Chapter 13

Direct and Inverse Proportions

Let’s explore a concept from everyday life: proportionality! If a change in one physical phenomenon creates a predictable change in another phenomenon, they are related. The change happens proportionally, either in the same direction or opposite directions. Learn more about this in NCERT Maths Chapter 13 – Direct and Inverse Proportions.

Topics Covered in Chapter 13 – Direct and Inverse Proportions

Introduction

 

Direct Proportion

 

Inverse Proportion

  

NCERT Solutions

  

 

Introduction

The physical world is run by scientific laws and phenomena. There are ample examples of proportions in physics and chemistry. For instance, sound is a wave that travels through air. That’s why we can hear sounds. As the temperature increases, sound waves travel faster than usual. This is because the air gets lighter during the summer, and its density decreases. Conversely, as the temperature decreases, sound waves travel slower than usual. This is because the air gets heavier in colder seasons, and its density increases.

Against this background, many questions arise: what is the relationship between the speed of sound and the temperature of air? What is the relationship between the speed of sound and the density of air? Finally, how does temperature change affect the density of air?

In the natural world, phenomena like speed, time, heat level, and air density are all examples of quantities. In science, a quantity is measurable on a specific scale made for it. Weight can be measured in grams or pounds; time is measured in milliseconds, minutes, hours, and even years, decades, and centuries; and heat is measured in Celsius or Fahrenheit.

These are the kinds of questions that help further explore direct and inverse proportions.

The learning app MSVgo explores such concepts in detail, simply and vividly. Let’s study the class 8 concept of direct and inverse proportions in depth.

Take “z” and “y” as two quantities. They are in direct proportion if an increase in z leads to an increase in y, and a decrease in z also leads to a decrease in y.

Going back to the example of sound, as the air gets hotter, the speed of sound increases. This is a direct proportion.

Direct proportionality is shown as z α y.

Therefore, z = ky.

Thus, k = z/y.

“K” is called the constant of proportionality. You can use any English or Greek letter in place of “k.” Please note that this constant number is not a quantity like weight, speed, or time. It cannot be measured in units like grams or seconds. However, it is always a positive number.

Moving on to an example:

A circle has a radius of 3 cm or a diameter of 6 cm. To find its circumference, the formula would be:

C = 2πr or πd. Take the value of π = ~3.14

C = 6 x 3.14 = 18.84 cm

The formula can be re-written as π = C/d

In this example, π = 18.84/6 = 3.14

Obviously, if the diameter increases, so will the circumference.

Let d = 10 cm. C = 3.14 x 10 = 31.4 cm

And C/d = 31.4/10 = 3.14

No matter what the new value of the diameter, the circumference will increase or decrease in a certain proportion.

The ratio of C/d will always be constant. Its value will always be ~3.14, which is the value of π.

In direct proportion, the ratio (division) of the quantities—z/y—is always a constant numerical value. If z/y = k, then z and y are in direct proportion.

In the case of inverse proportion, when the measure of one quantity “y” increases, the measure of another quantity “z” decreases. Further, as the measure of y decreases, the measure of z increases.

Going back to the example of sound, as the air density increases (air gets heavier), the speed of sound decreases. When the air gets lighter or less dense, the speed of sound increases.

The inverse relationship is shown as z α 1/y or y α 1/z

Thus, z = k/y or y = k/z

Again, “k” is the constant of proportionality.

To take an example:

If ten construction workers can build a house in ten days, in how many days can 50 construction workers build the same house?

Here, it is clear that when there are more workers, the work can be done in less time. There is an inverse relationship between the number of workers (N) and the time taken (T) to do the work of constructing the house. An increase in N will result in a decrease in T.

N = 10 workers and T = 10 days

N = k/T. Thus, k = (N)(T) = 10 x 10 = 100

k = 100. This is a constant, even if the values of N and T change.

Therefore, when N = 50,

T = k/N. Thus, T = 100/50 = 2 days

If there are 50 workers on the task, the house can be constructed in just two days.

Consequently, if there are five workers,

T = 100/5 = 20 days. With five workers it would take double the time, or 20 days, to construct the same house.

And with 20 workers, it would take half the time, or five days, to construct the same house.

In inverse proportion, the product (multiplication) of the quantities, (z)(y) is always a constant numerical value. If zy = k, then z and y are in inverse proportion.

Not all changes occur in proportion. Children grow in size with age, but not in direct proportion. The ageing continues, but physical growth accelerates in certain early years, slowing down in young adulthood. As ageing advances, there is a decline in strength and other capacities. This again is not a proper inverse proportion.

1.  A device in a cold drink factory fills 987 bottles in seven hours. How many bottles will it fill in nine hours?

No. of bottles filled (N) = 987

Time taken (T) = 7 hours

This is an instance of direct proportion. If the given time increases, more bottles will be filled.

N/T = k (constant)

987/7 = 141. The constant k = 141 in this case.

If the new time (T2) is 9 hours, how many bottles (N2) can be filled?

N2/T2 = 141. Thus, N2/9 = 141. N2 = 141 x 9 = 1269

Thus, in nine hours, about 1269 bottles can be filled.

2.  Julia is on a road trip. She has a map with a scale of 1 cm representing 21 km. After driving for 84 km, what distance is covered on the map?

1 cm represents 21 km on the map. 1 km = 1,00,000 cm. Thus, their ratio is 1:21,00,000 which is the constant of proportionality.

Let’s find the distance on the map when 84 km is covered on the road:

x:84,00,000 = 1:21,00,000.

x = 84,00,000/21,00,000 = 4 cm

Thus, for 84 km, the distance covered on the map is 4 cm.

3.  If a box of candies is divided among 24 children, they get six each. How many would each get if there are eight fewer children?

As the number of children reduces, each child gets more candies. This relation is an inverse proportion.

Hence, 24 x 6 = 144. This product remains constant.

Eight children reduced from 24 leaves 16 children. Let the number of sweets be “y”

16y = 144. Thus, y = 144/16 = 9

Each of the 16 children gets nine candies.

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     1. How many problem sets and sums are there in the NCERT Solutions Class 8 Maths Chapter 13 – Direct and Inverse                           Proportions?

         Along with many well-solved examples, there are two problem sets. The first set has 10 direct proportion-related questions,             while the second has 11 inverse proportion-related questions.

      2. What is meant by proportions in maths? What are direct and inverse relationships?

         If a change in one quantity causes a predictable change in another, the change is occurring in proportion. If y and z increase           or decrease together, they are directly proportional. If an increase in y causes a decrease in z or vice versa, they are inversely           proportional.

      3. Can I get Class 8 Maths Chapter 13 – Direct and Inverse Proportions online?

          Yes. The NCERT textbooks are available online. The NCERT Solutions for Class 8 Chapter 13 is also available on MSVgo,                      providing in-depth interactive learning to clarify concepts and for quick revisions.

      4. Where can I learn more about NCERT Solutions Direct and Inverse Proportions for free?

          Get started for free with MSVgo to explore CBSE learning videos, learn concepts in minutes, and solve maths problems                    quickly.

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