Math Ncert Exemplar 2019 Solutions for Class 8 Maths Chapter 10 Direct & Inverse Proportions are provided here with simple step-by-step explanations. These solutions for Direct & Inverse Proportions are extremely popular among class 8 students for Maths Direct & Inverse Proportions Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the Math Ncert Exemplar 2019 Book of class 8 Maths Chapter 10 are provided here for you for free. You will also love the ad-free experience on Meritnation’s Math Ncert Exemplar 2019 Solutions. All Math Ncert Exemplar 2019 Solutions for class 8 Maths are prepared by experts and are 100% accurate.
Page No 315:
Question 1:
In questions âthere are four options out of which one is correct.
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:
Given that u and v vary directly with each other.
If u = 10, v = 15, then
Option (a),
Option (b),
Option (c),
Option (d),
Hence, the correct answer is option C.
Page No 315:
Question 2:
In questions âthere are four options out of which one is correct.
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
Answer:
Given that x and y vary inversely with each other.
If x = 10, y = 6, then
Option (a),
Option (b),
Option (c),
Option (d),
Hence, the correct answer is option D.
Page No 315:
Question 3:
In questions âthere are four options out of which one is correct.
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:
If the land to be uniformly fertile, then the area of land and the yield on it vary directly with each other.
Hence, the correct answer is option A.
Page No 315:
Question 4:
In questions âthere are four options out of which one is correct.
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.
Answer:
We cannot predict about the number of teeth with exactly the age of a person.
The number of teeth and the age of a person vary sometimes directly and sometimes inversely with each other.
It changes with person-to-person.
Hence, the correct answer is option D.
Page No 316:
Question 5:
In questions âthere are four options out of which one is correct.
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 truck requires 54 L of diesel to cover a distance = 297 km
In 1 L, a truck can cover =
Diesel required for 550 km =
Hence, the correct answer is option A.
Page No 316:
Question 6:
In questions âthere are four options out of which one is correct.
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:
Given that, speed of car = 48 km/h
Time taken by car = 10 h
Speed of the car, if 480 km is to cover in 8 h =
Hence, the correct answer is option A.
Page No 316:
Question 7:
In questions âthere are four options out of which one is correct.
In which of the following case, do the quantities vary directly with each other?
(a)
x | 0.5 | 2 | 8 | 32 |
y | 2 | 8 | 32 | 128 |
(b)
p | 12 | 22 | 32 | 42 |
q | 13 | 23 | 33 | 43 |
(c)
r | 2 | 5 | 10 | 25 | 50 |
s | 25 | 10 | 5 | 2 | 0.5 |
(d)
u | 2 | 4 | 6 | 9 | 12 |
v | 18 | 9 | 6 | 4 | 3 |
Answer:
In option (a),
x = 0.5,2, 8, 32 and y = 2, 8, 32,128
Thus, x and y vary directly from each other.
In all other options, x and y do not vary directly with each other.
Hence, the correct answer is option A.
Page No 316:
Question 8:
In questions âthere are four options out of which one is correct.
Which quantities in the previous question vary inversely with each other?
(a) x and y
(b) p and q
(c) r and s
(d) u and v
Answer:
In option (d),
when u increases, v decreases.
u | 2 | 4 | 6 | 9 | 12 |
v | 18 | 9 | 6 | 4 | 3 |
Thus, u and v vary inversely each other.
Hence, the correct answer is option D.
Page No 316:
Question 9:
In questions there are four options out of which one is correct.
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:
We know that, when we increase the speed, then the time taken by the vehicle decreases.
Thus, the speed and time taken vary inversely with each other.
Hence, the correct answer is option D.
Page No 316:
Question 10:
In questions âthere are four options out of which one is correct.
Both x and y are in direct proportion, then are
(a) in direct proportion
(b) in inverse proportion
(c) neither in direct nor in inverse proportion
(d) sometimes in direct and sometimes in inverse proportion
Answer:
Both x and y are in directly proportion.
then are in direct proportion.
Hence, the correct answer is option A.
Page No 317:
Question 11:
In questions âthere are four options out of which one is correct.
Meenakshee cycles to her school at an average speed of 12 km/h and takes 20 minutes to reach her school. If she wants to reach her school in 12 minutes, her average speed should be
(a) km/h
(b) 16 km/h
(c) 20 km/h
(d) 15 km/h
Answer:
Given that the speed of the cycle is 12 km/h.
Time taken to reach school = 20 min =
Total distance cover =
Speed of the cycle to reach in 12 min =
Hence, the correct answer is option C.
Page No 317:
Question 12:
In questions âthere are four options out of which one is correct.
100 persons had food provision for 24 days. If 20 persons left the place, the provision will last for
(a) 30 days
(b) days
(c) 120 days
(d) 40 days
Answer:
100 persons had food provision for 24 days.
1 person had food provision for 24 × 100 = 2400 days
As 20 persons left the place then 80 persons remained.
∴ Food provision for remaining 80 persons =
Hence, the correct answer is option A.
Page No 317:
Question 13:
In questions âthere are four options out of which one is correct.
If two quantities x and y vary directly with each other, then
(a) remains constant
(b) x – y remains constant
(c) x + y remains constant
(d) x × y remains constant
Answer:
If two quantities x and y vary directly with each other, then
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, the correct answer is option A.
Page No 317:
Question 14:
In questions âthere are four options out of which one is correct.
If two quantities p and q vary inversely with each other, then
(a) remains constant
(b) p + q remains constant
(c) p × q remains constant
(d) p – q remains constant
Answer:
If two quantities p and q vary inversely with each other, then
Since, in inverse proportion, an increase in p cause a proportional decrease in q and vice-versa.
Hence, the correct answer is option C.
Page No 317:
Question 15:
In questions âthere are four options out of which one is correct.
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
Answer:
Distance traveled in one hour = 10 km
Distance traveled in 60 min = 10 km
Distance traveled in 1 min = km
Distance traveled in 1 min =
Hence, the correct answer is option D.
Page No 318:
Question 16:
In questions âthere are four options out of which one is correct.
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
Answer:
Given that x and y vary directly with each other.
If x = 10, y = 14, then
Option (a),
Option (b),
Option (c),
Option (d),
Hence, the correct answer is option B.
Page No 318:
Question 17:
Fill in the blanks to make the statements true:
If x = 5y, then x and y vary ______ with each other.
Answer:
Given: x = 5y
Hence, x and y vary directly with each other.
Page No 318:
Question 18:
Fill in the blanks to make the statements true:
If xy = 10, then x and y vary ______ with each other.
Answer:
Given: xy = 10
Hence, x and y vary inversely with each other.
Page No 318:
Question 19:
Fill in the blanks to make the statements true:
When two quantities x and y are in ______ proportion or vary ______ they are written as x ∝ y.
Answer:
When two quantities x and y are in direct proportion or vary directly, they are written as x ∝ y.
Page No 318:
Question 20:
Fill in the blanks to make the statements true:
When two quantities x and y are in _______ proportion or vary ______they are written as x ∝ .
Answer:
When two quantities x and y are in inverse proportion or vary inversely, they are written as x ∝ .
Page No 318:
Question 21:
Fill in the blanks to make the statements true:
Both x and y are said to vary ______ with each other if for some positive number k, xy = k.
Answer:
Both x and y are said to vary inversely with each other, if for some positive number k, xy = k.
Page No 318:
Question 22:
Fill in the blanks to make the statements true:
x and y are said to vary directly with each other if for some positive number k, ______ =k.
Answer:
Given,
Hence, x and y are said to vary directly with each other, if for some positive number k, = k.
Page No 318:
Question 23:
Fill in the blanks to make the statements true:
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.
Answer:
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.
Page No 318:
Question 24:
Fill in the blanks to make the statements true:
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.
Answer:
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.
Page No 318:
Question 25:
Fill in the blanks to make the statements true:
If 12 pumps can empty a reservoir in 20 hours, then time required by 45 such pumps to empty the same reservoir is ______ hours.
Answer:
Time taken by 12 pumps can empty a reservoir is 20 h.
1 pump can empty a reservoir is .
Then, 45 pumps can empty a reservoir is
The time required by 45 such pumps to empty the same reservoir is 5 hours 20 min.
Page No 319:
Question 26:
Fill in the blanks to make the statements true:
If x varies inversely as y, then
â
x | – | 60 |
y | 2 | 10 |
Answer:
If x varies inversely as y, then
If x = 60 and y = 10
When y = 2, then
Page No 319:
Question 27:
Fill in the blanks to make the statements true:
If x varies directly as y, then
â
x | 12 | 6 |
y | 48 | – |
Answer:
If x varies directly as y, then
If x = 12 and y = 48
When x = 6, then
Page No 319:
Question 28:
Fill in the blanks to make the statements true:
When the speed remains constant, the distance travelled is ______ proportional to the time.
Answer:
We know that
When the speed remains constant, the distance travelled is directly proportional to the time.
Page No 319:
Question 29:
Fill in the blanks to make the statements true:
On increasing a, b increases in such a manner that remains______ and positive, then a and b are said to vary directly with each other.
Answer:
We have,
Hence, on increasing a, b increases in such a manner that remains constant and positive, then a and b are said to vary directly with each other.
Page No 319:
Question 30:
Fill in the blanks to make the statements true:
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.
Answer:
We have,
Hence, 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.
Page No 319:
Question 31:
Fill in the blanks to make the statements true:
If two quantities x and y vary directly with each other, then ______ of their corresponding values remains constant.
Answer:
We have,
If two quantities x and y vary directly with each other, then ratio of their corresponding values remains constant.
Page No 319:
Question 32:
Fill in the blanks to make the statements true:
If two quantities p and q vary inversely with each other then ______ of their corresponding values remains constant.
Answer:
We have,
If two quantities p and q vary inversely with each other then product of their corresponding values remains constant.
Page No 319:
Question 33:
Fill in the blanks to make the statements true:
The perimeter of a circle and its diameter vary _______ with each other.
Answer:
Let the radius of the circle be r.
Diameter of a circle = 2r
Perimeter of a circle =
Hence, the perimeter of a circle and its diameter vary directly with each other.
Page No 319:
Question 34:
Fill in the blanks to make the statements true:
A car is travelling 48 km in one hour. The distance travelled by the car in 12 minutes is _________.
Answer:
Distance travelled by a car in 1 h = 48 km
Distance travelled by the car in 60 min = 48 km
Distance travelled by the car in 1 min = km
Distance travelled by the car in 12 min = km
Hence, the distance travelled by the car in 12 minutes is 9.6 km.
Page No 319:
Question 35:
Fill in the blanks to make the statements true:
An auto rickshaw takes 3 hours to cover a distance of 36 km. If its speed is increased by 4 km/h, the time taken by it to cover the same distance is __________.
Answer:
An auto rickshaw takes 3 hours to cover a distance of 36 km.
Speed of the auto rikshaw is
If speed is increased by 4 km/h, then the new speed of auto rikshaw is 12 + 4 = 16 km/h.
Time taken to cover distance of 36 km is
Hence, the time taken by it to cover the same distance is 2 h 15 min.
Page No 319:
Question 36:
Fill in the blanks to make the statements true:
If the thickness of a pile of 12 cardboard sheets is 45 mm, then the thickness of a pile of 240 sheets is _______ cm.
Answer:
The thickness of a pile of 12 cardboard sheets = 45 mm
The thickness of a pile of 1 cardboard sheet = mm
The thickness of a pile of 240 cardboard sheets = mm = 90 cm
Hence, the thickness of a pile of 240 sheets is 90 cm.
Page No 319:
Question 37:
Fill in the blanks to make the statements true:
If x varies inversely as y and x = 4 when y = 6, then when x = 3 the value of y is _______.
Answer:
If x varies inversely as y, then
If x = 4 and y = 6
When x = 3, then
Hence, when x = 3 the value of y is 8.
Page No 319:
Question 38:
Fill in the blanks to make the statements true:
In direct proportion, ____________ .
Answer:
We have, a is directly proportional to b.
Page No 320:
Question 39:
Fill in the blanks to make the statements true:
In case of inverse proportion,
Answer:
We have, a is inversely proportional to b.
Thus, the statement is False.
Page No 320:
Question 40:
Fill in the blanks to make the statements true:
If the area occupied by 15 postal stamps is 60 cm2, then the area occupied by 120 such postal stamps will be _______.
Answer:
The area occupied by 15 postal stamps = 60 cm2
The area occupied by 1 postal stamp = cm2
The area occupied by 120 postal stamps = cm2
Hence, the area occupied by 120 such postal stamps will be 480 cm2.
Page No 320:
Question 41:
Fill in the blanks to make the statements true:
If 45 persons can complete a work in 20 days, then the time taken by 75 persons will be ______ hours.
Answer:
We have,
45 persons can complete a work = 20 days
1 person can complete a work = days
75 persons can complete a work = days
Hence, the time taken by 75 persons will be 12 hours.
Page No 320:
Question 42:
Fill in the blanks to make the statements true:
Devangi travels 50 m distance in 75 steps, then the distance travelled in 375 steps is _______ km.
Answer:
Distance travel by Devangi in 75 steps = 50 m
Distance travel by Devangi in 1 step = m
Distance travel by Devangi in 375 steps = m = 0.25 km
Hence, the distance travelled in 375 steps is 0.25 km.
Page No 320:
Question 43:
State whether the statements are true (T) or false (F).
Two quantities x and y are said to vary directly with each other if for some rational number k, xy = k.
Answer:
False,
If , then
Page No 320:
Question 44:
State whether the statements are true (T) or false (F).
When the speed is kept fixed, time and distance vary inversely with each other.
Answer:
False,
As,
When the speed is kept fixed, time and distance vary directly with each other.
Page No 320:
Question 45:
State whether the statements are true (T) or false (F).
When the distance is kept fixed, speed and time vary directly with each other.
Answer:
False
As,
When the distance is kept fixed, speed and time vary indirectly/inversely with each other.
Page No 320:
Question 46:
State whether the statements are true (T) or false (F).
Length of a side of a square and its area vary directly with each other.
Answer:
False
Length of a side of a square and its area does not vary directly with each other.
Ex. 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.
Page No 320:
Question 47:
State whether the statements are true (T) or false (F).
Length of a side of an equilateral triangle and its perimeter vary inversely with each other.
Answer:
False
Length of a side of an equilateral triangle and its perimeter vary directly with each other.
Ex. Let a be the side of an equilateral triangle.
So, perimeter = 3 × (Side) = 3 × a = 3a .
So, if we increase the length of the side of the equilateral triangle, then their perimeter will also increase.
Page No 321:
Question 48:
State whether the statements are true (T) or false (F).
If d varies directly as t2, then we can write dt2 = k, where k is some constant.
Answer:
False
If d varies inversely as t2, then we can write dt2 = k, where k is some constant.
Page No 321:
Question 49:
State whether the statements are true (T) or false (F).
If a tree 24 m high casts a shadow of 15 m, then the height of a pole that casts a shadow of 6 m under similar conditions is 9.6 m.
Answer:
True
Height of a tree = 24 m
Length of its shadow = 15 m
If the length of the shadow of a pole is 6 m then let the height of the pole be x m.
Since, length and shadow vary directly to each other.
Thus, the height of the pole is 9.6 m.
Page No 321:
Question 50:
State whether the statements are true (T) or false (F).
If x and y are in direct proportion, then (x – 1) and (y – 1) are also in direct proportion.
Answer:
False
Consider x and y are in direct proportion.
Let x = 12 and y = 4
Then, x – 1 = 11 and y – 1 = 3
Hence, if x and y are in direct proportion, then (x – 1) and (y – 1) cannot be in direct proportion.
Page No 321:
Question 51:
State whether the statements are true (T) or false (F).
If x and y are in inverse proportion, then (x + 1) and (y + 1) are also in inverse proportion.
Answer:
False
If x and y are in inverse proportion, then xy = k (constant)
Let x = 4 and y = 2
∴ xy = 4 × 2 = 8 = k.
Now, x + 1 = 4 + 1 = 5 and y + 1 = 2 + 1 = 3
Then, (x + 1)(y +1) = 5 × 3 = 15 ≠ k [not in inverse proportion]
Hence, if x and y are in direct proportion, then (x + 1)and (y + 1) cannot be in inverse proportion.
Page No 321:
Question 52:
State whether the statements are true (T) or false (F).
If p and q are in inverse variation then (p + 2) and (q – 2) are also in inverse proportion.
Answer:
False
If p and q are in inverse proportion, then pq = k (constant)
Let p = 5 and q = 6
Then, pq = 5 × 6 = 30 = k
Now, p + 2 = 5 + 2 = 7 and q – 2 = 6 – 2 = 4
Thus, (p + 2) (q – 2) = 7 × 4 = 28 ≠ k [not in inverse proportion]
Hence, if p and q are in inverse variation then (p + 2) and (q – 2) cannot be in inverse proportion.
Page No 321:
Question 53:
State whether the statements are true (T) or false (F).
If one angle of a triangle is kept fixed then the measure of the remaining two angles vary inversely with each other.
Answer:
False
If one angle of a triangle is kept fixed, then the measure of the remaining two angles cannot vary inversely with each other.
Ex. In âBC, we have
∠A + ∠B + ∠C = 180°
If ∠A = 60°, then ∠B + ∠C = 180° − 60° = 120°
So, the remaining two angles cannot depend on any proportion.
Page No 321:
Question 54:
State whether the statements are true (T) or false (F).
When two quantities are related in such a manner that, if one increases, the other also increases, then they always vary directly.
Answer:
True
When two quantities are related in such a manner that if one increases, the other also increases, then they always vary directly.
Then both quantities are in direct proportion.
Page No 321:
Question 55:
State whether the statements are true (T) or false (F).
When two quantities are related in such a manner that if one increases and the other decreases, then they always vary inversely.
Answer:
True
When, two quantities are related in such a manner that if one increases and the other decreases, then they always vary inversely.
Then both quantities are in inverse proportion.
Page No 321:
Question 56:
State whether the statements are true (T) or false (F).
If x varies inversely as y and when x = 6, y = 8, then for x = 8 the value of y is 10.
Answer:
False
If x varies inversely as y, then
If x = 6 and y = 8
When x = 8, then
Hence, when x = 8 the value of y is 6.
Page No 321:
Question 57:
State whether the statements are true (T) or false (F).
The number of workers and the time to complete a job is a case of direct proportion.
Answer:
False
The number of workers and the time to complete a job is a case of indirect proportion.
Ex. If 50 workers can complete a work in 8 days.
Then, 100 workers can complete the same work in 4 days.
Page No 321:
Question 58:
State whether the statements are true (T) or false (F).
For fixed time period and rate of interest, the simple interest is directly proportional to the principal.
Answer:
True
Page No 321:
Question 59:
State whether the statements are true (T) or false (F).
The area of cultivated land and the crop harvested is a case of direct proportion.
Answer:
True
The area of cultivated land and the crop harvested is a case of direct proportion.
Since the quantities of crop harvested is depended upon the area of cultivated land.
Page No 321:
Question 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.
Answer:
(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 60 km in 1 h with a speed of 60 km/h.
Then, the same train cover 60 km in 30 min with a speed of 120 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 15 km in 1 kg of CNG.
Then, the same CNG bus travelled 30 km in 2 × 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 10 workers can complete a work in 1 day.
Then, 5 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 5 lakh annual income.
Then, he pays 5% income tax on his income.
But if Mr. X have 6 lakh annual income, then he has to pay 10% 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 1 h to travel 10 km.
Then, it will take 3 h to travel 30 km.
Page No 322:
Question 61:
Which of the following vary directly and which vary inversely with each other and which are neither of the two?
(i) Number of students in a hostel and consumption of food.
(ii) Area of the walls of a room and the cost of white washing the walls.
(iii) The number of people working and the quantity of work.
(iv) Simple interest on a given sum and the rate of interest.
(v) Compound interest on a given sum and the sum invested.
Answer:
(i) Number of students in a hostel and consumption of food are directly proportional to each other.
(ii) Area of the walls of a room and the cost of white washing the walls are directly proportional to each other.
(iii) The number of people working and the quantity of work are directly proportional to each other.
(iv) Simple interest on a given sum and the rate of interest are directly proportional to each other.
(v) Compound interest on a given sum and the sum invested are directly proportional to each other.
Page No 322:
Question 62:
Which of the following vary directly and which vary inversely with each other and which are neither of the two?
(i) The quantity of rice and its cost.
(ii) The height of a tree and the number of years.
(iii) Increase in cost and number of shirts that can be purchased if the budget remains the same.
(iv) Area of land and its cost.
(v) Sales Tax and the amount of the bill.
Answer:
(i) The quantity of rice and its cost are directly proportional to each other.
e.g. Let 1 kg of rice price = â¹60
Then, 5 kg of rice price = â¹5 × 60 = â¹300
(ii) The height of a tree and the number of years are neither directly nor inversely proportional to each other.
(iii) Increase in cost and number of shirts that can be purchased, if the budget remains the same are inversely proportional to each other.
e.g. Let the price of 2 shirts be â¹1500.
After increasing the price of each shirt, the price became â¹1000.
(iv) Area of land and its cost are directly proportional to each other.
e.g. Let the cost of 100 m2 land be â¹2500.
Then the cost of 200 m2 land be â¹5000.
(v) Sales Tax and the amount of the bill are directly proportional to each other.
e.g. Let amount of bill be â¹800. and sales tax be 10%.
â¹80
If bill amount be â¹2400.
â¹240
Page No 322:
Question 63:
Solve the following :
If x varies inversely as y and x = 20 when y = 600, find y when x = 400.
Answer:
If x varies inversely as y, then
If x = 20 and y = 600
When x = 400, then
Hence, when x = 400 the value of y = 30.
Page No 322:
Question 64:
Solve the following :
The variable x varies directly as y and x = 80 when y is 160. What is y when x is 64?
Answer:
If x varies directly as y, then
If x = 80 and y = 160
When x = 64, then
Hence, when x is 64 the value of y is 128.
Page No 322:
Question 65:
Solve the following :
l varies directly as m and l is equal to 5, when m = . Find l when m = .
Answer:
If l varies directly as m, then
If l = 5 and m =
When m = , then
Hence, when m = , the value of l is 40.
Page No 322:
Question 66:
Solve the following :
If x varies inversely as y and y = 60 when x = 1.5. Find x. when y = 4.5.
Answer:
If x varies inversely as y, then
If x = 1.5 and y = 60
When y = 4.5, then
Hence, when y = 4.5, the value of x = 20.
Page No 322:
Question 67:
Solve the following :
In a camp, there is enough flour for 300 persons for 42 days. How long will the flour last if 20 more persons join the camp?
Answer:
Flour enough for 300 persons for 42 days.
Therefore, flour is enough for 1 person for 42 × 300 days i.e., 12600 days.
If 20 more persons join the camp, then the total number of persons in camp become 320.
Thus, for 320 persons flour is enough for .
Hence, for days the flour will last after 20 more persons join the camp.
Page No 322:
Question 68:
Solve the following :
A contractor undertook a contract to complete a part of a stadium in 9 months with a team of 560 persons. Later on, it was required to complete the job in 5 months. How many extra persons should he employ to complete the work?
Answer:
A work of stadium can complete in 9 months = 560 persons
A work of stadium can complete in 1 month = 560 × 9 = 5040 persons
A work of stadium can complete in 5 months = persons
Thus, the number of extra persons required are 1008 − 560 = 448.
Page No 322:
Question 69:
Solve the following :
Sobi types 108 words in 6 minutes. How many words would she type in half an hour?
Answer:
In 6 mins, Sobi can type 108 words.
In 1 min, she can type .
Thus, in 30 mins, she can type (18 × 30) words i.e., 540 words.
Page No 323:
Question 70:
Solve the following :
A car covers a distance in 40 minutes with an average speed of 60 km/h. What should be the average speed to cover the same distance in 25 minutes?
Answer:
Speed of car to cover a distance in 40 mins = 60 km/h = m/min
Speed of car to cover a distance in 1 min = m/min
Speed of car to cover a distance in 25 mins = m/min = km/h
Hence, the average speed to cover the same distance in 25 minutes is 96 km/h.
Page No 323:
Question 71:
Solve the following :
It is given that l varies directly as m.
(i) Write an equation which relates l and m.
(ii) Find the constant of proportion (k), when l is 6 then m is 18.
(iii) Find l, when m is 33.
(iv) Find m when l is 8.
Answer:
Given that l varies directly as m.
(a)
(b) If l = 6, then m = 18
(c) when m = 33,
(d) when l = 8,
Page No 323:
Question 72:
Solve the following :
If a deposit of Rs 2,000 earns an interest of Rs 500 in 3 years, how much interest would a deposit of Rs 36,000 earn in 3 years with the same rate of simple interest?
Answer:
Interest earned in 3 years at Rs 2000 = Rs 500
Interest earned in 3 years at Rs 1000 = Rs
Interest earned in 3 years at Rs 36,000 = Rs
Page No 323:
Question 73:
Solve the following :
The mass of an aluminum rod varies directly with its length. If a 16 cm long rod has a mass of 192 g, find the length of the rod whose mass is 105 g.
Answer:
Given that the mass of an aluminum(m) rod varies directly with its length(l).
If l = 16 cm and m = 192 g
When, m = 105 g, then
Hence, the length of the rod whose mass is 105 g is 8.75 cm.
Page No 323:
Question 74:
Solve the following :
Find the values of x and y if a and b are in inverse proportion:
a | 12 | x | 8 |
b | 30 | 5 | y |
a | 12 | x | 8 |
b | 30 | 5 | y |
Answer:
Given that a and b are in inverse proportion.
∴ ab = k(constant)
a | 12 | x | 8 |
b | 30 | 5 | y |
If a = 12 and b = 30, then
When a = x and b = 5, then
When a = 8 and b = y, then
Page No 323:
Question 75:
Solve the following :
If Naresh walks 250 steps to cover a distance of 200 metres, find the distance travelled in 350 steps.
Answer:
Distance travelled to 250 steps = 200 m
Distance travelled to 1 step = m
Distance travelled to 350 steps =
Page No 323:
Question 76:
Solve the following :
A car travels a distance of 225 km in 25 litres of petrol. How many litres of petrol will be required to cover a distance of 540 kilometres by this car?
Answer:
Petrol needed to travels 225 km by the car = 25 L
Petrol needed to travels 1 km by the car = L
Petrol needed to travels 540 km by the car = L
Page No 323:
Question 77:
Solve the following :
From the following table, determine if x and y are in direct proportion or not.
(i)
x | 3 | 6 | 15 | 20 | 30 |
y | 12 | 24 | 45 | 60 | 120 |
(ii)
x | 4 | 7 | 10 | 16 |
y | 24 | 42 | 60 | 96 |
(iii) â
x | 1 | 4 | 9 | 20 |
y | 1.5 | 6 | 13.5 | 30 |
Answer:
For direct proportion,
(i)
x | 3 | 6 | 15 | 20 | 30 |
y | 12 | 24 | 45 | 60 | 120 |
Hence, (i) is not in direct proportion.
(ii)
x | 4 | 7 | 10 | 16 |
y | 24 | 42 | 60 | 96 |
Hence, (ii) is in direct proportion.
(iii)
x | 1 | 4 | 9 | 20 |
y | 1.5 | 6 | 13.5 | 30 |
Hence, (iii) is in direct proportion.
Page No 324:
Question 78:
Solve the following :
If a and b vary inversely to each other, then find the values of p, q, r ; x, y, z and l, m, n.
(i)
a | 6 | 8 | q | 25 |
b | 18 | p | 39 | r |
(ii)
a | 2 | y | 6 | 10 |
b | x | 12.5 | 15 | z |
(iii)
a | l | 9 | n | 6 |
b | 5 | m | 25 | 10 |
Answer:
Given that a and b vary inversely to each other.
(i)
a | 6 | 8 | q | 25 |
b | 18 | p | 39 | r |
When a = 8, b = p
When a = q, b = 39
When a = 25, b = r
(ii)
a | 2 | y | 6 | 10 |
b | x | 12.5 | 15 | z |
When a = 2, b = x
When a = y, b = 12.5
When a = 10, b = z
(iii)
a | l | 9 | n | 6 |
b | 5 | m | 25 | 10 |
When a = 9, b = m
When a = l, b = 5
When a = n, b = 25
Page No 324:
Question 79:
Solve the folowing :
If 25 metres of cloth costs Rs 337.50, then
(i) What will be the cost of 40 metres of the same type of cloth?
(ii) What will be the length of the cloth bought for Rs 810?
Answer:
Cost of 25 m cloth = â¹337.50
Cost of 1 m cloth =
(i) Cost of 40 m cloth =
(ii) Length of cloth bought for â¹1 =
Length of cloth bought for â¹810 =
Page No 324:
Question 80:
Solve the following :
A swimming pool can be filled in 4 hours by 8 pumps of the same type. How many such pumps are required if the pool is to be filled in hours?
Answer:
A swimming pool can be filled in 4 hours by 8 pumps.
Pumps required to fill the swimming pool in 1 h = 4 × 8 = 32 pumps
Pumps required to fill the swimming pool in h i.e. h = pumps
Page No 324:
Question 81:
Solve the folowing :
The cost of 27 kg of iron is Rs 1,080, what will be the cost of 120 kg of iron of the same quality?
Answer:
Cost of 27 kg of iron = â¹1080
Cost of 1 kg of iron =
Cost of 120 kg of iron =
Hence, the cost of 120 kg of iron is â¹4800.
Page No 324:
Question 82:
Solve the following :
At a particular time, the length of the shadow of Qutub Minar whose height is 72 m is 80 m. What will be the height of an electric pole, the length of whose shadow at the same time is 1000 cm?
Answer:
Given that, the length of Qutub Minar = 72 m
Length of its shadow = 80 m
Length of shadow of electric pole = 1000 cm = 10 m
Thus, the length of the electric pole =
Page No 324:
Question 83:
Solve the folowing :
In a hostel of 50 girls, there are food provisions for 40 days. If 30 more girls join the hostel, how long will these provisions last?
Answer:
For 50 girls food is available for 40 days.
For 1 girl food is available for (50 × 40) days i.e., 2000 days.
If 30 more girls join the hostel, then the total number of girls in the became 80.
For 80 girls food is available for i.e., 25 days.
Page No 324:
Question 84:
Solve the folowing :
Campus and Welfare Committee of school is planning to develop a blue shade for painting the entire school building. For this purpose various shades are tried by mixing containers of blue paint and white paint. In each of the following mixtures, decide which is a lighter shade of blue and also find the lightest blue shade among all of them.
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If one container has one litre paint and the building requires 105 litres for painting, how many container of each type is required to paint the building by darkest blue shade?
Answer:
(i) In mixture A,
Number of blue containers = 3
Number of white containers = 4
Thus, ratio of blue to white containers =
In mixture B,
Number of blue containers = 3
Number of white containers = 3
Thus, ratio of blue to white containers =
Clearly, mixture A is a lighter shade of blue.
(ii) In mixture C,
Number of blue containers = 3
Number of white containers = 3
Thus, ratio of blue to white containers =
In mixture D,
Number of blue containers = 2
Number of white containers = 5
Thus, ratio of blue to white containers =
Clearly, mixture D is a lighter shade of blue.
(iii) In mixture E,
Number of blue containers = 6
Number of white containers = 1
Thus, ratio of blue to white containers =
In mixture F,
Number of blue containers = 4
Number of white containers = 2
Thus, ratio of blue to white containers =
Clearly, mixture F is a lighter shade of blue.
(iv) In mixture G,
Number of blue containers = 3
Number of white containers = 3
Thus, ratio of blue to white containers =
In mixture H,
Number of blue containers = 4
Number of white containers = 3
Thus, ratio of blue to white containers =
Clearly, mixture G is a lighter shade of blue.
From the above mixtures, mixture D is the lightest among all mixtures.
Total number of containers required for painting = 105
Number of blue containers required for painting =
Number of white containers required for painting =
Page No 325:
Question 85:
Posing a question
Work with a partner to write at least five ratio statements about this quilt, which has white, blue, and purple squares.
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How many squares of each colour will be there in 12 such quilts?
Answer:
From the figure,
Purple = 12, Blue = 20 and White = 16
Total squares = 12 + 20 + 16 = 48
Five ratio statement:
I: Purple : Total = 12 : 48 = 1 : 4
II: Blue : Total = 20 : 48 = 5 : 12
III: White : Total = 16 : 48 = 1 : 3
IV: Purple : Blue = 12 : 20 = 3 : 5
V: Purple : White = 12 : 16 = 3 : 4
Page No 325:
Question 86:
A packet of sweets was distributed among 10 children and each of them received 4 sweets. If it is distributed among 8 children, how many sweets will each child get?
Answer:
Total number of children = 10
If each child received 4 sweets, then the total number of sweets = 10 × 4 = 40 sweets
If 40 sweets are distributed between 8 children, then each get i.e. 5 sweets.
Page No 326:
Question 87:
44 cows can graze a field in 9 days. How many less/more cows will graze the same field in 12 days?
Answer:
Given that, cows that can graze a field in 9 days = 44
Number of cows that can graze the same field in 1 day = 44 × 9 = 396 cows
Number of cows that can graze the same field in 12 days = cows
Hence, (44 − 33) i.e., 11 more cows will graze the same field in 12 days.
Page No 326:
Question 88:
30 persons can reap a field in 17 days. How many more persons should be engaged to reap the same field in 10 days?
Answer:
In 17 days, persons required to reap a field = 30
In 1 day, persons required to reap a field = 30 × 17
In 10 days, persons required to reap a field = persons
Page No 326:
Question 89:
Shabnam takes 20 minutes to reach her school if she goes at a speed of 6 km/h. If she wants to reach school in 24 minutes, what should be her speed?
Answer:
Distance covered by Subham in 20 mins = 6 km/h =
Shabnam's speed to cover 2000 m distance in 24 mins = m/min
= km/h
Hence, the speed of Shabnam to cover the same distance in 24 minutes is 5 km/h.
Page No 326:
Question 90:
Ravi starts for his school at 8:20 a.m. on his bicycle. If he travels at a speed of 10 km/h, then he reaches his school late by 8 minutes but on travelling at 16 km/h he reaches the school 10 minutes early. At what time does the school start?
Answer:
Let the total distance be x km.
Let the time taken by Ravi to reach school be t min.
If the speed of the bicycle is 10 km/h, then he reaches his school late by 8 minutes.
If the speed of the bicycle is 16 km/h, then he reaches his school early by 10 minutes.
From (1) and (2), we get
Substitute x = 8 in (2), we get
Since starting time of school is (8 : 20 a.m. + 40 min) i.e. 9:00 a.m.
Page No 326:
Question 91:
Match each of the entries in Column I with the appropriate entry in Column II.
Column I | Column II | ||
1. | x and y vary inversely to each other | A. | = constant |
2. | Mathematical representation of inverse variation of quantities p and q | B. | y will increase in proportion |
3. | Mathematical representation of direct variation of quantities m and n | C. | xy = constant |
4. | When x = 5, y = 2.5 and when y = 5, x = 10 | D. | |
5. | When x = 10, y = 5 and when x = 20, y = 2.5 | E. | y will decrease in proportion |
6. | x and y vary directly with each other | F. | x and y are directly proportional |
7. | If x and y vary inversely then on decreasing x | G. | m n |
8. | If x and y vary directly then on decreasing x | H. | x and y vary inversely |
I. | p q | ||
J. |
Answer:
Column I | Column II |
x and y vary inversely to each other | xy = constant |
Mathematical representation of inverse variation of quantities p and q | |
Mathematical representation of direct variation of quantities m and n | m n |
When x = 5, y = 2.5 and when y = 5, x = 10 | x and y are directly proportional |
When x = 10, y = 5 and when x = 20, y = 2.5 | x and y vary inversely |
x and y vary directly with each other | = constant |
If x and y vary inversely then on decreasing x | y will increase in proportion |
If x and y vary directly then on decreasing x | y will decrease in proportion |
Column I
Column II1. x and y vary inversely to each otherA. = constant2. Mathematical representation of inverse variation of quantities p and qB. y will increase in proportion3. Mathematical representation of direct variation of quantities m and nC. xy = constant4. When x = 5, y = 2.5 and when y = 5, x = 10D. 5. When x = 10, y = 5 and when x = 20, y = 2.5E. y will decrease in proportion6. x and y vary directly with each otherF. x and y are directly proportional7. If x and y vary inversely then on decreasing xG. m n8. If x and y vary directly then on decreasing xH. x and y vary inversely
I. p q
J.
Page No 326:
Question 92:
There are 20 grams of protein in 75 grams of sauted fish. How many grams of protein is in 225 gm of that fish?
Answer:
In 75 g of sauted fish, protein = 20 g
In 1 g of sauted fish, protein = g
In 225 g of sauted fish, protein = g
Page No 326:
Question 93:
Ms. Anita has to drive from Jhareda to Ganwari. She measures a distance of 3.5 cm between these villages on the map. What is the actual distance between the villages if the map scale is 1 cm = 10 km?
Answer:
The distance between Jhareda to Ganwari in the map = 3.5 cm
Given scale, 1 cm = 10 km
So, actual distance between the villages = 3.5 × 10 = 35 km
Page No 327:
Question 94:
A water tank casts a shadow 21 m long. A tree of height 9.5 m casts a shadow 8 m long at the same time. The lengths of the shadows are directly proprotional to their heights. Find the height of the tank.
Answer:
Given that, the height of the tree = 9.5 m
Shadow of the tree = 8 m
Since the lengths of the shadows are directly proprotional to their heights.
Shadow of the water tank = 21 m
Hence, the height of the water tank is 24.9 m.
Page No 327:
Question 95:
The table shows the time four elevators take to travel various distances. Find which elevator is fastest and which is slowest.
Distance (m) | Time (sec.) | |
Elevator- A | 435 | 29 |
Elevator- B | 448 | 28 |
Elevator- C | 130 | 10 |
Elevator- D | 85 | 5 |
How much distance will be travelled by elevators B and C seperately in 140 sec? Who travelled more and by how much?
Answer:
Elevator A takes 29 s to cover a distance of 435 m.
Thus, the distance covered by elevator A in 1 s =
Elevator B takes 28 s to cover a distance of 448 m.
Thus, the distance covered by elevator B in 1 s =
Elevator C takes 10 s to cover a distance of 130 m.
Thus, the distance covered by elevator C in 1 s =
Elevator D takes 5 s to cover a distance of 85 m.
Thus, the distance covered by elevator D in 1 s =
Hence, in 1 s, elevator D covers more distance than any other elevator.
So, elevator D is fastest, while elevator C is slowest.
Now, elevator B cover distance in 140 s = 140 × 16 = 2240 m
Elevator C cover distance in 140 s = 140 × 13 = 1820 m
Thus, elevator B covers more distance than C = 2240 − 1820 = 420 m
Page No 327:
Question 96:
A volleyball court is in a rectangular shape and its dimensions are directly proportional to the dimensions of the swimming pool given below. Find the width of the pool.
Answer:
Length of the volleyball court = 18 m
Breadth of the volleyball court = 9 m
Length of the swimming pool = 75 m
Let, breadth of the swimming pool be x m.
Since the dimensions of the volleyball court and swimming pool are directly proportional.
Hence, the width of the pool is 37.5 m.
Page No 328:
Question 97:
A recipe for a particular type of muffins requires 1 cup of milk and 1.5 cups of chocolates. Riya has 7.5 cups of chocolates. If she is using the recipe as a guide, how many cups of milk will she need to prepare muffins?
Answer:
A muffins requires 1 cup of milk and 1.5 cups of chocolates.
Riya has 7.5 cups of chocolates.
Number of cups of milk required = cups
Page No 328:
Question 98:
Pattern B consists of four tiles like pattern A. Write a proportion involving red dots and blue dots in pattern A and B. Are they in direct proportion? If yes, write the constant of proportion.
Answer:
In pattern A,
Number of red dots = 4
Number of blue dots = 2
Proportion involving red dots and blue dots in pattern A =
Since pattern B consists of four tiles like pattern A.
In pattern B,
Number of red dots = 4 × 4 = 16
Number of blue dots = 2 × 4 = 8
Proportion involving red dots and blue dots in pattern B =
Yes, they are in direct proportion and the constant of proportion is 2.
Page No 328:
Question 99:
A bowler throws a cricket ball at a speed of 120 km/h. How long does this ball take to travel a distance of 20 metres to reach the batsman?
Answer:
We have,
Time taken by the ball to cover 20 m =
Page No 328:
Question 100:
The variable x is inversely proportional to y. If x increases by p%, then by what percent will y decrease?
Answer:
The variable x is inversely proportional to y.
Since, we know that 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.
So, we can say y decrease by p%.
Page No 328:
Question 101:
Here is a key board of a harmonium:
(a) Find the ratio of white keys to black keys on the keyboard.
(b) What is the ratio of black keys to all keys on the given keyboard.
(c) This pattern of keys is repeated on larger keyboard. How many black keys would you expect to find on a keyboard with 14 such patterns.
Answer:
(a) Total number of black keys = 7
Total number of white keys = 10
Thus, the ratio of white keys to black keys = = 10 : 7
(b) Total number of keys = 10 + 7 = 17
Thus, the ratio of black keys to all keys = = 7 : 17
(c) Number of black keys in 1 keyboard = 7
Number of black keys in 14 keyboard = 14 × 7 = 98 keys
Page No 329:
Question 102:
The following table shows the distance travelled by one of the new eco-friendly energy-efficient cars travelled on gas.
Litres of gas | 1 | 0.5 | 2 | 2.5 | 3 | 5 |
Distance (km) | 15 | 7.5 | 30 | 37.5 | 45 | 75 |
Which type of properties are indicated by the table? How much distance will be covered by the car in 8 litres of gas?
Answer:
From the given table, the distance travelled by one of the new eco-friendly energy-efficient earns travelled on gas.
The car travelled 15 km In 1 L of gas.
The car travelled 7.5 km in 0.5 L of gas.
The car travelled 30 km in 2 L of gas.
This rate shows a direct proportion between litres of gas and the distance covered.
The car can cover the distance in 8 L of gas = 8 × 15 = 120 km
Page No 329:
Question 103:
Kritika is following this recipe for bread. She realises her sister used most of sugar syrup for her breakfast. Kritika has only cup of syrup, so she decides to make a small size of bread. How much of each ingredient shall she use?
Bread recipe
1 cup quick cooking oats
2 cups bread flour
cup sugar syrup
1 tablespoon cooking oil
cups water
3 tablespoons yeast
1 teaspoon salt.
Answer:
Kratika's sister used cup of sugar syrup.
Kratika left with cup of syrup.
She need cup of syrup for one piece of bread.
Thus, ingredient will be in proportion of .
Now, the bread recipe will be
cup quick cooking oats
1 cup bread flour
cup sugar syrup
tablespoon cooking oil
cups water
tablespoons yeast
teaspoon salt.
Page No 329:
Question 104:
Many schools have a recommended students-teacher ratio as 35 : 1. Next year, school expects an increase in enrollment by 280 students. How many new teachers will they have to appoint to maintain the students-teacher ratio?
Answer:
Students-teacher ratio = 35 : 1
In school, students increased = 280
Thus, the teacher required for 280 students = teachers
Page No 329:
Question 105:
Kusum always forgets how to convert miles to kilometres and back again. However she remembers that her car’s speedometer shows both miles and kilometres. She knows that travelling 50 miles per hour is same as travelling 80 kilometres per hour. To cover a distance of 200 km, how many miles Kusum would have to go?
Answer:
50 miles per hour is the same as travelling 80 kilometres per hour.
Thus, 1 km = mile
⇒ 200 km = mile
Hence, Kusum has to go 125 miles for 200 km.
Page No 330:
Question 106:
The students of Anju’s class sold posters to raise money. Anju wanted to create a ratio for finding the amount of money her class would make for different numbers of posters sold. She knew they could raise Rs 250 for every 60 posters sold.
(a) How much money would Anju’s class make for selling 102 posters?
(b) Could Anju’s class raise exactly Rs 2,000? If so, how many posters would they need to sell? If not, why?
Answer:
(a) For 60 posters, amount raised = â¹250
For 1 poster, amount raised =
For 102 posters, amount raised =
Hence, Anju’s class make â¹425 for selling 102 posters.
(b) For 1 poster, amount raised =
For â¹2000, posters need to sell = posters
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