Which of the following vary directly and which vary inversely with each other and which are neither of the two?
(i) The time taken by a train to cover a fixed distance and the speed of the train.
(ii) The distance travelled by CNG bus and the amount of CNG used.
(iii) The number of people working and the time to complete a given work.
(iv) Income Tax and the Income
(v) Distance Travelled by an auto-rickshaw and time taken

Solution:

(i) The time taken by a train to cover a fixed distance and the speed of the train vary inversely with each other. In other words, to cover the same distance the time taken is lesser if the speed increases. If the speed decreases then the time taken to cover the given distance increases. Hence it is a direct variation.

(ii) The distance travelled by CNG bus varies directly with the amount of CNG used. The more the CNG used the more the distance travelled. Hence it is a direct variation.

(iii) The number of people working and the time to complete a given work are always inversely related. In other words the more the people working the less the time taken to complete a given work, the fewer the number of people, the more the time will be taken.This is an inverse variation.

(iv) Income Tax and Income are directly related. It is so because as income increases one has to pay more tax provided the tax rate remains same.

(v) Distance travelled by an autorickshaw and time taken are directly related i.e the longer the distance travelled by the autorickshaw the longer the time taken if the auto travels at fixed or given speed.

✦ Try This: Which of the following vary directly and which vary inversely with each other and which are neither of the two?

(i) By how much less time it will take to cover a distance of 440km if the average speed of the car increases from 66 km/hour to 77 km/hour

(ii) If a car can travel 240 km with 20 gallons of gasoline then how much would it travel with 25 gallons of gasoline.

(iii) The time taken to dig a trench 200m long by 8 workers is 5 days. How many days will it take to dig the same trench by 10 workers?

(iv) The income tax rate is 35%. What will be the increase in tax paid if the income rises from Rs.18,00,000 to Rs.30,00,000?

(v) The average speed of an auto-rickshaw is 35km/hour. How much longer will it take to cover a distance of 17.5 km than a distance of 21 Km?

(i) Time taken to cover 440 km at an average speed of of 66 km/hour

Time Taken = 440/66 = 40/6 = 6 2/3 hours

Time taken to cover 440km at an average speed of 77km/hour is:

Time taken = 440/77 = 40/7 = 5 5/7 hours

The time taken to travel 440 km reduces by = 40/6 - 40/7

= (280 - 240)/42 = 40/42hours = 20/21 hours or 57.14 minutes

(ii) Distance Travelled(Km) Amount of Gasoline (Gallons)

240 20

x = ? 25

x = 240 × 25/20

x = 12 × 25

x = 300km

Hence the distance traveled is higher with 25 gallons of gasoline.

(iii) No. of Workers Time Taken(Days)

8 5

10 x = ?

Hence the number of days taken to dig the trench are:

x = 5 × 8/10

x = 4 days

The more the number of workers, the lesser is the time taken as evident by the above example.

(iv) Income Tax on an income of Rs.18,00,000 @35%

Income Tax = 35% × 18,00,000 = Rs. 6,30,000

Income Tax on an income of Rs. 30,00,000 @35%

Income Tax = 35% × 30,00,000 = Rs. 10,50,000

Increase in Tax = Rs. 10,50,000 - Rs. 6,30,000 = Rs. 4,20,000

(v) Time taken(T) to cover 17.5 km @ 35 km/hour

T = 17.5/35 = 1/2 hour = 30 minutes

Time taken(T) to cover 21km = 21/35 = 3/5 hour = 36 minutes

The autorickshaw takes 6 minutes longer to cover 21km than a distance of 17.5 km.

☛ Also Check: NCERT Solutions for Class 8 Maths Chapter 13

NCERT Exemplar Class 8 Maths Chapter 10 Problem 60

Which of the following vary directly and which vary inversely with each other and which are neither of the two? (i) The time taken by a train to cover a fixed distance and the speed of the train. (ii) The distance travelled by CNG bus and the amount of CNG used.(iii) The number of people working and the time to complete a given work. (iv) Income Tax and the Income (v) Distance Travelled by an auto-rickshaw and time taken

Summary:

(i) The time taken by a train to cover a fixed distance and the speed of the train vary inversely with each other.
(ii)The distance traveled by CNG bus varies directly with the amount of CNG used.
(iii)The number of people working and the time to complete a given work are always inversely related
(iv) Income Tax and Income are directly related.
(v) Distance traveled by an autorickshaw and time taken are directly related.

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