**Category : **8th Class

**Introduction**

Numbering is something that is sequential Follows a fixed order whereas ranking is a relationship between a set a items such that, for any two items, the first is either ‘ranked higher than’. ‘ranked lower than ‘or’ ranked equal to ‘the second.

**NUMBER TEST **

In this type of question, generally a set, group or series of numerals is given and the candidate is required to find out how many times a number satisfying the conditions specified in the question occurs.

**EXAMPLE 1: **

In the series given below, how many 8s are there each of which is exactly divisible by its immediate preceding as well succeeding numbers?

2 8 3 8 2 4 8 2 4 8 6 8 2 8 2 4 8 3 8 2 8 6

(a) One (b) Two

(c) Three (d) Four

**Sol.** (b) Clearly, we may mark such sets of 3 numbers, in which the middle number is 8 and each of the two numbers on both sides of it is a factor of 8, as shown.

2 8 3 8 2 4 8 2 4 8 6 8 2 8 2 4 8 3 8 2 8 6 so there are two such 8s.

**EXAMPLE 2:**

How many 7s are there in the following number series which are preceded by 9 and also followed by 6?

7 8 9 7 6 5 3 4 2 8 9 7 2 4 5 9 2 9 7 6 4 7

(a) Two (b) Three

(c) Four (d) Five

**Sol.** (b) The 7s satisfying the given conditions are shown below.

7 8 9 7 6 5 3 4 2 8 9 7 2 4 5 9 2 9 7 6 4 7

Hence, there are only two such 7.

**RANKING TEST **

Generally, the number of persons are arranged in either ascending or descending order of their performance in a certain activity.

Let there be n persons who qualified in a certain event. A particular man Rohan is one of them whose rank from the top i.e, top rank \[{{T}_{r}}\]. is 14 th and his rank from the other side i.e, from the bottom \[({{B}_{r}})\] is 26th. Then clearly the number of persons who qualify is \[{{T}_{r}}+{{B}_{r}}-1\]. i.e, \[14+26-1=39.\]

\[\Rightarrow \] \[n={{T}_{r}}+{{B}_{r}}-1\] .......(a)

\[{{T}_{r}}=(n+1)-{{B}_{r}}\] ........(b)

and \[{{B}_{r}}=(n+1)-{{T}_{r}}\] ........(c)

these rules (a), (ii) and (iii) are very useful in problems on ranks

**EXAMPLE 3:**

Sunita ranked 11 th from txhe top and 27th from the bottom in a class. How may student are in the class?

(a) 38 (b) 28

(c) 40 (d) 37

**Sol.** (d) \[{{T}_{r}}=11,{{B}_{r}}=27.\].

Number of students in the class \[=11+27-1=37.\]

**EXAMPLE 4:**

In a row of boys, Suresh is seventh from the left and Rohit is twelfth from the right. If they interchange their positions, Suresh becomes twenty- second from the left. How many boys are there in a row?

(a) 19 (b) 31

(c) 33 (d) Cannot be found

**Sol.** (c) Suresh's new position is 22nd from left. But it is the same as Rohit’s earlier position which is 12th from right.

\[\Rightarrow \] The number of persons in a row.

\[=(22+12-1)=33\]

**EXAMPLE 5:**

In a queue, Vijay is fourteenth from the front and jack is seventeenth from the end, while Mary is in between Vijay and Jack. If Vijay be ahead of Jack and there be 48 persons in the queue, How may persons are there between Vijay and Mary?

(a) 8 (b) 7

(c) 6 (d) 5

**Sol.** (a) Number of persons between Vijay and Jack

\[=48-(14+17)=17\]

Now, Mary lies in the middle of these 17 persons i.e, at the ninth position.

\[\therefore \]Number of persons between Vijay and Mary = 8

**USEFUL TIPS**

- Position of person from top

= [Total no. of persons -position of person from down] +1.

- Position of person from bottom

= [Total no. of persons - position of person from up]+l.

- Position of person from right

= [Total no. of persons - position of person from left]+l

- Position of person from left

= [Total no. of persons - position of person from right] + 1.

- Total no. of persons

= [Position of person from top/ right + position of person from bottom/left]-!.

- If two persons are on a definite position from up and down or left and right) and they interchange their ranks, then

(a) Total no. of persons in order

= [First position of first person + second position of second person ] - 1

(b) Second position of first person or second position of second person = Difference of the two positions of second person + first position of second person.

= Difference of the two positions of first person + First position of second person.

**Miscellaneous Solved Examples**

** **

**EXAMPLE 1:**

In a row of 40 girls, when Komal was shifted to her left by 4 places her number from the left end of the row became 10.

What was the number of Swati from the right end of the row if Swati was three places to the right of komal's original position?

(a) 22 (b) 23

(c) 25 (d) 24

**Sol.** (d) On shifting 4 to the left, Komal is 1 Oth from the left end of the row. Thus, komal's original position was 14^{th} from the left end.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17..............

Swati is 3 places to the right of Komal’s original position.

Clearly, swati is 17 th from the left end.

Number of girls to the right of swati \[=(40-17)=23.\].

Thus, Swati is 24th from the right end of the row.

**EXAMPLE 2:**

In the first and second digits in the sequence 5 9 8 1 3 2 7 4 3 8 are interchanged. Also the third and fourth digits, the fifth and sixth digits and so on, which digit would be the seventh counting to your left?

(a) 1 (b) 4

(c) 7 (d) 8

**Sol. **(d) The new sequence becomes 9 5 1 8 2 3 4 7 8 3.

Counting to the left, the seventh number is 8.

**EXAMPLE 3:**

Raman is 7 ranks ahead of Suman in a class of 39. If Suman's rank is seventeenth from the last, what is Raman's rank from the start?

(a) 14th (b) 15th

(c) 16th (d) 17th

**Sol.** (c) Suman is 17th from the last and Raman is 7 ranks ahead of Suman. So Raman is 24 th from the last.

\[\therefore \]Raman rank from the start is \[39+1-24\] i.e, 16 th

**EXAMPLE 4:**

Mohan and Ramesh are ranked seventh and eleventh respectively from the top in a class of 41 students. What will be their respective ranks from the bottom in the class

(a) 30th and 34th (b) 34th and 30th

(c) 35th and 31st (d) 36th and 32nd

**Sol.** (c) Rank of Mohan from the bottom \[=(41+1)-7=35\]th.

Rank of Ramesh from the bottom \[=(41+1)-11=31\] st.

**EXAMPLE 5:**

If you count 21 letters in the english alphabet from the end and 20 letters from the beginning, which letter will exactly appear in the middle of the sequence thus formed?

(a) N (b) L

(c) K (d) M

**Sol.** (d) By counting 21 letters from the end and 20 letters from the beginning we get the following sequence

F G H I J K L M N O P Q R S T

Obviously, the letter M appears exactly in the middle of the sequence thus formed.

**EXAMPLE 6:**

Ravi is 7 ranks ahead of Sumit in a class of 39. If Sumit's rank is seventeenth from the last, what is Ravi's rank from the start?

(a) 14th (b) 15th

(c) 16th (d) 17th

**Sol.** (c) Sumit is 17th from the last and Ravi is 7 rank ahead of Sumit, so Ravi is 24th from the last. Thus Ravi's rank from the start: \[(39-24+1)=16\]

** **

**EXAMPLE 7:**

The average of three numbers is 24.

The average of two of these three numbers is \[22\frac{1}{2}\].

What is the value of the third number?

**Sol.** (27) The total of three numbers is \[72(24\times 3)\]. The total of two numbers is \[45(22{\scriptstyle{}^{1}/{}_{2}}\times 2).\].

The third number must be \[27(72-45).\].

**EXAMPLE 8:**

How many numbers from 1 to 100 are there each of which is not only exactly divisible by 4 but also has 4 as a digit'

(a) 7 (b) 10

(c) 20 (d) 21

**Sol.** (a) The numbers from 1 to 100 which are exactly divisible by 4 are

4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100.

But each number should have 4 as its digit.

The required numbers are 4, 24, 40, 44, 48, 64, 84.

Clearly, there are 7 such numbers.

** **

**EXAMPLE 9:**

Manish is fourteenth from the right end in a row of 40 b( What is his position from the left end ?

(a) 24th (b) 25th

(c) 26th (d) 27th

**Sol.** (d) Clearly, number of boys towards the left of Man is

\[=(40-14)=26.\].

So, Manish is 27th from the left end.

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