quantities are said to be in proportional if the values of two quantities are related in such a way that changes in one results in a corresponding change in another quantity. You must have observed that when you increase the speed of the car it takes lesser time to cover the same distance. So speed is inversely proportional to time with a constant distance.
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In NCERT solutions for Class 8 Maths chapter 13 Direct and Inverse Proportions, you are going to deal with problems based on direct and indirect proportions. In this chapter, there are 2 exercises with 21 questions. The first exercise of this chapter cover problems based on the direct proportions and the second exercise cover problems based on the inverse proportions. All these questions are prepared in NCERT solutions for Class 8 Maths chapter 13 Direct and Inverse Proportions in a step-by-step manner. It will be very easy for you to understand the concept. There are solved examples, exercises, and daily life activities in the textbook for a better understanding of this chapter.
In solutions for Class 8 Maths chapter 13 Direct and Inverse Proportions, you will deal with problems related to properties and applications of direct proportion and inverse proportion. You can find NCERT Solutions from Classes 6 to 12 of Science and Maths by clicking on the above link. Here you will get the detailed NCERT Solutions for Class 8 by clicking on the link.
NCERT Solutions for Class 8 Maths Chapter 13 Direct and Inverse Proportions Topic 13.2 Direct Proportion
Question:1(i). Observe the following tables and find if x and y are directly proportional.
If we want to know that they are directly proportional, then we calculate :
Hence we can say that x and y are directly proportional as cames out to be a constant equals .
Question:2. Principal = Rs.1000, Rate = 8% per annum. Fill in the following table and find which type of interest (simple or compound) changes in direct proportion with time period.
Given that Principal (P) = 1000 and Rate (r) = 8% per annum(per year).
Calculating the Simple Interest:
The formula for the simple interest is = .
So, for 1 year :
for 2 years :
similarly for 3 years:
Calculating the Compound Interest :
The formula for the compound interest is
So for 1 year :
for 2 years :
similarly for 3 years:
Hence we have
Simple Interest(in rupees)
Compound Interest(in rupees)
In case of simple interest
Simple interest is directly proportional with time.
While in case of compound interest:
does not give the same constant.
Compound interest is not directly proportional with time.
Question:2 A mixture of paint is prepared by mixing 1 part of red pigments with 8 parts of base. In the following table, find the parts of base that need to be added.
Suppose the parts of red pigment is x and parts of base, is ' y '.
As the requirement of the number of parts of base for 1 part of red pigment is 8,
and as the parts of red pigment is increases, parts of the base also increase in the same ratio. It is a case of direct proportion.
we can assume other parts of the base that will be required for red pigments as y1, y2, y3 and y4 for parts of pigment 4, 7, 12 and 20 respectively.
We make use of the relation of type
That gives for the (i) case or .
32 parts of the base will be required for the 4 parts of the red pigment.
(ii) If parts of red pigment used is 7 then parts of base used will be
that gives or .
56 parts of the base will be required for the 7 parts of the red pigment.
(iii) for 12 parts of red pigment :
we have or .
so, 96 parts of the base will be required for the 12 parts of the red pigment.
(iv) for 20 parts of red pigment:
we have or .
Hence for the following parts of red pigment, parts of base are given :
Parts of red pigment
Parts of base
Question:5 A photograph of a bacteria enlarged 50,000 times attains a length of 5 cm as shown in the diagram. What is the actual length of the bacteria? If the photograph is enlarged 20,000 times only, what would be its enlarged length?
We can calculate it easily,
Given that for 50,000 times enlarged, the photograph of a bacteria attains a length of 5cm.
So, we have to calculate the actual length of the bacteria(assume it to be ' x 'cm) that is when the photograph is enlarged to 1 time
Knowing that the microscope's zoom has a direct relation with the length of bacteria observed So, we get the relation ;
Solving the equation for x we get;
The actual length of the bacteria is which is so small to be observed through naked eyes.
Now, calculating the length of bacteria when it the photograph is enlarged 20,000 times,
assume it as ' y '.
So, we get this relation
solving for y we get;
So, if the photograph is enlarged 20,000 times only then the enlarged length would be 2cm.
Question:7 Suppose 2 kg of sugar contains crystals. How many sugar crystals are there in (i) 5 kg of sugar? (ii) 1.2 kg of sugar?
Here given that 2 kg of sugar contains crystals.
So, we have to find the number of crystals in 5 kg of sugar as well as in 1.2 kg of sugar:
As here we will assume that all crystals have the same dimensions i.e., length, breadth, and width. Then the weight of sugar follows the direct proportion with the number of crystals as increasing the number of crystals there will be an increase in the weight also.
(i) For 5 kg sugar:
let the number of crystals be 'x' then,
we have the relation:
, calculating x from this relation,
Therefore 5 kg of sugar contains sugar crystals.
(ii) For 1.2 kg sugar:
Let the number of cystals be 'y' then,
we have the relation:
. calculating similarly for y we get
Therefore 1.2 kg of sugar contains sugar crystals.
Question:9 A 5 m 60 cm high vertical pole casts a shadow 3 m 20 cm long. Find at the same time
(i) the length of the shadow cast by another pole 10 m 50 cm high
(ii) the height of a pole which casts a shadow 5m long.
Consider there is a direct proportion relation of pole height with pole shadow.
So we have 5m 60cm high verticle pole that casts a shadow of 3m 20cm long .
(i) for the length of the shadow cast by another pole of 10 m 50cm high would be ' x ' cm;
finding x from the equation we get;
Therefore for a 10m 50cm pole we would get a shadow of 600cm or 6m length.
(ii) The height of a pole which casts a shadow of 5m long let it be 'y'
similar relation holds here also, so we can apply it once more
we get .
Thus, the height of a pole which casts a shadow of 5m long is 8m 75cm.
Finding if x and y are in inverse proportion:
Two quantities x and y are said to vary in inverse proportion, if there exists a relation of the type xy = k between them, k being a constant
so, calculating xy for each case,
clearly xy is not equal to a constant value 'k',
hence x and y are not inversely proportional.
NCERT Solutions for Class 8 Maths Chapter 13 Direct and Inverse Proportions-Exercise: 13.2
Question:1 Which of the following are in inverse proportion?
(i) The number of workers on a job and the time to complete the job.
(ii) The time taken for a journey and the distance travelled in a uniform speed.
(iii) Area of cultivated land and the crop harvested.
(iv) The time taken for a fixed journey and the speed of the vehicle.
(v) The population of a country and the area of land per person.
(i) As the number of workers on a job increases the time taken to complete the job decreases, hence it is an inverse proportion .
(ii) Distance and time are directly proportional to each other as time increases you could travel more distance compared to if you get less time to travel. Hence it is not an inverse proportion.
(iii) Both area of cultivated land and crop arvested are directly proportional, more the area cultivated more crop harvest. Hence it is not an inverse proportion.
(iv) With more speed if you are travelling lesser the time taken for a fixed journey to complete. Hence it is an inverse proportion.
(v) The population of a country if increases then there would be lesser area available per person, Hence it is an inverse proportion.
Question:2 In a Television game show, the prize money of ` 1,00,000 is to be divided equally amongst the winners. Complete the following table and find whether the prize money given to an individual winner is directly or inversely proportional to the number of winners?
Number of winners
Prize for each winner (in Rs.)
Let us assume that for the number of winners 4, 5, 8, 10 and 20 are x1, x2, x3, x4, and x5 respectively and given that the prize money of Rs. 1,00,000 is to be divided equally amongst the winners.
Thus we have,
For 4 number of winners:
So Rs. 25,000 to be distributed among each.
For 5 number of winners:
So Rs. 20,000to be distributed among each.
For 8 number of winners:
So Rs. 12,500 to be distributed among each.
For 10 number of winners:
So Rs. 10,000 each would get.
similarly for 20 number of winners:
So Rs. 5000 each would get.
Hence we have;
Number of winners (x)
Prize for each winner (in ? ) (y)
Two quantities x and y are said to be in inverse proportional if they satisfy the given relation;
xy=k; where k is a constant.
Clearly, we can see that the prize money given to an individual winner is inversely proportional.
Question:4 If a box of sweets is divided among children, they will get sweets each. How many would each get, if the number of the children is reduced by ?
Given that the box of sweets is divided among 24 children, getting 5 sweets each.
If the number of children is reduced by 4 then the number of children now is .
As here if the number of children increases then the number of sweets they will get decreases hence we can say that there exists an inverse relationship between them.
if we assume:
Number of children before (x1) = 24
Number of sweets each would get before (y1) = 5
and number of children after reduction (x2) = 20
and the number of sweets each would get after reduction is y2
Then the relation holds;
or or .
Hence each child will get 6 number of sweets .
Question:10(i) Two persons could fit new windows in a house in 3 days.
One of the persons fell ill before the work started. How long would the job take now?
Here, given that 2 persons could fit new windows in a house in 3 days.
(i) 1 person has fallen ill so, now the number of persons remaining is only one. Assume that the only person which is working takes the time of 'x' days.
hence we could write the inverse relation as;
One person will take 6 days to complete that window job.
Question:10(ii) Two persons could fit new windows in a house in 3 days.
How many persons would be needed to fit the windows in one day?
(ii) So, now we are calculating the number of persons that would be needed to fit the windows in one day. Let it be 'y'.
So from previous part (i) we have the relation; .
hence the required number of persons would be 6.
Direct and Inverse Proportions Class 8 Chapter 14-topics
- Direct Proportion
- Inverse Proportion
What is a Direct Proportion?
Two quantities 'a' and 'b' are said to be in direct proportion if the increase (decrease) in 'a' results in the increase (decrease) in 'b' in such a manner that the ratio of their corresponding values remains constant. That is if a/b= k
What is Inverse Proportional?
Two values are Inversely proportional it means decreases in one value results in a gradual increase in other value. Example- If the distance is fixed, speed and time are inversely proportional to each other.
( If Distance is fixed)
NCERT Solutions for Class 8 Maths: Chapter-Wise
NCERT Solutions for Class 8: Subject-Wise
Benefits of NCERT Solutions for Class 8 Maths Chapter 13 Direct and Inverse Proportions-
You will also know different ways to solve the problems.
You will also get solutions to the practice questions given below every topic which will give you conceptual clarity.
You will also get some short tricks and tips to solve some specific problems.
In one of the questions of NCERT solutions for Class 8 Maths chapter 13 Direct and Inverse Proportions, you will learn to calculate simple and compound interest which may be useful in real life too.
- It is going to help you with your homework as all the practice questions including questions given below every topic are covered in this article.
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