DISTANCE FORMULA (BASIC)

EXERCISE -1

Fill in the blanks:

1) If a point lies on x-axis, then y coordinates is ______

2) If a point lies on y-axis, then x coordinates is ______

3) If two points have same absicca, the line joining them is parallel to ____

4) If two points have same ordinate, the line joining them is parallel to ____

5) The line joining (2,3) and (-2,-3) passes through the ____.

6) The distance along the x-axis is called ___

7) The distance along the y-axis is called ___

8) XOX' is called the ____

9) YOY' is called the _____

EXERCISE -B

1) Find the distance between the points

A) P(-2,3), Q(6,-3). 10

B) A(2,3), B(-2,3). 4

C) X(3,5), Y(3,1). 4

D) P(2,-), Q(5,2). 5

E) M(3,7), N(-2,-5). 13

F) P(2/5,2/5), Q(-2/5,-1/5). 1

G) P(5/10,1/10), Q(-1/10,-7/10). 1

H) P(√3,1), Q(0,0). 2

I) X(√2+1, 2), Y(1, 2 - √2). 2

J) (a,b),(-a,-b). 2√(a²+b²)

2) Show that (3,3) is the centre of the circle passing through the points (6,2) (0,4) and (4,6).

EXERCISE -C

1)a) If A(4,2) and B(1,y), find the possible values of y so that AB= 5. -2,.6

b) If A(2,-3) and B(10,y), find the possible values of y so that AB= 10. 3,9

c) If A(2,5) and B(x,-7), find the possible values of x so that AB= 13. -3, 7

d) Find the value of a, if the distance between the points A(-3,-14) and B(a,-5) is 9 units. -3

e) A line segment is of length 10 units and one of its end is (-2,3). If the ordinate of the other end is 9, find the abscissa of the other end? 6 or -10

f) ) The distance between A(1,3) and B(x,7) is 5. Calculate the possible value of x. 4or -2

2) a) Find the coordinates of the points on the x-axis which are at a distance of 5 units from the point (5,-4). (2,0),(8,0)

b) Find the coordinates of the points on the x-axis which are at a distance of 2√5 units from the point (7,-4). How many such points are there? (9,0),(5,0), 2

c) Find the coordinates of the points on the x-axis which are at a distances from the point (2,3) and (3/2, -1) are in the ratio 2:1. (0,0),(8/3,0)

d) ) Calculate the distance between A(7,3) and B on the x-axis whose abscissa is the 11. 5 units

3)a) Find the coordinates of the points on the y-axis which are at a distance of 13 units from the point (12,9). (0,4),(0,14)

 b) Find the coordinates of the points on the y-axis which are at a distance of 10 units from the point (8,8). (0,2),(0,14)

4) a) A is a point on y-axis whose ordinate is 4 and B is a point on x-axis whose abscissa is -3. Find the length of the line segment AB. 5 units

b) Find the point/s which are at a distance of √10 units from the point (4,3) given that the ordinate of the point/s is twice the abscissa. (1,2) or (3,6)

c) A is a point on the y-axis whose ordinate is 5 and B is the point (-3,1), Calculate the length of AB. 5 units

d) What point(or points) in the y axis are at a distance of 10 units from the point (8,8). (0,2),(0,14)

5)a) The centre of a circle is (2a, a-7). Find the values of a if the circle passes through the point (11,-9) and has a diameter 10√2 units. 3,5

b) The centre of a circle is (2a-1, 3a+1) and passes through the point (-3,-1) and has a diameter 20 units, find the value/s of a. 2, -46/13

c) The centre of a circle is (x+2 , x-1). Find if the circle passes through (2,-2) and (8,-2). 3

d) Points A(-1,y) and B (5,7) lie on the circle with centre O(2, -3y). Find the value of y. Hence, find the radius of the circle. 7,-1, √793, 5

e) If the points A(4,3) and X(x,5) are on a circle with centre C(2,3), find the value of x. 2

EXERCISE -D

1) a) Show that the points A(6,6), B(2,3) and C(4,7) form a right angled triangle, Whose hypotenuse is AB.

b) A(6,0), B(-2,6) and C(12,8), Show that BC²= AB² + AC². Assign a special name to the triangle ABC.

c) Show that the points A(0,3), B(-2,1) and C(-1,4) form a right angled triangle. 

d) Find a if the triangle formed by A(8,-10), B(7,-3) and C(0,a) is a right angled triangle at B. -4

e) Find a if the triangle formed by A(0,3), B(-2,a) and C(-1,4) is a right angled triangle at A. 1

f) The points (2,9),(a,5),(5,5) are the vertices of a triangle ABC right angled at B. Find the value/s of a and hence find the area of ∆ ABC. 2,5, 6sq. units 

g) Show that the points (3,3),(9,0) and (12,21) are the vertices of a right angle triangle

2) a) Do the points P (3,2), Q(-2,-3) and (2,3) form a triangle? If so, name the type of triangle formed. triangle

b) Name the type of triangle formed by the points X(-5,6), Y(-4,-2) and C(7,5).

2)a) Show that the points A(2,-4), B(7,-1) and C(5,1) are vertices of an isosceles triangle. Name the equal sides. AB= AC

b) Check whether the points A(5,-2), B(6,4) and C(7,-2) are vertices of an isosceles triangle. 

c) A(7,5), B(2,4) and C(6,10), show that AB= AC. Assign special name to the triangle ABC. Isosceles triangle

d) A(2,2), B(- 2,4) and C(2,6) are the vertices of a triangle ABC, Prove that ABC is an isosceles triangle.

e) If A(-3,2), B(x,y) and C(-1,4) are vertices of an isosceles triangle. Prove that x+ y= 1, given AB= BC 

3) a) Show that the points X(1,1), Y(-1,-1) and Z(-√3,√3) form an equilateral triangle.

b) If two vertices of an equilateral triangle are (0,0),and (3,0), find the third vertex. (3/2, ±3√3/2)

4) a) Show that the points A(7,10), B(-2,5) and C(3,-4) are vertices of an isosceles right angled triangle.

b) Show that the points A(3,0), B(6,4) and C(-1,3) are vertices of an isosceles right angled triangle.

EXERCISE -E 

1) Show that the point (4,4) is equidistance from the point A(1,0) and B(-1,4).  

2) a) Find the value of K if the point P(2,4) is equidistant from the points A(5,K) and B(K,7). 3

b) If the point A(2,-4) is equidistant from the points P(3,8) and Q(-10,y), find the values of y. Also find distance PQ. -3, -5, √290, 13√2

c) If A(0,1) is equidistant from B(5,-3) and C(x,6), find the value of x. 4 or -4

d) If A(0,2) is equidistant from B(3,p) and C(p,5), find the value of p. 1

e) Find a relation between x and y such that the point (x,y) is equidistant from the points (7,1) and (3,5). x-y -2= 0

f) If the point P(x,y) is equidistant from the points A(a+ b, b -a) and B(A- b, a+ b), prove that bx = ay.

3)a) Find the point on the x-axis, which is equidistant from (2,-5) and (-2,9). (-7,0)

4)a) Find the coordinates of a point on the y-axis, which is equidistant from (4,0) and (4,12). (0,6)

b)Find the coordinates of a point on the y-axis, which is equidistant from A(6,5) and B(-4,3). (0,9)

5) Find the value of x such that PQ= QR where the coordinates P, Q and E are (6,-1),(1,3) and (x, 8). 5 or -3

6) The x-coordinate of a point X is twice its x-coordinate. If X is equidistant from the points Y(2,-5) and Z(-3,6), then find the coordinates of X. (16,8)

7) a) Find a relation between x and y such that the point (x,y) is equidistant from the points (3,6) and (-3,4). 3x+y -5= 0

8)a) By using distance formula, show that the points (4,2),(7,5), and (9,7) are collinear.

b) By using distance formula, show that the points (3,1),(6,4), and (8,6) are collinear.

EXERCISE - F

1) a) Show that the four points (2,-2),(8,4),(5,7) and (-1,1) are the vertices of a rectangle.

b) Show that the four points (0,-1),(-2,3),(6,7) and (8,3) are the vertices of a rectangle. Also find its area. 40

2)a) Show that the four points (0,5),(-2,-2),(5,0) and (7,7) are the vertices of a rhombus.

b) Show that the four points (3,0),(4,5),(-1,4) and (-2,-1) are the vertices of a rhombus. Also find the area of of the rhombus. 24 sq. units

c) Show that the four points (2,-1),(3,4),(-2,3) and (-3,-2) are the vertices of a rhombus. Also find its area. 24

3) Show that the four points (1,2),(-2,-1),(1,0) and (4,3) are the vertices of a parallelogram.

4) a) Show that the four points (2,1),(0,3),(-2,1) and (0,-1) are the vertices of a square.

b) Show that the four points (1,7),(4,2),(-1,-1) and (-4,4) are the vertices of a square.

c) Show that the four points (2,3),(-2,2),(-1,-2) and (3,-1) are the vertices of a square.

d) Show that the four points (2,1),(0,3),(-2,1) and (0,-1) are the vertices of a square. Also find its area. 8 

5) A(-3,2), B(-5,-5), C(2,-3) and D(4,4) are the 4 points in the plane. Show that the ABCD is a rhombus but not a square.

6) Name the type of quadrilateral formed by the following points and give reasons for your answer:   

i) (-1,-2),(1,0),(-1,2) and (-3,0). Square

ii) (4,5),(7,6),(4,3) and (1,2). ||gm

iii) (2,-2),(7,3),(11,-1) and (6,-6).

EXERCISE - G

1)a) Find the coordinates of the circumcentre of the triangle whose vertices are (4,6),(0,4),(6,2). (3,3)

b) Find the coordinates of the circumcentre of the triangle whose vertices are (8,6),(8,-2),(2,-2). Also, find its circumradius. (5,2), 5 units 

c) Find the coordinates of the circumcentre of the triangle whose vertices are (5,1),(-3,-7),(7,-1). Hence find the circumradius. (2,-4), √34

d) Find the coordinates of the circumcentre of the triangle whose vertices are (2,-2),(8,-2),(8,6). (5,2)

e) Find the co-ordinates of the circumcentre of ∆ ABC with the vertices at A(3,0), B(-1,-6) and C(4,-1). Also, find its circum-radius. (1,-3), √13 units 

2)a) Find the centre of a circle passing through the points (6,-6),(3,-7) and (3,3). Also find its radius. (3,-2),5 units

b) Find the centre of a circle passing through the points (5,1),(-3,-7) and (7,-1). (2,-4)

c) Find the coordinate of the centre of a circle which passes through the point A(0,0), B(-3,3) and C(5,-1). Also find the radius of the circle. (4,7), √65

3)a) The two opposite vertices of a square are (-1,2) and (3,2). Find the coordinates of the other two vertices. (1,0) and (1,4)

b) The ends of a diagonal of a square have coordinates (-2, p) and (p, 2). Find p if the area of the square is 40 sq. units. 6 or -6

Miscellaneous Exercise 

1) KM is a straight line of 13 units. If K has the coordinates (2,5) and M has the coordinates (x,-7), find the possible values of x. 7 and -3

2) Find the point (or points) which are distance of √10 from the point (4,3), given that ordinate of the point (or points) is twice the abscissa . (1,2),(3,6)

3) Show that the point (7,3),(3,0),(0,-4) and (4,-1) are the vertices of a rhombus. Also find the area of the rhombus.

4) What point on the x-axis is equidistance from A(5,4) B(- 2,3)? (2,0)

5) What points on x-axis are at a distance of 17 units from the point (11,- 8)? (26,0) and (-4,0)

6) What points on y-axis are at a distance of 10 units from the point A(-8,4)? (0,10) and (0,-2) 

7) A point P is at distance of √10 units from the point A (4,3). Find the coordinates of P, it being given that its ordinate is twice its abscissa . (3,6) and (1,2)

8) Show that the given points are collinear .

a) A(-2,3), B(1,2) and B(7,0).

b) A(3,-2), B(5,2) and C(8,8).

c) A(1,1),B(-2,7) and C(3,-3).

e) Show by distance formula that the points (-1,-1),(2,3) and (8,11) are collinear .   

9) Show that the points A(7,10), B(- 2,5) and C(3,4) are the vertices of an isosceles right angled triangle. Also, find the area of the triangle. 53 sq. units 

10) Show that the points P(1,1), Q(-1,1) and R(-√3, √3) are the vertices of an equilateral.

11) Show that the points L(2a, 4a), M(2a, 6a) and N(2a+ √3 a, 5a) are the vertices of an equilateral triangle with each side 2a.

12) Find a point equidistant from the point A(6,2), B(-1,3) and C(-3,-1). (2,-1)

13) Find the circumcentre of the triangle whose vertices are (-2,-3),(-1,0) and (7,-6). (3,-3)

14) Show that P(11,2) is the centre of the circle which passes through A(1,2), B(3,-4) and C(5,-6).       

MIXED MENSURATION

Sphere/Cylinder 

1) A copper sphere having a radius of 6cm is melted and then drawn into a cylindrical wire of radius 2mm. Calculate the length of the wire. 90m

2) A solid metallic cylinder of base radius 2 cm and height 9cm is melted to form a sphere. Calculate:

A) the radius of the sphere. 3

B) the surface of the sphere. (π= 3.14). 113.04

3) The diameter of a copper sphere is 6cm. The sphere is melted and is drawn into a long wire of uniform cross-section. If the length of the wire is 36 cm, find its radius. (Use π=3.14).            1cm

4) The diameter of a sphere is 6 cm. It is melted and down into a wire of a diameter 0.2 cm. Find the length of wire. 36m

5) The diameter of the copper sphere is 18 cm. The sphere is melted and is drawn into a long wire of uniform circular cross section. If the length of wire is 108 m, find its radius.       0.3cm

6) A hemispherical bowl of radius 3cm is full of water. The water is emptied into a cylindrical can of radius 2cm. Find the depth of water in the can. 4.5cm

7) A cylindrical vessel 60cm in diameter is partially filled with water. A sphere of diameter 36cm is dropped into it and fully submerged in water. Find the increase in the level of water in the vessel. 8.64cm

8) A building is in the form of a cylinder surmounted by hemispherical vaulted dome and contains 880/21 m³ of air. If the internal diameter of the building is equal to its total height above the floor; find the height of the building. 4m

9) From a sphere of radius 10cm, a right circular cylinder of base diameter 12cm is carved out. Calculate the volume of the right circular cylinder correct to 2 decimal places. 1810.29cm³

10) A cylindrical can whose base is horizontal and of radius 3.5 cm contains sufficient water so that when a sphere is placed in the can, the water just covers the sphere. Given that the sphere just fits into the can, Calculate:

A) The total surface area of the can in contact with water when the sphere is in it. 383/2, 7/3

B) The depth of water in the can before the sphere was put into can. Take π to be 22/7 and give your answer as proper fractions.

11) The internal and external radii of a hollow sphere are 3cm and 5cm respectively. The sphere is melted to form a solid cylinder of height 8/3cm. Find the diameter and curved surface area of the cylinder. 14, 352/3

12) Spherical marbles of diameter 1.4 cm are dropped into cylindrical beaker containing some water and her fully submerged . The diameter of the beaker is 7 cm. Find how many marbles have been dropped in it if the water rises by 5.6 cm. 150

13) A metallic disc, in the shape of a right circular cylinder, is a height 2.5 m.m and base radius 12cm. Metallic disc is melted and made into a sphere. Calculate the radius of the sphere. 3

14) A solid sphere of radius 6cm is melted into a hollow cylinder of uniform thickness. If the external radius of the base of the cylinder is 4 cm and its height is 72cm, find the uniform thickness of the cylinder.

15) What is the least number of solid metallic spheres , each of 6cm diameter, that should be melted and recast to form a solid metal cylinder whose height is 45cm and diameter 4cm.  

16) A cylinder boiler, 2m high, is 3.5m in diameter. It has a hemispherical lid. Find the volume of its interior, including the part covered by the lid.    

17) The volume of a sphere and a right circular cylinder are equal and the diameter of the sphere equals the diameter of the base of the cylinder. Determine the height of the cylinder in terms of the diameter of the sphere.

18) A solid is composed of a cylinder with hemispherical ends . If the whole length of the solid is 105cm and the diameter of the hemisphere ends is 36 cm, find the cost of policy the surface of the solid at the rate of 21 paise per sq. cm.

19) A solid metal cylinder of diameter 28cm the height 42 cm is melted down and recast into sphere of diameter 7cm. Calculate the number of spheres that can be made.   

20) A sphere of diameter 14cm is dropped into a cylinder of base radius 10.5cm containing water. Calculate the rise in the lvel of the water in the cylinder, if the sphere is completely submerged, and no water flows out.

21) A spherical shell of lead whose external diameters 18 cm, is melted into a right circular cylinder, 8 cm high and 12cm in diameter. Find the inner diameter of the shell.    16.01 cm

22) a sphere has the same volume as a cylinder whose height is equal to the diameter of its cross section. Find the ratio between their radii.       ³√3: ³√2

23) A cylindrical tub of radius 16cm contains water to a depth of 30 cm. A spherical iron ball is dropped into a tub and thus the level of water is raised by 9 cm. What is the radius of the ball ?    12cm

24) A cylinder jar of radius 6cm contains oil. Iron spheres each of radius 1.5cm are immersed in the oil. How many spheres are necessary to raise the level of the oil by 2 cms?      16

25) A sphere of radius 3 cm is dropped into a cylindrical vessel partly filled with the water. The radius of the vessel is 6cm. If the sphere is submerged completely, by how much will the surface of the water be raised.      1 cm

26) A cylindrical tub of radius 12cm contains water to depth of 20cm. A spherical iron ball is dropped into the tub and thus the level of the water is raised by 6.75 cm. What is the radius of the ball.     9cm

27) A vessel is in the form of hemispherical bowl mounted by hollow cylinder. The diameter of the sphere is 14 cm and the total height of The vessel is 13 cm. Find its capacity.      1642.66cm³

Sphere/Cube/Cuboid:

1) A spherical ball of diameter 21 cm is melted and recast into cubes of side 1cm. Find the number of cubes thus formed. 4851

2) What is the ratio of the volume of the cube to that of the sphere which will fit inside of the cube ? 6:π

3) the largest sphere is carved out of a cube of edge 7cm. Find the volume of the sphere. 539/3cm³

4) A piece of butter 3cm by 5cm by 12cm is placed in a hemispherical bowl of diameter 6.5cm. Will the butter overflow when it melts completely? Yes

5) A solid rectangular block of metal 49cm by 44cm by 18cm is melted and formed into a solid sphere. Find the radius of the sphere. 21cm

6) The diameter of a sphere is 8 cm. Find the length of the edge of a cube which has the same surface area of that surface.

7) How many spherical bullets can be made out of a cube of lead whose edge measures 22 cm, each bullet being 2 cm in diameter.       2541

8) How many spherical lead shot each 0.6 cm in diameter can be made from a rectangular solid 9cm x  11 cm x 12 cm.       10500

9) The largest sphere is carved out of a cube of side 7 cm. Find the volume of the sphere.     179.5 cm³

10) The surface area of a cube is equals to the surface area of a sphere. Find the ratio of volume of cube that of the sphere.     √π: √6

Sphere/Cone

1) the radius of a sphere and the base radius of a cone are equal, each being 8cm. If the volumes of these two solids are also equal, find the slant height of the cone. 32.98

2) A cone and a hemisphere have equal bases and equal volumes. find the ratio between the height of the cone and the radius of the hemisphere. 2:1

3) The curved surface of a solid cone with base radius 3cm is 300/7cm². Three such cones are melted and recast into a sphere. Find the radius and surface of the sphere. Assume that there is no loss of metal in melting and recasting. 3cm, 792/7cm²

4) An inverted conical vessel of radius 6cm and height 8cm is filled with water. A sphere is lowered into the vessel. find:

A) the radius of the sphere, given that when it touches the sides, the highest point of the sphere is in level with the base of the cone.  

B) volume of water that flows out of conical vessel consequent to lowering the sphere in it. 3 113.04

5) A toy is in the form of a cone mounted on a hemisphere with the same radius. The diameter of the base of the conical portion is 12cm and its height is 8cm. Determine the surface area and volume of the toy. (π=3.14). 413.48cm², 753.6cm³

6) A vessel is in the form of an inverted cone. Its height is 11cm and the radius of its top which is open is 2.5cm. It is filled with water upto the rim. When lead shots, each of which is a sphere of radius 0.25 cm and dropped into the vessel, 2/5 of the water flows out. Find the number of lead shots are dropped into the vessel. 440

7) A hollow sphere of internal and external dimension 4cm and 8cm respectively, is melted into a cone of base diameter 8cm. Find the height of the cone. 14

8) The surface area of a solid metallic sphere is 1256cm². It is melted and recast into solid right circular cone of radius 2.5cm and height 8cm. Calculate:

A) the radius of the solid sphere.

B) the number of cones recast. Take π=22/7. 10, 80

9) A sphere has the same curved surface as the curved surface of a cone of height 36cm and base radius 15cm. Find the radius of the sphere. 12.09cm

10) A sphere of radius 8 cm is melted and recast into a right circular cone of a height 32 cm. Find the radius of the base of the cone. 8cm

11) A hollow sphere of internal and external diameter 4cm and 8 cm respectively is melted into a cone of base diameter 8cm. Find the height of the cone. 14cm

12) A vessel is in the form of an inverted cone. its height is 8 cm and the radius of its top, which is open, is 5 cm. It is filled with water up to the rim. When lead shots, each of which is a sphere of radius 0.5cm are dropped into the vessel, one-fourth of the water flows out. Find the number of lead shots dropped in the vessel.

13) a buoy is made in the form of hemisphere surmounted by a right circular base coincides exactly with the plane surface of hemisphere. The radius of the base of the cone is 3.5m and its volume is two-third of the hemisphere. Calculate the height of the cone and the surface area of the buoy, correct to two decimal places.

14) A cone of height 15cm and diameter 7cm is mounted on a hemisphere of same diameter. Determine the volume of the solid thus formed .

15) A hollow sphere of internal and external diameters is 4cm and 8 cm respectively is melted into a cone of base diameter 8 cm. Find the height of cone. 14cm

16) A cone and a hemisphere have equal bases and equal volumes . Find the ratio of their lengths.     2:1

17) A toy is in the form of a cone mounted on hemisphere of radius 3.5cm. The total height of the toys 15.5 cm. Find the total surface area.      214.5 cm²

18) A toys in the form of a cone mounted on a hemisphere. The diameter of the base the cone is 6 cm and its height is 4 cm. Calculate the surface area of the toy(π= 3.14) 103.62cm²

   

Cone/Sphere/Cylinder 

1) A cone, a hemisphere and a cylinder stand on equal bases and have the same height, the height being equal to the radius of the circular base. find the ratio of the circular base. Find the ratio of their whole surface.      (√2+1):3:4

2) A toys hemispherical base, cylindrical middle portion and conical top. The radius of the hemisphere, cylinder, the base of the cone and heights of all the three portions are equal. If the number of units in the volume of the toy is the same as the number of units on its surface area. Find the total height of the toy.   8.121 cm

Cylinder/Sphere/Cone

1) Find the ratio of the volumes of a cylinder, a cone and sphere, if each has the same radius and the same height. 3:1:2

2) A cone, a hemisphere and a cylinder stand on equal bases and have the same height, the height being equal to the radius of the circular base. Find the ratio of their whole surfaces. (√2+1):3:4

3) A solid consisting of right circular cone, standing on a hemisphere is placed upright in a right circular cylinder full of water and touched the bottom. Find the volume of water left in the cylinder, having given that the radius of the cylinder is 3 cm and its height is 6cm, the radius of the hemisphere is 2 cm and the height of the cone is 4cm. Give your answer to the nearest cubic centimetre. (π=22/7). 2860/21

Diagrammatic Questions:

1) With reference to the figure given alongside,

 a metal container in the form of a cylinder id surmounted by a hemisphere of the same radius. the internal eight of the cylinder is 7 m and the internal radius is 3.5 m. calculate

a) The total area of the internal surface, excluding the base.   231m²

b) The internal volume of the container in m³.          1078/3 m3

2) The figure shows the cross-section of an ice cream cone consisting of a cone surmounted by a hemisphere.

The radius of the hemisphere is 3.5cm and the height of the cone is 10.5cm. The outer shell ABCDEF is shaded and is not filled with ice cream. AF= DC=0.5 cm, AB|| FE and BC|| ED. calculate :

a) the volume of the ice-cream in the cone( the unshaded portion including the hemisphere) in cm³.

b) the volume of the outer shell (the portion) in cm³.

give your answer correct to the nearest cm³.    175cm³, 50cm³

3) In the given figure , a cylinder is surmounted by a cone at one end and a hemisphere at the other end, 

Given that common radius=3.5 cm, the height of the cylinder is 6.5 cm and the total height 12.8cm, calculate the volume of the solid correct to the nearest integer.     376 cm³

4) A solid wooden toy is in the shape of a right circular cone mountained on a hemisphere. If the radius of the hemisphere is 4.2 cm and the total height of the toy is 10.2 cm, find the volume of the wooden toy . Also, find its surface area.     96.36cm²

5) A child rocket toy has a hemispherical base, a cylindrical body and a conical nose cap the rocket is 20 cm long. The hemisphere, the cylinder and the cone have a base diameter of 3 cm each and the body of the cylindrical boxe is 8cm long. Find the volume of the toy rocket.    88.3125 cm³

TIME AND WORK

) A alone can finish a piece of work in 12 days and B alone can do it in 15 days. If both 

of them work at it together, how much time will they take to finish it.      20/3 days 

) A and B together can do a piece of work in 12 days, while B alone can finish it in 30 days. In how many days can A alone finish the work?      20 days

) A can do a piece of work in 25 days and B can finish it in 20 days. They work together for 5 days and then A leaves. In how many days will B finish the remaining work?       11 days

) A and B can do it in 12 days, B and C can do it in 15 days while C and A can finish it in 20 days. In how many days will A, B and C finish it, working together ? In how many days will each one of them finish it, working alone ?    30 20,60 days

) A tap A can fill a cistern in 8 hours while tap B can fill it in 4 hours. In how much time will the cistern be filled if in how much time the cistern be filled if both A and B are opened together?       12 hours

) A cistern can be filled by two taps A and B in 12 hours and 16 hours respectively. The full cistern can be emptied by a third tap C in 8 hours. If all the taps are turned on at the same time. In how much time will the empty cistern be filled up completely?       48 hours

) Rajan can do a piece of work in 24 days while Amit can do it in 30 days. In how many days can they complete it, if they work together ? 40/3 days 

) Ravi can do a piece of work in 15 hours while Raman can do it in 12 hours. How long will both take to do it, working together ? 6 hrs 40 mins

) A and B working together can finish a piece of work in 6 days, while A alone can do it in 9 days. How much time will B alone take to finish it ?   18 days 

) Two motor mechanics , Raju and Siraj, working together can overhaul a scooter in 6 hours. Raju alone can do the job in 15 hours. In how many hours can Siraj alone do it ?    10 hours

) A, B and C can do a piece of work in 10 days, 12 days and 15 days respectively. How long will they take to finish it if they work together ?     4 days

) A can do a piece of work in 24 hours, while B alone can do it in 16 hours. If A, B and C working together can finish it in 8 hours. In how many hours can C alone finish the work?  48 hours

) A, B and C working together can finish a piece of work in 8 hours. A alone can do it in 20 hours and B alone can do it in 24 hours. In how many hours will C alone do the same work ?    30 hours 

) A and B can finish a piece of work in 16 days and 12 days respectively. A started the work and worked at it for 2 days. He was joined by B. Find the total time taken to finish the work.      8 days 

) A can do a piece of work in 14 days while B can do it in 21 days. They began together worked at it for 6 days. Then, A fell ill and B had to complete the remaining work alone.  In how many days was the work completed ?    12 days

) A can do 2/3 of a certain work in 16 days and B can do 1/4 of the same work in 3 days. Un how many days can both finish the work, working together ?     8 days 

) A, B and C can do a piece of work in 15,12 and 20 days together. They started the work together, but C left after 2 days. In how many days will the remaining work be completed by A and B .     4 days

) A and B can do a piece of work in 18 days ; B and C can do it in 24 days while C and A can finish it in 36 days. In how many days can A, B and C finish it, if they all work together ?     16 days

) A and B can do a piece of work in 12 days, B and C in 15 days, and C and A in 20 days. How much time will A alone take to finish the job.    30 days

) Pipes A and B can fill an empty tank in 10 hours and 15 hours respectively. If both are opened together in the empty tank, how much time will they take to fill it completely .    6 hours

) Pipe A can fill an empty tank in 5 hours while pipe B can empty the full time in 6 hours. If both are open at the same time in the empty tank. How much time will they take to fill it up completely ?     30 hours

) Three taps A, B and C can fill an overhead tank in 6 hours, 8 hours and 12 hours respectively. How long would it the three taps take to fill the empty tank, if all of them are opened together ?     2 hours 40 minutes

) A cistern has two outlets A and B which can fill it in 12 minutes and 15 minutes respectively. An outlet C can empty the full cistern in 10 minutes. If all the three pipes are opened together in the empty tank, how much time will take to fill the tank completely .     20 minutes

) A pipe can fill a cistern in 9 hours. Due to a leak in its bottom, the cistern fills up in 10 hours. If the cistern is full , in how much time will it be emptied by the leak.    90 hours

) pipes A can fill a cistern in 6 hours and pipe B can fill it in 8 hours. Both the pipes are opened and after 2 hours, pipe A is closed. How much time will B take to fill the remaining part of the tank ?     3 hrs 20 mins

) A alone can do a piece of work in 10 days and B can do it in 15 days. In how many days A and B together do the same work ?

a) 5 days b) 6 days c) 8 days d) 9 days 

) A man can do a piece of work in 5 days. He and his son working together can finish it in 3 days. In how many days can the son do it alone.

a) 13/2 days b) 7 days c) 15/2 days d) 8 days

) A can do a job in 16 days and B can do the same job in 12 days. With the help of C they can finish the job in 6 days only. Then , C alone can finish it in

a) 34 days b) 22 days c) 36 days d) 48 days 

) To complete a work. A takes 50% more time then B. If together they take 18 days to complete the work. How much time shall B take to do it ?

a) 30 days b) 35 days c) 40 days d) 45 days 

) A work twice as fast as B. If both of them can together finish a piece of work in 12 days, then B alone can do it in

a) 24 days b) 27 days c) 36 days d) 48 days

) A alone can finish a piece of work in 10 days which B alone can do it in 15 days. If they work together finish it, then out of total wages of ₹3000, A will get

a) 1200 b) 1500 c) 1800 d) 2000

) The ratio of the working of A and B are in the ratio 3:4. The number of days taken by them to finish the work are in the ratio 

a) 3:4 b) 9: 16 c) 4 : 3 d) 16: 9

) A and B together can do a piece of work in 12 days; B and C can do it in 20 days while C and A can do it in 15 days. A, B, C all working together can do it in

a)  6 days b) 9 days c) 10 days d) 21/2 days 

) three men or 5 women can do a work in 12 days. How long will six men and five women take to do it?

a) 6 days b) 5 days c) four days d) three days 

) A can do a piece of work in 15 days. B is 50% more efficient than A. B can finish it in

a) 10 days b) 15/2 days c) 12 days d) 21/2 

) A does 20% less work than B. if A can finish a piece of work in 15/22 hour, then B can finish it in

a) 5 hours b) 11/2 hour c) 6 hours d) 13/2 hours 

) A can do a piece of work in 20 days which B alone can do in 12 days. B worked at it for 9 days. A can finish the remaining work in 

a) 3 days b)  5 days c) 7 days d) 11 days

) A can do a piece of work in in 25 days, which B alone can do in 20 days. A started the work and was joined by B after 10 days. The work lasted for 

a) 25/2 days b) 15 days c) 50/3 days d) 14 days

) 2 pipes can fill a tank in 20 minutes and 30 minutes respectively. If both the pipes are opened simultaneously, then the tank will be filled in 

a) 10 mins b) 12 minutes c) 15 minutes d) 25 minutes

) A tap can fill cistern in 8 hours and another tap can empty the full cistern in 16 hours. If both the taps are open,  the time taken to fill the cistern is

a) 16/3 hours b) 10 hours c) 16 hours d) 20 hours

) A pump can fill a tank in 2 hours. Due to leak in the tank it takes 7/3 hours to fill the tank. The leak can empty the full tank in

a) 7/3 hours  b) 7 hours c) 8 hours d) 14 hours

)  Two pipes can fill a tank in 10 hours and 12 hours respectively, while a third pipe empties the tank in 20 hours. If all the three pipes operates simultaneously, in how much time will the tank be full.

a) 7 hours 15 minutes b)  7 hour 30 minutes c) 7 hours 45 minutes d) 8 hours 

) A can do a piece of work in 10 days while B alone can do it in 15 days. In how many days can both finish the same job ?     6 days 

) A and B can do a piece of work in 15 days; B and C  in 12 days,  C and A in 20 days. How many days will be taken by A,  and B and C working together to finish the work?     10 days

) Tap A can fill a cistern in 8 hours and tap B can empty it in 12 hours. How long will it take to fill the cistern if both of them are opened together.      24 hours

) Two men or three women can do a piece of work in 16 days. In how many days can 4 men and 6 women do the same work ?          4 days

) A pipe can fill a cistern in 9 hours. Due to a leak in its bottom, the system fills up in 10 hours. If the cictern is full, In how much time will it be emltied by the leak.   90 hours 

) The rates of working of two taps A and B are in the ratio 2:3. The ratio of the time taken by A and B respectively to fill a given cistern is

a) 2:3 b) 3:2 c) 4:9 d) 9:4

) A can finish a piece of work in 12 hours while B can finish it in 15 hours. How long will both take to finish it, working together ?

a) 9 hours b) 20/3 hours c) 27/4 d) 25/3 hours 

) A can do a piece of work in 14 days and B is 40% more efficient than A . In how many days can B finish it .   

a) 10 days b) 15/2 days c) 21/4 days d) 28/5 days

) A pump can fill a cistern in 3 hours. Due to leak in the tank it takes 7/3 hours to fill it. The leak can empty the full tank in

a) 7 hours b) 14 hours c) 8 hour d) 3 hours

) A work twice as fast as B. If both of them can together finish the piece of work in 12 hours. then B alone can do it in 

a) 24 hours b) 27 hour c) 36 hour d) 18 hours

) A tap can fill a tank in 6 hours . The part of the tank filled in one hour is ___

) A and B working together can finish a piece of work in 6 hours while A alone can do it in 9 hours. B alone can do it in ____hours.

) A can do a work in 16 hours and B alone can do it in 24 hours. If aA, B and C working together can finish it in 8 hours, then C alone can finish it in____ hours.

) if A's one day's work is 3/20, then A can finish the whole work in _______days.

) Ram can do his work in 15 days. How much work can he do in one day ?  1/15th work 

) Mohan can complete his course in 20 days. What part of the course will be completed in 5 days.      1/4 th part

) If A takes 12 days to finish a work while B takes 15 days to complete the same amount of work then find the number of days both A and B will take if they are working together.     20/3 days

) If A and B can complete a work in 12 days while B takes 15 days to complete the same amount of work then find the number of days A will take to complete the work.   60 days 

) if A can do a work in 12 hours. B can do the same work in 15 hours and C can do the same work in 20 hours , then how many days will they take to complete the work if they work working together ?      5 days

) if A+ B can do a work in 20 days, B + C can do the same piece of work in 24 days and A+C can do the same work in 30 days then how many days working together they can finish the work ?       16 days 

) A can do a piece of work in 80 days. He works for 10 days and then B alone finishes the remaining work in 42 days. In how much time will A and B working together, finish the work ?    30 days

) Riya and Priya can do a piece of work in 9 days and 12 days respectively. If they work for a day alternatively and Riya begins the work first, in how many days the work will be completed ?        41/4 days

) Namitha can make 1/5 jar of pickle in 4 days. whereas Nilu can make 1/5 jar of pickle in 6 days . If they work together how long will they take to make a jar of pickle?    12 days

) If A+ B can do a work in 20 days, B+ C can do the same piece of work in 24 days and A+ C can do the same work in 30 days then in how many days A alone can finish the work ?      48 days

) A trains crosses a 120m bridge with a speed of 36 kmph in 20 seconds then find the length of train.   80m

) Two trains having length 140 m and 160m with their respective speed 36kmph and 54 kmph are moving in opposite direction crosses each other. Find the time taken to cross to each other.   12s

) Two trains of length 200m and 175m crosses each other moving in opposite direction in 15/2 seconds whereas it takes 55/2 seconds to cross each other moving in same direction. Find the speed of the train .      30, 20 m/s

) A thief is spotted by a policeman from a distance of 100 m. When the policeman starts the chase the thief also starts running. If the speed of the thief be 8 kmph and that of the police is 10 kmph, how far the thief will run before he is overtaken ?   400 m

)  Leela  walks at the rate of 5 kmph and reaches his school 7 minutes late, however if he wolks at the rate of 6 kmph he reaches the school 5 minutes before the schedule time. Find the distance Leela covers to reach the school.         6 km

) If two pipes A and B can fill a tank and 12 hours and 15 hours respectively. In how many hours will they fill the tank if both the pipes are opened simultaneously.    20/3 hours

) If pipes A and B can fill a tank in 6 hours while pipe B can fill the tank in 8 hours. Find how many hours will pipe A fill the tank .     24 hours

) 2 pipes can fill the tank in 10 hours and 12 hours respectively. While a third pipe can empty the tank in 20 hours. If all the three pipes are open simultaneously then in how much time will the time be filled ?       7 hours 30 minutes

) workers A takes 8hours to do a job. Worker B takes 10 hours to do a job. How long should it take both A and B, working together to do the same job ?   40/9 hours

) Rajan can do a piece of work in 24 days while Amit can do it in 34 days. In how many days can they complete the work ?    40/3 days

) Tap A fills a tank in 4 hours and tap B fills the tank in 8 hours. How long will it take to fill the tank if both the taps are opened simultaneously .    8/3 days

) If A and B can complete a peace of work in 4 days while A alone can do the same work in 12 days, in how many days can B alone complete the work.    6 days 

) X and Y working together plough a field in 6 days. If X alone can plough the same field in 8 days, how long would Y take to plough the field?    24 days

) A and B be can do a piece of work in 18 days. B and C can do it in 24 days. A and C can do it in 36 days. In how many days will A, B and C complete it, working together?  48,144/5, 144 days 

UNITARY METHOD SUMS

EXERCISE - A

1) If 6 bats cost ₹96, find the cost of 4 bats.      Rs64

2) If the cost of 5 balls is ₹75 then find the cost of 7 such balls.   Rs105

3) ) If 2 pencils cost ₹6, find the cost of 5 such pencils?      Rs15

4) If 7 kg of sugar costs Rs140, find the cost of 10 kg of sugar.     Rs200

5) If 14m of cloth costs Rs3864, how much will 24 m cloth cost?   Rs 6624

6) If 30 pens cost Rs750, how many pens will cost Rs1000 ?     Rs 40

7) if 6 copies of a newspaper cost Rs21. How many copies will cost Rs70?   Rs20

8) if 18 dolls cos Rs 630, how many dolls can be bought for Rs 445.     Rs13

9) The cost of 140 tennis balls is Rs4900. Find the cost of two dozen such balls.  Rs840

10) The railway fare for 61 km is Rs183. Find the fare of 53 km.        Rs159

11) A dry-cleaner charges Rs40 to wash 2 shirts. What will be the charge for washing 11 shirts?      Rs220

12) A taxi charges a fare of Rs250 for a journey of 150 km. How much would it charge for a journey of 1024 km?  ₹2108

13) It takes 60.2 kg of paper to make 700 exercise books. How much paper would be required to make 1550 exercise books?      133.3 kg

14) If 9 kg of sugar costs Rs238.50. How much sugar can be bought for Rs371? 14 kg

15) The cost of 15m of cloth is Rs981. What length of this cloth be purchased for Rs1308?     20m

16) A car covers 432 km in 36 litres of petrol. How much distance would it cover in 25 litres of petrol?      300km

17) A truck travels a distance of 510 km in 34 litres of diesel. How much distance would it cover in 20 litres of diesel? 300 km

18) If 50m of cloth costs Rs3725, how much cloth can be purchased for Rs1788?    24m

19) ) A hotel charges Rs3500 for a stay of 10 days. How much will it charge for a stay of 1 week ?   ₹2450

20) Rakesh takes 100 equal steps to cover a distance of 80m. What distance would he cover if he took 375 equal steps of the same size?      300m

21) A man buys 10 L of petrol for Rs420. How much petrol could he have bought for the same amount if the price had been Rs2 less per litre?       10.5L

EXERCISE - B

1) If 12 identical bags of sugar weight 100 kg then how many similar bags of sugar will weight 150 kgs?    18

2) 5 bags of flour of equal size weight 35 kg. How many such bags of flour will weight 56 kg?       8

3) If the weight of 9 sheets of thick paper is 30 grams, how many sheets of the same paper would weight 5/4 kilograms ?    375

4) 350 boxes can be placed in 25 cartoons . How many boxes can be placed in 16 cartoons ?     224

5) A boy is 1.6 metre tall when he is 12 years old. What will be his height when he is 24 years old ?      

6) A watch loses 20 seconds in four days. How many seconds does it lose in a week?        35

EXERCISE - C

1) If 6 men take 12 hours to complete a job, how long would 9 men take to do it, provided all of them work at the same rate?       8 days 

2) ) 30 men can do a job in 21 days. In how many days would 45 men finish the job?     14days

3) three men can do a job in 8 days. How long will 4 men take to finish the job?     6days

4) 15 workers can repair a road in 24 days. How long will it take for 9 workers to repair the road ?    40days

5) If 20 men complete a job in 4 days, how long will 16 men take to do it?     5 days

6) A military camp has enough food to feed 1000 soldiers for 36 days. If 500 additional soldiers join the camp, how long will the food last?      24 days

7) If 45 men can do a piece of work in 49 days. In how many days will 39 men do it?      63 days 

8) If 35 men can reap a field in 8 days, in how many days can 20 men reap the same field?      14 days

9) 30 man can finish a piece of work in 28 days. How many days will be taken by 21 men to finish it?      40 days 

10) A hostel has enough food to feed 250 students for 30 days. How long will the food last for 300 students.    25 days

11) 64 children at a camp have provisions for 9 days. Eight more children join the camp. How long will the provisions last ?        8 days

12) Adityi has enough money for six weeks for holidaying if she spends Rs1400 per week. How long will the money last if she spends Rs1050 per week .      8 weeks

13) In a camp, provisions for food was made for 400 people for 30 days. If 500 people turned up for the camp, how long would the food last?         24 days 

14) A garrison of 200 men had provisions for 45 days. After 15 days, 40 more men join the garrison. Find the number of days for which the remaining food will last?      25 days

15) A garrison of 900 men had provision for 42 days. However, a reinforcement of 500 men arrived. For how many days will the food last now ?     27 days

16) In a fort, 300 men had provision for 90 days. After 20 days, 50 mem left the fort. How long would the food last at the same rate?         84

17) A farmer has enough food to feed 28 animals in his cattle for 9 days. How long would the food last . If there were 8 more animals in his cattle ?   7 days 

18) In a hostel, 75 students had food provisions for 24 days. If 15 students leave the hostel, for how many days would the food provision last ? 30 days

19) 7 taps of the same size fill a tank in 1 hour 36 minutes. How long will 8 taps of the same size take to fill the tank? 1 hr 24 mins

20) 8 tapes of the same size fill a tank in 27 minutes. If two taps go out of the order, how long would the remaining taps take to fill the tank?     36 mins

21) 6 pipes can fill a tank in 1 hour 24 minutes. How long will it take to fill the tank if 7 pipes of the same type are used?      1 hr 12 mins

EXERCISE - D

1) If 12 men can plough a field in 6 days, how many men will be required to plough the field in 4 days?      18

2) 12 men can complete a job in 15 days. How many men will be required to finish the work in 30 days ?       20

3) 8 people consume a stock of food in 30 days. How many people will consume the stock of food in 20 days.      12

4) If 12 men complete a job in 30 days, how many men will be needed to complete it in 40 days ?         9 men

5) If 14 workers can build a wall in 45 hours. How many workers will be required to do the same work in 35 hours ?      18

6) 12 men can dig a pond in 8 days. How many men can dig it in 6 days.     16

7) 10 people can dig a trench in 6 days. How many people can dig it in 4 days?  15

8) 6 cows can graze a field in 28 days. How long would 14 cows take to graze the same field.      12

9) 11 men can dig 27/4 metres long trench in one day. How many men should be employed for digging 27 m long trench of the same type in one day?     44men

Miscellaneous Exercise - 1

1) A car consumes 15L of petrol to cover 270km

a) How much petrol will it consume to cover 180 km ?     10L

b) If the cost of petrol is Rs42 per litre. What will be the expenditure on petrol when the car covers 336 km ?     Rs840

2) A train is moving at a uniform speed of 75 kmph .

a) How far will it travel in 24 minutes ?       30km

b) In how much time will it cover 175 km ?      2 hrs 20mins

3) A verticle pole 5m 60 cm high casts shadow a 3m 20cm long . Find at the same time

a) the length of shadow csst by another pole 10m 50 cm high.          6m

b) the height of a pole which cssts a shadow 5 m long.       8m 75cm

4) If 20 copies of a book of R3700 then what will be the cost of 35 copies of the book? Also, find the number of copies of the book that can be bought for Rs2775.    Rs6475, 15

5) six towels take 30 minutes to dry in the sun. How long do 2 towels take to dry in the same situation?      30

6) A car takes 5 hours to reach a destination by traveling at the speed of 60 kmph . How long will it take when the car travels at the speed of 75 kmph?     4 hrs

7) A factory requires 42 machines to produce a given number of articles in 56 days . How many machines would be required to produce the same number of articles in 48 days.      49

8) If 5 men or 7 women can earn Rs875 per day, how much would 10 men and 5 women earn per day.     Rs2375

9) A loaded truck travels 16km in 25 minutes. At the same speed, how far can it travel in 5 hours.      192km

10) In a model of a ship, the mast is 9cm high, while the mast of the actual ship is 15m high. If the length of the ship is 35 m, how long is the model ship?     21cm

11) A car is traveling at the average speed of 50 kmph. How much distance would it travel in 1 hour 12 minutes ?      60km

12) Ravi walks at the uniform rate of 5 kmph . What distance would he cover in 2 hours 24 minutes ?    12km

13) If the thickness of a pile is 12 cardboards is 65mm, find the thickness of a pile of 312 such cardboards.       1m 69cm

14) Reena types 540 words during half an hour. How many words would she type in 8 minutes?    144 words

15) The scale of a map is 1:3 x10⁷. Two cities are 5cm apart on the map. Find the actual distance between them in kilometres .     1500 km

16) In 8 days, the earth picks up (6.4 x 10⁷) kg of dust from the atmosphere. How much dust will it pick up in 15 days?      1.2 x 10⁸ kg

17) A school has 9 periods a day each of 40 minutes duration. How long would each period be, if the school has 8 periods a day, assuming the number of school hours to be the same?  45 mins

18) if x and vary inversely and x= 15 when y=6, find y when x=9.           10

19) if x and vary inversely and x= 18 when y=8, find x when y=16.             9

CHOOSE THE CORRECT OPTION:

1) If 14 kg of pulses cost Rs882, what is the cost of 22 kgs of pulses ?

a)Rs 1254 b) Rs1298 c) Rs 1342 d) Rs1386 

2) If 8 oranges cost Rs52, how many oranges can be bought for Rs 169

a) 13 b) 18 c) 26 d) 24 

3) A machine fills 420 bottles in 3 hours. How many bottles will it fill in 5 hours .

a) 252 b) 700 c) 504 d) 300

4) A car is traveling at a uniform speed of 75 kmph . How much distance will it cover in 20 minutes.

a) 25km b) 15km c) 30km d) 20 km

5) The weight of 12 sheets of a thick paper is 40 grams. How many sheets would weigh 1 kg?

a) 480 b) 360 c) 300  d) none

6) A pole 14 m high casts shadow of 10m. At the same time, what will be the height of a tree, the length of whose shadow is 7m ?

a) 20m b) 9.8m c) 5m d) none

7) A photographer of a bacteria enlarged 50000 times attens a length of 5 cm. The actual length of bacteria is 

a) 1000 CM b) 1/10³ cm c) 1/10⁴ cm d) 1/10² cm

8) 6 pipes fill a tank in 120 minutes , then 5 pipes will fill it in 

a) 100 minutes b) 144 minutes c) 140mins d) 108 mins

9) 3 persons can build a wall in 4 days, then 4 persons can build it in 

a) 16/3 days  b) 3 days c) 13/3 days d) none

10) A car takes 2 hours to reach a destination by travelling at 60 kmph. How long will it take while travelling at 80 kmph ?

a) 1 hr 30min b) 1 hr 40min c) 2 hr 40min d) none

11) 6 pipes can fill a tank in 24 minutes. One pipe can fill it in.

a) 4 minutes b) 30 minutes c) 72 minutes d) 144 minutes 

12) 14 workers can build a wall in 42 days. One worker can build it in

a) 3 days b) 147 days c) 294 days d) 588 days 

13) 35 men can reap a field in 8 days. In how many days can 20 men reap it?

a) 14 days b) 28 days c) 175/2 days d) none

14) A car is traveling at an average speed of 60 kmph . How much distance will it cover in 1 hour 12 minutes ?

a) 50 km b) 72km c) 63 km d) 67.2 km 

15) Rashmi types 510 words and half an hour. How many words would she type in 10 minutes.

a) 85 b) 150 c) 170 d) 153 

16) x and y vary directly. when x =3 then y= 36 . What will be the value of x when y= 96.   a) 18 b) 12 c) 8 d) 4

17) x and y vary inversely. When x= 15, then y= 6. What will be the value of y when x=9?

a) 10 b) 15 c) 54 d) 135

FILL IN THE BLANKS:

1) If 3 persons can do a piece of work in 4 days, then 4 persons can do it in ___ days.

2)  if 5 pipes can fill a tank in 144 minutes, then 6 pipes can fill it in___ minutes.

3) a car covers a certain distance in one hour 30 minutes at 60 kmph. If it moves at 45 kmph , it will take____ hours.

4)  if 8 oranges cost Rs20.80, the cost of 5 orange is Rs------ 

5) The weight of 12 sheets of a paper is 50 grams. How many sheets weigh 500 grams

) A car covers 432 km 36 litres of petrol, how much would it cover in 25 litres of petrol?      300 km

) If the weight of 12 sheets of thick paper is 54 GM's. How many sheets of the same paper would weight 5/2 kg.      555.5 GM's

) The cost of 5 metre particular quality of cloth is Rs200. Find the cost of 2, 4, 10 and 12 m of cloth .     

) A vertical pole 5m 60 cm high casts shadow of 3 m 20 cm long. Find at the same

a) the length of the shadow cast by another pole 10 m 50 cm high.m.     600m

b) height of a pole that casts a shadow 5m long.      875m

) The scale of a map is given as 1: 30000000. Two cities are 4 cm apart on the map. Find the actual distance between them .      1200 km

) A private taxi charges of fare of Rs960 journey for a journey of 160 km. How much would it travel for Rs600.         100 km

) if 100 dollars can be had in exchange of Rs6600. How many dollars can one get in exchange for Rs3557.40.            Rs53.90

) The diameter of a cell in a photograph enlarged 10000 times in 10 cm. What is the actual diameter of the cell ? If the photograph is enlarged 2000 times only, what would be the enlarged diameter of the cell ?    2 cm

) If 3 men or 4 women can earn Rs900 per day. How much would 6 men and 6 women earn per day.          3150

) A bus travels 15 km in 30 minutes. If the speed remains the same, how far will it travel in 6 hours.    180km

) A train is moving at a uniform speed of 75 kmph.

a) How far will it travel in 20 minutes.      25km

b) Find the time required to cover a distance of 250 km ?      3hrs 20mins

) A private taxi charges a fare of Rs1320 for a journey of 160 km. How much will it travel for Rs1031.25 ?        125km

) If 9 kg of sugar costs Rs166.50, how much sugar can be bought for Rs259?    14kg

) A machine in a soft factory fills 840 bottles in 6 hours. How many bottles will it fill in 5 hours ?       700 bottles

) A taxi charges fare of Rs1275 for a journey of 150 km. How much would it charge for a journey of 124 km ?      Rs1054

) If the cost of 8 kg of potatoes is Rs88, find 

a) the weight of the potatoes bought for Rs60.50.         5.5 kg

b) the cost of 17 kg of potatoes.           Rs187

) Suppose 2kg of sugar contains 9x10⁶ crystals . How many sugar crystals are there in 5 kg of sugar?   2.25 x 10⁷

) If 2/3 piece of work iron has a mass of 6kg, then what would be the mass of 5/6 of the iron piece?    7.5 kg

) On a birthday, on an average, 5 bottles of coke are served for a group of 6 children. How many friends were present at Mishra's party if bottles of coke were used?     18

) The monthly earning and monthly savings of Bhupendra are in the ratio of 4:1. If Bhupendra's monthly income is Rs75000, how much is his savings ?  Rs18750

) A loaded truck travels 14 km 25 minutes. If the speed remains the same, how far can it travel in 5 hours.      168km

) A tractor can plough 12 hectares in 4 hours. Moving at the same speed, how much time does it require to plough 13.5 hectares.    4hrs 30mins

) If 11 men can dig 27/4 metres long trench in one day. How many men would be employed for digging 27 metres long trench of the same type in one day?   44 men

) A car covers a distance of 175.5 km in 13 litres of petrol. How much distance will it cover in 20 litres of petrol ?    270km

) In 8 days, the earth picks up 6.4 x10⁹ kg of dust from the atmosphere. How much dust will it pick up in 15 days.    12 x 10⁹

)6 pipes are required to fill a tank in 1 hour 20 minutes. How long will it take if only 5 pipes of the same types are used ?     96 mins

) if 40 workers can do a job in 15 hours. How many workers will be required to do the same job in 8 hours?      75

) In a fort, 300 men had provisions for food for 90 days. After 20 days, 50 men left the fort. How long would the food last at the same rate?          84days

) A school has 9 periods in a day each of 40 minutes duration. How long would each person be, if the school has 6 periods a day, assuming that the number of hours to be the same.    60mins

) There are 90 students in a hostel. Food provision for them is for 20 days. How long will their food provision last if 30 more students join the group ?    15days

) Nishtha has enough money to send 12 parcels each weighing 500 gms. Find the weight of each parcel if she can send 30 parcels for the same money.    200gms

) If 45 men can do a piece of work in 49 days. In how many days will 35 men do it ?    63

) if 15 workers can build a wall in 48 hours , how many workers will be required to do the same work in 30 hours.    24

) A farmer has enough food to feed 20 animals in his cattle for 6 days. How long would the food last if there were 10 more animals in his cattle?    4

) A farmer has enough food to feed 28 horses for 9 days. How long would the food last, if there were 8 more horses.     7 days

) Amit cycles to his school at an average speed of 14 kmph . It takes him 30 minutes to reach the school in time. At what speed should he cycle, if he wants two minutes earlier.    15 kmph

) A garrison of 900 men had  provision for 42 days, however a reinforcement of 500 men arrived . For how many days will the food last  now?    27 days

) A train takes 3 hours to reach a destination by travelling at a speed of 90 kmph. How long will it take the train when it travels at the speed of 120 kmph.     9/4 hours

) A factory requires 42 machines to produce a given number of articles in 56 days. How many machines would be required to produce the same articles in 48 days.   49

) At constant temperature the pressure of a gas is inversely proportion to its volume. If  certain amount of a gas occupies a volume of 30 cc and exerts a pressure of 85 g/cm²,  then find the difference in its pressure when the same amount of gas occupies a volume of 75 cc.    51 g/cm²

) A sprinter jogs and covers 5 rounds of the race track taking 12 minutes per round . Find the he should take to cover each round if he wants to complete 6 rounds, assuming the time for which he jogs remains the same.    10 mins

) If x and y axis are inversely proportional and x= 15 when y=6 then find y when x= 9.   10

) If x and y axis are inversely and x= 40, find y when constant of variation= 8.   1/5

Mix

) 15 men can dig 20 m long trench in one day. How many men should be employed for digging 140 metres long trench of the same type in one day ?     Directly

) The shadow of the Qutub Minar, which is 72m high is 80m at a particular time on a day. Find the height of an electric pole that cast a shadow of 10m under similar condition.    9m

) If 721 men constructed bridge in 48 days then in how many days 1442 men can do this work?   24 days

) Ashish takes 20 minutes to reach his school at an average speed of 6 kmph. If he is required to reach school in 24 minutes, what should be his speed ?    5 kmph 

) In a military camp there is food for 30 days for 50 soldiers. Assuming that average meal of every soldiers is same. If 25 more soldiers join them, how many days this food will last ?     20 days 

) Mahesh goes 5 km with a speed of 10 kmph. If he double his speed, find the time to cover the same distance.     15 mins

) Amar weaves 35 seats of chairs in 7 days. How many days will he take to weave 146 seats of chairs?     28 days

) if a minute hand makes an angle 30° in 5 minutes. Find the angle covered between 7:10pm to 7:30 pm.      120°

) A train is running at a speed of 18 kmph . If it crosses a pole in 35 second. Find the length of the train.     175m

) A vegetablea vendor has Rs2000 to buy potatoes available at the rate of Rs8 per kg. If the price of potatoes increases by 25%, find how much potatoes he can purchase at the same amount.    200 kg

) If 14kg of pulses cost Rs441, what is the cost of 22kg of pulses?    Rs693

) A car takes tp2 hours to reach a destination by travelling at 16 kmph . How long will it take while travelling at 80 kmph.        1 hour 30 minutes