**Direct and Inverse Proportions**

CBSE Class 8 - Mathematics

*NCERT Exemplar Problems and Solutions*

**Q1: If x and y are directly proportional and when x = 13, y = 39, which of the following is not a possible pair of corresponding values of x and y ?**

(a) 1 and 3

(b) 17 and 51

(c) 30 and 10

(d) 6 and 18

Answer: (c) 30 and 10

For direct variation,

x_{1} |
= | x_{2} |

y_{1} |
y_{2} |

Thus x/y = 13/39 = 1/3. Options (a), (b) and (d) have the same ratio while option (c) does not.

**Q2: Which of the following is in direct proportion?**

(a) One side of a cuboid and its volume.

(b) Speed of a vehicle and the distance travelled in a fixed time interval.

(c) Change in weight and height among individuals.

(d) Number of pipes to fill a tank and the time required to fill the same tank.

Answer: (b) Speed of a vehicle and the distance travelled in a fixed time interval.

∵ Speed / distance = fixed ratio.

**Q3: Both u and v vary directly with each other. When u is 10, v is 15, which of the following is not a possible pair of corresponding values of u and v?**

(a) 2 and 3

(b) 8 and 12

(c) 15 and 20

(d) 25 and 37.5

Answer: (c) 15 and 20

u/v = 2/3

Option (C) does not maintain the same ratio.

**Q4: Assuming land to be uniformly fertile, the area of land and the yield on it vary**

(a) directly with each other.

(b) inversely with each other.

(c) neither directly nor inversely with each other.

(d) sometimes directly and sometimes inversely with each other.

Answer: (d) sometimes directly and sometimes inversely with each other.

**Q5: A truck needs 54 litres of diesel for covering a distance of 297 km. The diesel required by the truck to cover a distance of 550 km is**

(a) 100 litres

(b) 50 litres

(c) 25.16 litres

(d) 25 litres

Answer: (a) 100 litres

More diesel means more distance covered. It is a case of direct variation.

∴ 54/297 = x/550

⇒ x = 54 × 550 / 297 = 100

**Q6: By travelling at a speed of 48 kilometres per hour, a car can finish a certain journey in 10 hours. To cover the same distance in 8 hours, the speed of the car should be**

(a) 60 km/h

(b) 80 km/h

(c) 30 km/h

(d) 40 km/h

Answer: (a) 60 km/h

Higher the speed, less time will it take to finish the journey. It is a case of inverse propotion.

∴ 48 × 10 = s × 8

⇒ s = 60

**Q7: Which of the following vary inversely with each other?**

(a) speed and distance covered.

(b) distance covered and taxi fare.

(c) distance travelled and time taken.

(d) speed and time taken.

Answer: (d) speed and time taken.

**Q8: Both x and y are in direct proportion, then 1/x and 1/y are**

(a) in indirect proportion.

(b) in inverse proportion.

(c) neither in direct nor in inverse proportion.

(d) sometimes in direct and sometimes in inverse proportion.

Answer: (b) in inverse proportion.

**Q9: 100 persons had food provision for 24 days. If 20 persons left the place, the provision will last for**

(a) 30 days

(b) 96/5 days

(c) 120 days

(d) 40 days

Answer: (a) 30 days

Less people, food will last more days. ⇒ It is an inverse proportion.

∴ 100 × 24 = 80 × days

⇒ days = 30

**Q10: (Fill in the blanks)**

**(a)**Amrita takes 18 hours to travel 720 kilometres. Time taken by her to travel 360 kilometres is _______.

**(b)**If x and y are inversely proportional then _____ = k where k is positive constant.

**(c)**If x = 5y, then x and y vary ______ with each other.

**(d)**If xy = 10, then x and y vary ______ with each other.

**(e)**x and y are said to vary directly with each other if for some positive number k, ______ =k.

**(f)**If 12 pumps can empty a reservoir in 20 hours, then time required by 45 such pumps to empty the same reservoir is ______ hours.

Answers:

(a) 9 hours [Direct proportion]

(b) xy

(c) directly (∵ x/y = 5 a fixed ratio)

(d) inversely

(e) x/y

(f) 16/3 hours. More pumps means less hours to empty the reservoir.

∴ 12 × 20 = 45 x h

⇒ h = 16/3

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