RS Aggarwal Solutions Class 10 Ex 9D

Q1: Find the median of the following data by making a ‘less than give’

Marks 0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80 80-90 90-100
No. of students 5 3 4 3 3 4 7 9 7 8

Sol:

The frequency distribution table of less than type is given as follows:

Maks (upper class limits) Cumulative frequency (cf)
Less than 10 5
Less than 20 5+3=8
Less than 30 8+4=12
Less than 40 12+3=15
Less than 50 15+3 = 18
Less than 60 18+4 = 22
Less than 70 22+7 =29
Less than 80 29+9 =38
Less than 90 38+7 =45
Less than 100 45+8 =53

Taking upper class limits of class intervals on x-axis and their respective frequencies on y-axis, its ogive can be drawn as follows:

https://lh4.googleusercontent.com/uvhuxFTA8Uam2iTwEYD2aBNDTXik7ctnMJtmusjNnLLA3CW2uHrTUs7Pn_Lz3coJM0wgOjJS6cVANhg_OWBIre0Iy4olav9NeUw55mUNzIszAawZ_ZHz09YMqI0lVPCYuXSLkb4q

Here, N=53,

\(\Rightarrow \frac{N}{2}= 26.5\)

Mark the point A whose ordinate is 26.5 and its x-coordinate is 66.4.

https://lh4.googleusercontent.com/oWhkcMkz4WgKtp7N0JVR7dliv2Xp2MdpvtpjMGkU9-J2er63COY-mOg_9RRZTXJb3_3_gMqPKUsQwwrvVY87jUh2lhoxLnWpyNXJ3lHi47oJKSjUail5G6ThvHzByvnjn-VkCz9M

The median of the data is 66.4.

Q2: The given distribution shows the number of wickets taken by the bowlers in one-day international cricket matches:

No. of wickets Less than 15 Less than 30 Less than 45 Less than 60 Less than 75 Less than 90 Less than 105 Less than 120
No. of bowlers 2 5 9 17 39 54 70 80

Sol:

Taking upper class limits of class intervals on x-axis and their respective frequencies on y-axis, its ogive can be drawn as follows:

https://lh3.googleusercontent.com/QxALtNTwmAKMK0_PlQDjwO-0m7olOgPIQYYcWyzyY0gd1BKDibX4OhSLarL1Q2PIt3dnnEDChb3jhrOFFCjpI0gxMnf9rWpxxyZaBqcj1XeM-syi9WuDyJrC535JvZ7L2FT4Z6ij

Here, N=80,

\(\Rightarrow \frac{N}{2}= 40 \)

Mark the point A whose ordinate is 40 and its x-coordinate is 76.

https://lh3.googleusercontent.com/dorsEnW_If1tIR6Sw-LKl4kt0cYnoIi1cV6xyjW8HpmfwCfVnFiliEqd06Fm8oVxOC4LiCeo85zY20ikaEbYLxJ3utm0iQ1ipvQN28_AqhK-fJS9dQanmBTJsuTB69xvxguiNLIZ

The median of the data is 76.

Q3: Draw a ‘more than’ ogive for the data given below which gives the marks of 100 students.

Marks 0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80
No. of students 4 6 10 10 25 22 18 5

Sol:

Taking upper class limits of class intervals on x-axis and their respective frequencies on y-axis, its ogive can be drawn as follows:

Marks (lower class limits) Cumulative frequency (cf)
More than 0 96+4 = 100
More than 10 90+6 = 96
More than 20 80+10=90
More than 30 70+10=80
More than 40 45+25=70
More than 50 23+22=45
More than 60 18+5=23
More than 70 5

Taking the lower class limits on x-axis and their respective cumulative frequencies on y-axis, its ogive can be obtained as follows:

https://lh3.googleusercontent.com/KqqMRmTactgyf_pA6Rl9qJTdCoURPhRUcbdjIAAFPGI57OmcmjQDFndyJrpSsSS1ZnaIgl1av-rdlqNDnprQ9cJWJ-1zKZ_cBUyLrhxeAgWrX-t1LXxHGDQSHwXKB4EFT0mzjmJu

Q4: The height of 50 girls of class X of a school are recorded as follows:

Height ( in cm) 135-140 140-145 145-150 150-155 155-160 160-165
No. of girls 5 8 9 12 14 2

Draw a ‘more than type’ ogive for the above data.

Sol:

The frequency distribution table of more than type is as follows:

Height (in cm) (lower class limits) Cumulative frequency (cf)
More than 135 45+5=50
More than 140 37+8=45
More than 145 28+9=37
More than 150 16+12=28
More than 155 14+2=16
More than 160 2

Taking the lower class limits on x-axis and their respective cumulative frequencies on y-axis, its ogive can be obtained as follows:

https://lh4.googleusercontent.com/1Fk1RW_Ey5iFKqgXzIEjllmCLCWdtoBkSg9Gs9G59FMO0KPLlfBplB5v6Of_-dcaZT76tEZHRZj60-uLhB_Zkbk8rxPPHh4zoldL_ZaDAJDP-r_X3IDimkmUo31uW8umuvPOMp6D

Q5: The monthly consumption of electricity ( in units) of some families of a locality is given in the following frequency distribution:

Monthly consumption (in units) 140-160 160-180 180-200 200-220 220-240 240-260 260-280
No. of families 3 8 15 40 50 30 10

Sol:

The frequency distribution table of more than type is as follows:

Height (in cm) (lower class limits) Cumulative frequency (cf)
More than 140 3+153=156
More than 160 8+145=153
More than 180 15+130=145
More than 200 40+90=130
More than 220 50+40=90
More than 240 30+10=40
More than 260 10

Taking the lower class limits on x-axis and their respective cumulative frequencies on y-axis, its ogive can be obtained as follows:

https://lh5.googleusercontent.com/ToE8jIUgalG-myLLF1nR-tSwx7DHGewL_7MwfGpYwSJMzF9ZU4EUJWSKrdChRr8FRfxJgHcjDrhydda6RGZzyJIborHhYGfpaHJT9R7n9kwzBkbOMNYdcNPY1d1mtI3lfAesY-6P

Q6: The following table gives the production yield per hectare of wheat of 100 farms of a village.

Production yield (kg/ha) 50-55 55-60 60-65 65-70 70-75 75-80
No. of farms 2 8 12 24 38 16

Change the distribution to ‘a more than type’ distribution and draw its ogive. Using ogive, find the median of the given data.

Sol:

The frequency distribution table of more than type is as follows:

Production yield (kg/ha)(lower class limits) Cumulative frequency (cf)
More than 50 2+98=100
More than 55 8+90=98
More than 60 12+78=90
More than 65 24+54=78
More than 70 38+16=54
More than 75 16

Taking the lower class limits on x-axis and their respective cumulative frequencies on y-axis, its ogive can be obtained as follows:

https://lh4.googleusercontent.com/En6i9Ts57j3o3PrHml6XvIjytb-4Uu3hEW0TJaBkL69pCK5pFDcxyNoiXT1LmSxZ3BMa-zdlR7JOFiCljK5shVHD40tveaP4V66aOQ4Gqbq8h7BLuX3N6pqvu3smO1qJjB0ImeOd

Here, N=100,

\(\Rightarrow \frac{N}{2}= 50 \)

Mark the point A whose ordinate is 50 and its x-coordinate is 70.5.

https://lh5.googleusercontent.com/HYlsw9CZbJhJVDlS_8inebSBooHelg5Jpmt4YwjU6wjIzu-jXxtYMIxRk6UKTQO245DhwJPMJUa66xva2pLqMz1JpnKzEmXKukSNZeUtAFu8DU7VdTyDIghBd2VlUGiGcsEUF9TQ

The median of the data is 70.5

Q7: The table below shows the weekly expenditures on food of some household in a locality:

Weekly introduction (in Rs.) No. of household
100-200 5
200-300 6
300-400 11
400-500 13
500-600 5
600-700 4
700-800 3
800-900 2

Draw a ‘less than type ogive’ and a ‘more than type ogive’ for this distribution.

Sol:

The frequency distribution table of less than type is as follows:

Weekly expenditure ( in Rs)

(Upper class limits)

Cumulative frequency (cf)
Less than 200 5
Less than 300 5+6=11
Less than 400 11+11=22
Less than 500 22+13=35
Less than 600 35+5=40
Less than 700 40+4=44
Less than 800 44+3=47
Less than 900 47+2=49

Taking the lower class limits on x-axis and their respective cumulative frequencies on y-axis, its ogive can be obtained as follows:

https://lh4.googleusercontent.com/L2AzDrhK0VUf4WQJC5nj4qSZ2vx8qzN_063SgqpEr5tE-wJLnD3iehLscDJJtkeu3BzIeVk8y8-Hphitmx6xew1FLxW_QU2IF6usLPisMEmXKNUdLvTWh9_zAqXHAAwzKfECaByb

Now,

The frequency distribution table of more than type is as follows:

Weekly expenditure (in Rs) ( lower class limits) Cumulative frequencies (cf)
More than 100 44+5=49
More than 200 38+6=44
More than 300 27+11=38
More than 400 14+13=27
More than 500 9+5=14
More than 600 5+4=9
More than 700 2+3=5
More than 800 2

Taking the lower class limits on x-axis and their respective cumulative frequencies on y-axis, its ogive can be obtained as follows:

https://lh5.googleusercontent.com/-B_X2dJaHl3krbJPwAK3lblzrH9Q0DYONYNCVbc6exvthmWONtSHiGbQFnYXKZ3acQv2maqYNan47MMmve6kH3eTSS3ihjMkZDs2u6Lxwn-hwWLNpqAVab_3PttR8xz7oxdLVaxq

Q8: From the following frequency distribution , prepare the ‘more than ogive’.

Score No. of candidates
400-450 20
450-500 35
500-550 40
550-600 32
600-650 24
650-700 27
700-750 18
750-800 34
Total 230

Also find the median.

Sol:

From the given table, we may prepare the ‘more than’ frequency table as shown below:

Score Number of candidates
More than 750 34
More than 700 52
More than 650 79
More than 600 103
More than 550 135
More than 500 175
More than 450 210
More than 400 230

We plot the points A(750,34), B(700,50), C(650,79), D(600,103), E(550,135), F(500,175), G(450,210) and H(400,230).

Join AB,BC,CD,DE,EF,FG,GH and HA with a free hand to get the curve representing the ‘more than types’ series.

https://lh3.googleusercontent.com/7lymDoYTKVcM1uaBNKzSin8t4e35HtVwMYQMhTJ3f4w1WXBlJ4bpuYB-CpFxMU1zHm22H3wn5bHxLNQkdrgsuzZjieD1Qge_2JsCr_S2Ojjpp3HvYo1slFNKwsEJt_65EUzwy5Yj

Here N= 230

\(\Rightarrow \frac{N}{2}= 115\)

From P(0,115), draw PQ meeting the curve at Q.

Draw QM meeting at M.

Clearly, QM = 590 units

Hence, median = 590 units

Q9: The marks obtained by 100 students of a class in an examination are given below:

Marks No. of students
0-5 2
5-10 5
10-15 6
15-20 8
20-25 10
25-30 25
30-35 20
35-40 18
40-45 4
45-50 2

Draw cumulative frequency curves by using

(i) ‘Less than’ series and

(ii) ‘More than’ series

Hence find the median.

Sol:

(i) From the given table, we may prepare the ‘less than’ frequency table as shown below:

Marks No. of students
Less than 5 2
Less than 10 7
Less than 15 13
Less than 20 21
Less than 25 31
Less than 30 56
Less than 35 76
Less than 40 94
Less than 45 98
Less than 50 100

We plot the points A(5,2), B(10,7), C(15,13), D(20,21), E(25,31), F(30,56), G(35,76), H(40,94), I(45,98), and J(50,100).

Join AB,BC,CD,DE,EF,FG,GH,HI,IJ and JA with a free hand to get the curve representing the ‘less than types’ series.

(ii) More than series:

Marks No. of students
More than 0 100
More than 5 98
More than 10 93
More than 15 87
More than 20 79
More than 25 69
More than 30 44
More than 35 24
More than 40 6
More than 45 2

Now on the same graph paper, we plot the points (0,100), (5,98), (10,94), (15,76), (20,56), (25,31), (30,21), (35,13), (40,6) and (45,2).

Join, with a free hand to get the ‘more than type’ series.

https://lh5.googleusercontent.com/aEVySFarYsfCuIcORXHG6uGWHt8MPDve4aSTAURQO_UbVCwDeMxCM05y8Gp7qe1qLsOu2WSj2WQUtzmLj0hOWVWdIpBli56WhcRaWpZky89AwyJwUN-f7rteyWt9FWstFcoMsi8M https://lh4.googleusercontent.com/7U-awFKHJVzEItsJ5JUgjYzK_5tbA_r6-KCIZRfvZjQezMnJ60XpC81QNRZUpmbR_qenXUN1oGQAZqDnQ32v0EAq_8ANvlQrrUu3BrSURiBb_g_ToA16A4keHLHDBjKjxO0qsn5w

The two curves intersect at point L. Draw \(LM \perp OX\) cutting the x-axis at M. Clearly, M= 29.5.

Hence, Median = 29.5

Q10: From the following data, draw two types of cumulative frequency curves and determine the median

Height ( in cm) Frequency
140-144 3
144-148 9
148-152 24
152-156 31
156-160 42
160-164 64
164-168 75
168-172 82
172-176 86
176-180 34

Sol:

(i) Less than series:

Marks No. of students
Less than 144 3
Less than 148 12
Less than 152 36
Less than 156 67
Less than 160 109
Less than 164 173
Less than 168 248
Less than 172 330
Less than 176 416
Less than 180 450

We plot the points A(144,3), B(148,12), C(152,36), D(156,67), E(160,109), F(164,173), G(168,248), H(172,330), I(176,416), and J(180,450).

Join AB,BC,CD,DE,EF,FG,GH,HI,IJ and JA with a free hand to get the curve representing the ‘less than types’ series.

(ii) More than series:

Marks No. of students
More than 140 450
More than 144 447
More than 148 438
More than 152 414
More than 156 383
More than 160 341
More than 164 277
More than 168 202
More than 172 120
More than 176 34

Now on the same graph paper, we plot the points  \(A_{1}(140,4510) ,B_{1}(144,447), C_{1}(148,447), D_{1}(152,414), E_{1}(156,383), F_{1}(160,341),G_{1}(164,277), H_{1}(168,202),I_{1}(172,120) and J_{1}(176,34)\)

Join, \(A_{1}B_{1}, B_{1}C_{1}, C_{1}D_{1},D_{1}E_{1},E_{1}F_{1},F_{1}G_{1},G_{1}H_{1},H_{1}I_{1},I_{1}J_{1}\) with a free hand to get ‘more than type’ series.

https://lh6.googleusercontent.com/Yb4OLOFuU4ZGix9BheLcBrWrfAnQ98mpdH6_ZjL3TLv1haaRrFbMyGCuM1P8cEu131TCc74kqEwNMfgBXJKZONujJ9oTwqQnpOni1-iE5_uCrgtoSpZSD6Gc26q86fkeyr9BzjpJ https://lh4.googleusercontent.com/P-kyT7x2q8UvZsRHSrme0BGBYpe1_mUeDCAt_DldyRsgTrNd_dRFpBCr3F0xQI9hpaNeXBac6JZYhUCVEL3JJAgAh3kuud-Hgr0O2KKEnh0EzeXB2PCAg-W_U-cdXBgNWI-pHcEw

The two curves intersect at point L. Draw \(LM \perp OX\) cutting x-axis at M. Clearly, M= 166 cm.

Hence, Median = 166 cm.


Practise This Question

On combining two aqueous solutions of soluble ionic compounds, a precipitate is formed when