SELINA Solution Class 9 Chapter 19 Means and median (For Unground Data Only) Exercise 19B

Question 1.1

Find the median of: 25, 16, 26, 16, 32, 31, 19, 28 and 35

Sol:

 Firstly arrange the numbers in ascending order
16, 16, 19, 25, 26, 28, 31, 32, 35

Now since
n = 9( odd )

Therefore the Median

= ${\displaystyle {\left(\frac{n+1}{2}\right)}^{\text{th}}}$  

= ${\displaystyle {\left(\frac{9+1}{2}\right)}^{\text{th}}}$  

= 5^th

Thus the median is 26.

Question 1.2

Find the median of:
241, 243, 347, 350, 327, 299, 261, 292, 271, 258 and 257

Sol:

Firstly arrange the numbers in ascending order

241, 243, 257, 258, 261, 271, 292, 299, 327, 347, 350

Now since n = 11 ( Odd )

Median = Value of ${\displaystyle {\left(\frac{n+1}{2}\right)}^{\text{th}}}$ term

             = 6^th term
             = 271
Thus the median is 271.

Question 1.3

Find the median of:

63, 17, 50, 9, 25, 43, 21, 50, 14 and 34

Sol:

 Firstly arrange the numbers in ascending order

9, 14. 17, 21, 25, 34, 43, 50, 50, 63

Now since n = 10( even )

Median =${\displaystyle \frac{1}{2}[\text{value of}{\left(\frac{n}{2}\right)}^{\text{th}}\text{term}+\text{value of} {(\frac{n}{2}+1)}^{\text{th}}\text{term}]}$

=${\displaystyle \frac{1}{2}[\text{value of}{\left(\frac{10}{2}\right)}^{\text{th}}\text{term}+\text{value of} {(\frac{10}{2} +1)}^{\text{th}}\text{term}]}$

= ${\displaystyle \frac{1}{2}[25+34]}$

= ${\displaystyle \frac{1}{2}\left[59\right]}$

= 29.5

Thus the median is 29.5.

Question 1.4

Find the median of:

 233, 173, 189, 208, 194, 204, 194, 185, 200 and 220.

Sol:

 Firstly arrange the numbers in ascending order

173, 185, 189, 194, 194, 200, 204, 208, 220, 223

Median =${\displaystyle \frac{1}{2}[\text{value of}{\left(\frac{n}{2}\right)}^{\text{th}}\text{term}+\text{value of} {(\frac{n}{2}+1)}^{\text{th}}\text{term}]}$

=${\displaystyle \frac{1}{2}[\text{value of}{\left(\frac{10}{2}\right)}^{\text{th}}\text{term}+\text{value of} {(\frac{10}{2} +1)}^{\text{th}}\text{term}]}$

= ${\displaystyle \frac{1}{2}[200+194]}$

= ${\displaystyle }$${\displaystyle \frac{1}{2}\left[394\right]}$

= 197

Thus the mention is 197.

Question 2

The following data have been arranged in ascending order. If their median is 63, find the value of x.
34, 37, 53, 55, x, x + 2, 77, 83, 89 and 100.

Sol:

Given numbers are 34, 37, 53, 55, x, x+2, 77, 83, 89, 100

Here n = 10(even)

Median = ${\displaystyle \frac{1}{2}[\text{value of}{\left(\frac{n}{2}\right)}^{\text{th}}\text{term}+\text{value of} {(\frac{n}{2}+1)}^{\text{th}}\text{term}]}$

= ${\displaystyle \frac{1}{2}[\text{value of}{\left(\frac{10}{2}\right)}^{\text{th}}\text{term}+\text{value of} {(\frac{10}{2}+1)}^{\text{th}}\text{term}]}$

= ${\displaystyle \frac{1}{2}}$ [ value of ( 5 )^th term + value of ( 5 + 1)^th term ]

= ${\displaystyle \frac{1}{2}}$ [ value of ( 5 )^th term + value of (6)^th term ]

63 = ${\displaystyle \frac{1}{2}}$ [  x + x + 2 ]

⇒ ${\displaystyle \frac{2+2x}{2}}$ = 63

⇒  x + 1 = 63

⇒ x = 62

Question 3

In 10 numbers, arranged in increasing order, the 7^th number is increased by 8, how much will the median be changed?

Sol:

For any given set of data, the median is the value of its middle term.

Here, total observations = n = 10 (even)

If n is even, we have

Median =${\displaystyle \frac{1}{2}[\text{value of} {\left(\frac{n}{2}\right)}^{\text{th}}\text{term}+\text{value of} {(\frac{n}{2}+1)}^{\text{th}}\text{term}]}$

Thus, for n = 10, we have

Median =${\displaystyle \frac{1}{2}[\text{value of}{\left(\frac{10}{2}\right)}^{\text{th}}\text{term}+\text{value of} {(\frac{10}{2}+1)}^{\text{th}}\text{term}]}$

=${\displaystyle \frac{1}{2}[\text{value of}{5}^{\text{th}}\text{term}+\text{value of}{6}^{\text{th}}\text{term}]}$

Hence, if 7^th number is diminished by 8, there is no change in the median value.

Question 4

Out of 10 students, who appeared in a test, three secured less than 30 marks and 3 secured more than 75 marks. The marks secured by the remaining 4 students are 35, 48, 66 and 40. Find the median score of the whole group.

Sol:

Here, total observations = n = 10 (even)
Thus, we have

Median =${\displaystyle \frac{1}{2}[\text{value of}{\left(\frac{10}{2}\right)}^{\text{th}}\text{term}+\text{value of}{(\frac{10}{2}+1)}^{\text{th}}\text{term}]}$

=${\displaystyle \frac{1}{2}[\text{valueof}{5}^{\text{th}}\text{term}+\text{value of}{6}^{\text{th}}\text{term}]}$

According to the given information, data in ascending order is as follows:

	1 st Term	2nd Term	3rd Term	4th Term	5th term	6th Term	7th Term	8th Term	9th term	10th term
Marks	Less than 30			35	40	48	66	More than 75

∴ Median =${\displaystyle \frac{1}{2}(40+48)=\frac{88}{2}=44}$

Hence, the median score of the whole group is 44.

Question 5

The median of observations 10, 11, 13, 17, x + 5, 20, 22, 24 and 53 (arranged in ascending order) is 18; find the value of x.

Sol:

Total number of observations = 9(odd)
Now, if n = odd
Median = ${\displaystyle {\left(\frac{n+1}{2}\right)}^{\text{th}} \text{term}}$

⇒ Median = ${\displaystyle {\left(\frac{9+1}{2}\right)}^{\text{th}}\text{term}}$ = 5^th term = x + 5

Now, Median = 18    ...(given)
∴ x + 5 = 18
⇒ x = 13.

S.chand class 6 Mathematics Chapter 19 Exercise 19B

 Exercise 19B

Question 1

1. Draw a line segment $P Q=10 \mathrm{~cm}$. Take a point $A$ on $P Q$ such that $P A=5.5 \mathrm{~cm}$. Draw a ray perpendicular to $P Q$ at $A$ by

(i) using ruler and compasses. (ii) using ruler and set-squares.

2. Draw any line segment $A B$ and take a point $P$ on it. Using ruler and set squares, construct a perpendicular $P Q$ on it. Verify by using a protractor whether $\angle A P Q=90^{\circ}$.

3. Draw a line $X Y$ and take a point $O$ not lying on it. Using set squares, construct a perpendicular from $O$ to the line $X Y$.

Use ruler and compasses only for the following constructions.

4. Draw a line segment $M N=7.5 \mathrm{~cm}$. Mark a point $P$ on $M N$ such that $M P=5.4 \mathrm{~cm}$. Draw a ray perpendicular to $M N$ at $P$.

5. Draw a line segment $X Y$ of length $6.5 \mathrm{~cm}$. At each end of the line segment $X Y$, draw a line perpendicular to the line $X Y$. Are these two lines parallel?

6. Draw a line segment $O A=8 \mathrm{~cm}$. Using set squares, construct $\angle A O B=90^{\circ}$, such that $O B=10 \mathrm{~cm}$. Join $A$ and $B$. Measure the length of $A B$.

RS Aggarwal solution class 8 chapter 19 Three Dimensional Figure Exercise 19B

Exercise 19B

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Q1 | Ex-19B | Class 8 | RS AGGARWAL | chapter 19 | Three Dimensional Figure

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Question 1:

Define Euler's relation between the number of faces, number of edges and number of vertices for various 3-dimensional figures.

Answer 1:

The Euler's relation for a three dimensional figure can be expressed in the following manner:

Q2 | Ex-19B | Class 8 | RS AGGARWAL | chapter 19 | Three Dimensional Figure

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Question 2:

How many edges are there in a
(i) cuboid
(ii) tetrahedron
(iii) triangular prism
(iv) square pyramid?

Answer 2:

(i) A cuboid has 12 edges, namely AD, DC, CB, BA, EA, FB, HD, DC, CG, GH, HE, and GF.

(ii) A tetrahedron has 6 edges, namely KL, LM, MN, NL , KM and KN.

(iii) A triangular prism has 9 edges, namely QR, RS, SQ, TU, UV, VT, RU, SV and QT.

(iv) A square pyramid has 8 edges, namely OW, OX, OY , OZ , WX, XY, YZ and ZW.

Q3 | Ex-19B | Class 8 | RS AGGARWAL | chapter 19 | Three Dimensional Figure

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Question 3:

How many faces are there in a
(i) cube
(ii) pentagonal
(iii) tetrahedron
(iv) pentagonal pyramid?

Answer 3:

(i) A cube has 6 faces, namely IJKL, MNOP, PLIM , OKJN, POKL and MNJI.

(ii) A pentagonal prism has 7 faces, i.e. 2 pentagons and 5 rectangles, namely ABCDE, FGHIJ, ABGF, AEJF , EDIJ, DCHI and CBGH.

(iii) A tetrahedron has 4 faces, namely KLM, KLN, LMN and KMN.

(iv) A pentagonal pyramid has 6 faces, i.e. 1 pentagon and 5 triangles, namely NOPQM, SNM, SOP, SNO, SMQ and  SQP.

Q4 | Ex-19B | Class 8 | RS AGGARWAL | chapter 19 | Three Dimensional Figure

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Question 4:

How many vertices are there in a
(i) cuboid
(ii) tetrahedron
(iii) pentagonal prism
(iv) square pyramid?

Answer 4:

(i) A cuboid has 8 vertices, namely A, B, C, D, E, F, G and H.

(ii) A tetrahedron has 4 vertices, namely K, L, M and N.

(iii) A pentagonal prism has 10 vertices, namely A, B, C, D, E, F, G, H, I and J.

(iv) A square pyramid has 5 vertices, namely O, W, X, Y and Z.

Q5 | Ex-19B | Class 8 | RS AGGARWAL | chapter 19 | Three Dimensional Figure

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Question 5:

Verify Euler's relation for each of the following:
(i) A square
(ii) A tetrahedron
(iii) A triangular prism
(iv) A square pyramid

Answer 5:

Euler's relation is:



(i) A square prism  

(There is an error in this question. It should have been a square prism rather than square.)                                        



(ii) A tetrahedron



(iii) A triangular prism



(iv) A square pyramid