Maharashtra Board Class 7 Maths part 2 Chapter 9 Direct Proportion and Inverse Proportion PDF Download

Direct Proportion And Inverse Proportion

Let's Discuss

Direct Proportion

In the previous class we have learnt how to compare two numbers and write them in the form of a ratio.

Example Look at the picture below. We see divisions of a circle made by its diameters.

Do you see any relationship between the number of diameters and the number of divisions they give rise to?

In figure (A) one diameter makes 2 parts of the circle.

In figure (B) two diameters make 4 parts of the circle.

In figure (D) four diameters make 8 parts of the circle.

No. of diameters / No. of divisions = 1/2 = 2/4 = 3/6 = 4/8. Here, the ratio of the number of diameters to the number of divisions remains constant.

Example The number of notebooks that the students of a Government School received is shown in the table below.

Number of children / Number of notebooks = 15/90 = 12/72 = 10/60 = 5/30 = 1/6

In other words, the ratio 1:6 remains the same or constant.

In the examples above, we see that when the number of diameters increases the number of divisions also increases. As the number of children decreases the number of notebooks also falls. The number of diameters and the number of divisions are in direct proportion as are the number of students and the number of notebooks.

Activity : Think: Are the amount of petrol filled in a motorcycle and the distance travelled by it, in direct proportion?

Discuss: Can you give examples from science or everyday life, of quantities that vary in direct proportion?

Teacher's Note

Direct proportion means when one thing increases, the other also increases. For example, if you buy more apples from the market, you will pay more money.

Exam Trick

Remember: In direct proportion, both things move together - when one goes up, the other goes up too. When one goes down, the other goes down too.

Points to Remember

Direct proportion means two quantities change together in the same way.
When one quantity increases, the other also increases.
When one quantity decreases, the other also decreases.
The ratio between them always stays the same.

Example If 10 pens cost 60 rupees, what is the cost of 13 such pens?

Solution: Let us suppose the cost of 13 pens is x rupees.

The number of pens and their cost vary in direct proportion. Let us express the ratios and obtain an equation.

10/60 = 13/x

∴ 10x = 780 (multiplying both sides by 60x)

∴ x = 78

Cost of 13 pens is ₹ 78.

Practice Set 37

1. If 7 kg onions cost 140 rupees, how much must we pay for 12 kg onions?

2. If 600 rupees buy 15 bunches of feed, how many will 1280 rupees buy?

3. For 9 cows, 13 kg 500 g of food supplement are required every day. In the same proportion, how much will be needed for 12 cows?

4. The cost of 12 quintals of soyabean is 36,000 rupees. How much will 8 quintals cost?

5. Two mobiles cost 16,000 rupees. How much money will be required to buy 13 such mobiles?

Let's Learn

Inverse Proportion

Some volunteers have gathered to dig 90 pits for a tree plantation programme. One volunteer digs one pit in one day. If there are 15 volunteers, they will take 90/15 = 6 days to dig the pits.

10 volunteers will take 90/10 = 9 days.

Are the number of pits and the number of volunteers in direct proportion?

If the number of volunteers decreases, more days are required; and if the number of volunteers increases, fewer days are required for the job. However, the product of the number of days and number of volunteers remains constant. We say that these numbers are in inverse proportion.

Suppose Sudha has to solve 48 problems in a problem set. If she solves 1 problem every day, she will need 48 days to complete the set. But, if she solves 8 problems every day, she will complete the set in 48/8 = 6 days and if she solves 12 problems a day, she will need 48/12 = 4 days. The number of problems solved in a day and the number of days needed are in inverse proportion. Their product is constant.

Thus, note that 8 × 6 = 12 × 4 = 48 × 1

Teacher's Note

Inverse proportion means when one thing increases, the other decreases. For example, if more workers dig a pit, they will finish faster.

Exam Trick

Remember: In inverse proportion, when one quantity increases, the other decreases. When more people work, it takes less time.

Points to Remember

Inverse proportion means two quantities change in opposite ways.
When one quantity increases, the other decreases.
When one quantity decreases, the other increases.
The product of the two quantities always stays the same.

Example Fifteen workers take 8 hours to build a wall. How many hours will 12 workers need to build the same wall?

Solution: As the number of workers increases, the number of hours decreases. The number of workers and number of hours are in inverse proportion. The product of the number of workers and the number of hours needed to build the wall is constant. Let us use the variable x to solve this problem.

Suppose, 12 workers take x hours.

15 workers take 8 hours.

12 workers take x hours.

12 × x = 15 × 8

∴ 12x = 120

∴ x = 10

Thus, 12 workers will take 10 hours to build the wall.

Example A 40-page class magazine is to be written. If one student would require 80 days to write it, how many would 4 students require?

Solution: If more students help to do the same task, fewer days will be required. That is, the number of students and number of days are in inverse proportion.

Suppose 4 students need x days.

4x = 80 × 1

x = 80/4

x = 20

∴ 4 students require 20 days.

Example Students of a certain school went for a picnic to a farm by bus. Here are some of their experiences. Say whether the quantities in each are in direct or in inverse proportion.

Each student paid 60 rupees for the expenses.

As there were 45 students, 2700 rupees were collected.

Had there been 50 students, 3000 rupees would have been collected.

The number of students and money collected are in direct proportion.

The sweets shop near the school gave 90 laddoos for the picnic.

If 45 students go for the picnic, each will get 2 laddoos.

If 30 students go for the picnic, each will get 3 laddoos.

The number of students and that of laddoos each one gets are in inverse proportion.

The farm is 120 km away from the school.

The bus went to the farm at a speed of 40 km per hour and took 3 hours.

On the return trip, the speed was 60 km per hour. Therefore, it took 2 hours.

The speed of the bus and the time it takes are in inverse proportion.

Teacher's Note

We use inverse proportion in our daily lives. For example, if you walk faster, you take less time to reach school.

Exam Trick

Remember: Inverse proportion = one increases, one decreases. More speed = less time. More workers = less time.

Points to Remember

Inverse proportion means two things work opposite to each other.
When speed increases, time decreases.
When workers increase, days decrease.
Multiply the two quantities - the answer is always the same.

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