NCERT Exemplar Solutions for Class 8 Maths Chapter 10 Direct & Inverse Proportions

Download NCERT Exemplar Solutions Class 8 Maths Chapter 10 Direct & Inverse Proportions gives solutions and explanations to all of the textbook’s exercise questions. Questions about an Direct & Inverse Proportions and examples of Direct & Inverse Proportions may be found in these NCERT Exemplar Solutions.

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NCERT Exemplar Class 8 Maths Chapter 10 Direct and Inverse Proportion

Multiple Choice Questions

Question. Both x and y vary inversely with each other. When x is 10, y is 6, which of the following is not a possible pair of corresponding values of x and y?
(a) 12 and 5 (b) 15 and 4 (c) 25 and 2.4 (d) 45 and 1.3

Solution.

Question. 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

Solution. (a) If land to be uniformly fertile, then the area of land and the yield on it vary directly with each other. Hence, option (a) is correct.

Note Two quantities x and y are said to be in direct proportion, if they increase or decrease together in such a manner that the ratio of their corresponding values remains constant.

Question. The number of teeth and the age of a person 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

Solution. (d) The number of teeth and the age of a person vary sometimes directly and sometimes inversely with each other, we cannot predict about the number of teeth with exactly the age of a person. It change with person-to-person.

Hence, option (d) is correct.

Question. In which of the following cases, do the quantities vary directly with each other?

Solution. (a) In option (a),

x = 0.5,2, 8, 32 and y = 2, 8, 32,128

If we multiply x with 4, we get the directly required result as same as shown in corresponding y. In this case, as the value of x increases, the value of y also increases. Hence, option (a) is correct.

Question. 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

Solution. (d) We know that, when we increases the speed, then the time taken by vehicle decreases. Hence, speed and time taken vary inversely with each other. So, option (d) is correct.

Question. Both x and y are in direct proportion, then 1x and 1y 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

Solution. (b) If both x arid y are in directly proportion, then 1x and 1y are in inverse proportion. Hence, option (b) is correct.

Note- Two quantities x and y are said to be in inverse proportion, if an increase in x cause a proportional decrease in y and vice-versa.

Question. If two quantities x and y vary directly with each other, then

(a) xy remains constant

(b) x – y remains constant

(c) x + y remains constant ‘

(d)ix y remains constant

Solution. (a) If two quantities x and y vary directly with each other, then xy = k = constant.
Since, in direct proportion, both x and y increases or decreases together such a manner that the ratio of their corresponding value remains constant. Hence, option (a) is correct.

Question. If two quantities p and q vary inversely with each other, then

(a) pq remains constant

(b) p + q remains constant

(c) p x q remains constant

(d) p – q remains constant

Solution. (c) If two quantities p and q vary inversely with each other, then p x q remains constant. Since, in inverse proportion, an increase in p cause a proportional decrease in q and vice-versa.
Hence, option (c) is correct.

Question. If the distance travelled by a rickshaw in one hour is 10 km, then the distance travelled by the same rickshaw with the same speed in one minute is
(a)2509m (b)5009m (c)1000m (d)5003m

Solution. (b)5009m

Question. Both x and y vary directly with each other and when x is 10, y is 14, which of the following is not a possible pair of corresponding values of x and y?

(a) 25 and 35

(b) 35 and 25

(c) 35 and 49

(d) 15 and 21

Solution. (b) 35 and 25

Fill in the Blanks

Question. When two quantities x and y are in——-proportion or vary——-they are written as x∝y

Solution. When two quantities x and y are in direct proportion or vary directly, they are written as x∝y [see definition of direct proportion]

Question. When two quantities x and y are in——-proportion or vary———-they are written as x∝1y

Solution. When two quantities x and y are in inverse proportion or vary inversely, they are written as x∝1y [see definition of inverse proportion]

Question. Both x and y are said to vary——–with each other, if for some positive number k, xy =k.

Solution. Both x and y are said to vary inversely with each other, if for some positive number k,xy = k. [see condition of inverse proportion]

Question. x and y are said to vary directly with each other, if for, some positive number k,———-= k.

Solution. x and y are said to vary directly wifh’ether, if for some positive number k, xy=k.

Question. Two quantities are said to vary——— with each other, if they increase (decrease) together in such a manner that the ratio of their corresponding values remains constant.

Solution. Two quantities are said to vary directly with each other, if they increase (decrease) together in such a manner that the ratio of their corresponding values remains constant.

Question. Two quantities are said to vary——–with each other, if an increase in one causes a decrease in the other in such a manner that the product of their corresponding values remains constant.

Solution. Two quantities are said to vary inversely with each other, if increase in one cause a decrease in the other in such a manner that the product of their corresponding values remains constant.

Question. When the speed remains constant, the distance travelled is——–proportional to the time.

Solution. When the speed remains constant, the distance travelled is directly proportional to the time. e.g. If 10 km cover in 10 min with uniform speed, then 20 km cover in 20 min with same speed.

Question. On increasing a, b increases in such a manner that ab remains——and positive, then a and b are said to vary directly with each other.

Solution. On increasing a, b increases in such a manner that ab remains constant and positive, then a and b are said to vary directly with each other.

Question. If on increasing a, b decreases in such a manner that—– remains——–and positive, then a and b are said to vary inversely with each other.

Solution. If on increasing a, b decreases in such a manner that ab remains constant and positive, then a and b are said to vary inversely with each other. [see definition of inverse proportion]

Question. If two quantities x and y vary directly with each other, then——— of their corresponding values remains constant.

Solution. If two quantities x and y vary directly with each other, then ratio of their corresponding values remains constant. [see definition of direct proportion]

Question. If two quantities p and q vary inversely with each other, then————- of their corresponding values remains constant.

Solution. If two quantities p and q vary inversely with each other, then product of their corresponding values remains constant.

In questions from 43 to 59, state whether the statements are True or False.

Question. Two quantities x and y are said to vary directly with each other, if for some rational number k, xy =k.

Solution. False, Two quantities x and y are said to vary directly with each other, if xy = k (constant)

Question. When the speed is kept fixed, time and distance vary inversely with each other.

Solution. False, When the speed is kept fixed, time and distance vary directly with each other.

Question. When the distance is kept fixed, speed and time vary directly with each other.

Solution. False, When the distance is kept fixed, speed and time vary indirectly/inversely with each other. Since, if we increase speed, then taken time will less and vice-versa.

Question. Length of a side of a square and its area vary directly with each other.

Solution. False, Length of a side of a square and its area does not vary directly with each other, e.g. Let a be length of each side of a square., So, area of the square = Side2 = a2. So, if we increase the length of the side of a square, then their area increases but not directly.

Question. Length of a side of an equilateral triangle and its perimeter vary inversely with each other.

Solution. False, Length of a side of an equilateral triangle and its perimeter vary directly with each other, e.g. Let a be the side of an equilateral triangle. So, perimeter = 3 x (Side) = 3 x a = 3a . So, if we increase the length of side of the equilateral triangle, then their perimeter will also increases.

Question. If d varies directly as t2, then we can write dt2 = k, where k is some constant.

Solution. False, If d varies inversely as t2, then we can write dt2 = k, where k is some constant. Since, two quantities x and y are said to be in Inverse proportion, if an increases in x cause a proportional decreases in y and vice-versa, in such a manner that the product of their corresponding values remains constant.

Question. If x and y are in inverse proportion, then (x +1) and (y +1) are also in inverse proportion.

Solution. False

If x and y are in inverse proportion, then xy = k (constant) e.g. Let x= 2 and y = 3

xy = 2 x 3= 6. Now, x + 1=2 + 1 = 3 and y+ 1 = 3 + 1 = 4

Then, (x + 1)(y+1) = 3 x 4 = 12 [not in inverse proportion]

Hence, (x+ 1)and (y + 1) cannot be in inverse proportion.

Question. If p and q are in inverse proportion, i.e. pq = k (constant), then (p + 2)and (q – 2) are also in inverse proportion.

Solution. False, If p and q are in inverse proportion, then

xy = k (constant)

e.g. Let p = 3andq = 4

Then, pq = 3×4 = 12

Now, p+ 2 = 3+ 2 = 5 and q-2 = 4-2 =2

(p + 2) (q – 2) = 5 x 2 = 10 [not in inverse proportion]

Hence, (p+2) and (q -2)cannot be in inverse proportion.

Question. When two quantities are related in such a manner that, if one increases, the other also increases, then they always vary directly.

Solution. True, When two quantities are related in such a manner that if, one increases the other also increases, then they always vary directly.
Above statement is correct for direct proportion. It is a basic properties of direct proportion.

Question. When two quantities are related in such a manner that if one increases and the other decreases, then they always vary inversely.

Solution. True, When, two quantities are related in such a manner that if one increases and the other decreases, then they always vary inversely. Above statement is correct for inverse proportion. It is a basic properties of inverse proportion.

Question. The number of workers and the time to complete a job is a case of direct proportion.

Solution. False, The number of workers and the time to complete a job is a case of indirect proportion, e.g. If 60 workers can complete a work in 10 days.
Then, 120 workers can complete the same work in 5 days.

Question. The area of cultivated land and the crop harvested is a case of direct proportion.

Solution. True, the area of cultivated land and the crop harvested is a case of direct proportion. Since, the quantities of crop harvested is depend upon area of cultivated land.

Question.

(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 are inversely proportional. ‘ e.g. Let a train cover 100 km in 1 h with speed 100 km/h.
Then, the same train cover 100 km in 30 min with speed 200 km/h.

(ii) The distance travelled by CNG bus and the amount of CNG used are directly proportional. e.g. Let a CNG bus can travelled 10 km in 1 kg of CNG. Then, the same CNG bus travelled 20 km in 2 x 1 = 2 kg of CNG.

(iii) The number of people working and the time to complete a given work are inversely proportional to each other. e.g. Let 20 workers can complete a work in 1day. Then, 10 workers can complete the same work in 2 days.

(iv) Income tax and the income are directly proportional to each other, e.g. Let Mr X have 4.5 lakh annual income. Then, he pay 10% income tax on his income. But if Mr X have 5.5 lakh annual income, then he has to pay 30% income tax on his salary/income.

(v) Distance travelled by an auto rickshaw and time taken are directly proportional to each other. e.g. Let an auto rickshaw takes 2 h to travel 10 km. Then, it will take 4 h to travel 20 km.