Series completion Questions in English

(D) Step $1$: Add $1$ to the last digit of each number:
$567 \rightarrow 568$
$284 \rightarrow 285$
$696 \rightarrow 697$
$865 \rightarrow 866$
$738 \rightarrow 739$
Step $2$: Reverse the order of digits for each number:
$568 \rightarrow 865$
$285 \rightarrow 582$
$697 \rightarrow 796$
$866 \rightarrow 668$
$739 \rightarrow 937$
Step $3$: Arrange the resulting numbers in ascending order:
$582, 668, 796, 865, 937$
Step $4$: Identify the fourth number in the sequence:
The numbers are $1^{st}: 582, 2^{nd}: 668, 3^{rd}: 796, 4^{th}: 865, 5^{th}: 937$.
The fourth number is $865$.

(D) According to the question,after interchanging the first two digits of each number,the new set of numbers is:
$230 \rightarrow 320$
$422 \rightarrow 242$
$328 \rightarrow 238$
$516 \rightarrow 156$
$138 \rightarrow 318$
The new set is: $320, 242, 238, 156, 318$.
Arranging these numbers in ascending order:
$156, 238, 242, 318, 320$.
The number in the middle is $242$,which corresponds to the original number $422$.

(D) The original number is $5726489$.
Arranging the digits in ascending order:
Original: $5, 7, 2, 6, 4, 8, 9$
Sorted: $2, 4, 5, 6, 7, 8, 9$
Comparing the positions:
Original: $5, 7, 2, 6, 4, 8, 9$
Sorted: $2, 4, 5, 6, 7, 8, 9$
Comparing digit by digit:
$1^{st}$ position: $5 \neq 2$
$2^{nd}$ position: $7 \neq 4$
$3^{rd}$ position: $2 \neq 5$
$4^{th}$ position: $6 = 6$ (Same)
$5^{th}$ position: $4 \neq 7$
$6^{th}$ position: $8 = 8$ (Same)
$7^{th}$ position: $9 = 9$ (Same)
The digits that remain at the same position are $6, 8,$ and $9$.
Therefore,there are $3$ digits that remain at the same position.

(D) Given number: $9, 4, 7, 8, 6, 3, 5, 1, 2$
Arrange the digits in descending order: $9, 8, 7, 6, 5, 4, 3, 2, 1$
Now,compare the gaps between pairs in the original number and the descending sequence:
$1.$ Pair $(9, 7)$: In original number,there is $1$ digit $(4)$ between them. In descending order,there is $1$ digit $(8)$ between them. (Match)
$2.$ Pair $(8, 6)$: In original number,there is $1$ digit $(7)$ between them. In descending order,there is $1$ digit $(7)$ between them. (Match)
$3.$ Pair $(7, 5)$: In original number,there are $3$ digits $(8, 6, 3)$ between them. In descending order,there is $1$ digit $(6)$ between them. (No match)
$4.$ Pair $(1, 2)$: In original number,there are $0$ digits between them. In descending order,there are $0$ digits between them. (Match)
$5.$ Pair $(9, 8)$: In original number,there are $2$ digits $(4, 7)$ between them. In descending order,there are $0$ digits between them. (No match)
$6.$ Pair $(8, 5)$: In original number,there are $3$ digits ($6, 3, 5$ is not correct,let's re-evaluate).
Re-evaluating pairs:
- $(9, 7)$: $1$ digit between in both. (Yes)
- $(8, 6)$: $1$ digit between in both. (Yes)
- $(1, 2)$: $0$ digits between in both. (Yes)
- $(7, 5)$: In original,$8, 6, 3$ ($3$ digits). In descending,$6$ ($1$ digit). (No)
- $(9, 6)$: In original,$4, 7, 8$ ($3$ digits). In descending,$8, 7$ ($2$ digits). (No)
Correct pairs are $(9, 7), (8, 6), (1, 2)$. Total $3$ pairs. Since the options are None,One,Two,More than three,the answer is 'More than three' if we include others or 'Two' if only specific ones. Re-checking: $(9, 7), (8, 6), (1, 2)$ are $3$ pairs. The question asks for pairs with same number of digits between them. The answer is $3$ pairs.

(C) To find the new numbers,we interchange the first and the last digits of each given number:
$389 \rightarrow 983$
$476 \rightarrow 674$
$635 \rightarrow 536$
$847 \rightarrow 748$
$568 \rightarrow 865$
Comparing the new numbers: $536, 674, 748, 865, 983$.
The smallest number is $536$,which corresponds to the original number $635$.

(D) Step $1$: Interchange the first two digits of each number:
$389 \rightarrow 839$
$476 \rightarrow 746$
$635 \rightarrow 365$
$847 \rightarrow 487$
$568 \rightarrow 658$
Step $2$: Arrange these new numbers in ascending order:
$365, 487, 658, 746, 839$
Step $3$: Identify the second number in the sequence.
The second number is $487$,which corresponds to the original number $847$.

(A) The given number is $7326985$.
First,arrange the digits of the number in ascending order: $2, 3, 5, 6, 7, 8, 9$.
Now,compare the number of digits between pairs in the original number $(7326985)$ and the rearranged sequence $(2356789)$:
$1$. Pair $(7, 3)$: In original,$0$ digits between them. In rearranged,$0$ digits between them ($2, 3, 5, 6, 7, 8, 9$ - $3$ and $7$ have $5, 6$ between them,so this is not a match).
Let's check all pairs:
Original: $7, 3, 2, 6, 9, 8, 5$
Sorted: $2, 3, 5, 6, 7, 8, 9$
Pairs in original:
$(7, 3): 0$ digits between. In sorted: $3, 5, 6, 7$ ($3$ digits between).
$(7, 2): 1$ digit between. In sorted: $2, 3, 5, 6, 7$ ($3$ digits between).
$(3, 2): 0$ digits between. In sorted: $2, 3$ ($0$ digits between). Match found: $(3, 2)$.
$(6, 9): 0$ digits between. In sorted: $6, 7, 8, 9$ ($2$ digits between).
$(6, 8): 1$ digit between. In sorted: $6, 7, 8$ ($1$ digit between). Match found: $(6, 8)$.
$(9, 8): 0$ digits between. In sorted: $8, 9$ ($0$ digits between). Match found: $(9, 8)$.
$(8, 5): 1$ digit between. In sorted: $5, 6, 7, 8$ ($2$ digits between).
The matching pairs are $(3, 2), (6, 8), (9, 8)$.
There are $3$ such pairs.

(C) We need to find the number $4$ such that it is followed by $5$ (pattern $4, 5$) and not preceded by $3$ (pattern $x, 4, 5$ where $x \neq 3$).
Let's analyze the series: $5, 4, 4, 3, 3, 4, 5, 4, 5, 5, 3, 4, 4, 3, 4, 5, 4, 3, 3, 5, 4, 5, 4, 3, 5, 4, 4, 3, 5, 4$.
$1$. Look for $4, 5$ patterns:
- $5, 4, 5$ (Preceded by $5$,which is not $3$. Valid)
- $4, 5$ (Preceded by $4$,which is not $3$. Valid)
- $4, 5$ (Preceded by $4$,which is not $3$. Valid)
- $4, 5$ (Preceded by $3$. Invalid)
- $4, 5$ (Preceded by $4$,which is not $3$. Valid)
Wait,let's re-examine the sequence carefully:
$5, 4, 4, 3, 3, 4, 5, 4, 5, 5, 3, 4, 4, 3, 4, 5, 4, 3, 3, 5, 4, 5, 4, 3, 5, 4, 4, 3, 5, 4$
Pairs of $4, 5$:
- $...3, 4, 5...$ (Preceded by $3$,invalid)
- $...4, 5, 5...$ (Preceded by $4$,valid)
- $...3, 4, 5...$ (Preceded by $3$,invalid)
- $...4, 5, 4...$ (Preceded by $4$,valid)
There are $2$ such occurrences.

(D) Original number: $5, 2, 3, 6, 1, 9, 8, 4$
Ascending order: $1, 2, 3, 4, 5, 6, 8, 9$
Comparing the positions:
Original: $5, 2, 3, 6, 1, 9, 8, 4$
Sorted: $1, 2, 3, 4, 5, 6, 8, 9$
Positions:
$1^{st}$ digit: $5 \neq 1$
$2^{nd}$ digit: $2 = 2$ (Match)
$3^{rd}$ digit: $3 = 3$ (Match)
$4^{th}$ digit: $6 \neq 4$
$5^{th}$ digit: $1 \neq 5$
$6^{th}$ digit: $9 \neq 6$
$7^{th}$ digit: $8 = 8$ (Match)
$8^{th}$ digit: $4 \neq 9$
The digits $2, 3,$ and $8$ remain in the same position. Therefore,there are $3$ such digits.

(C) The given numbers are $519, 368, 437, 246, 175$.
The highest number among these is $519$,and its middle digit is $1$.
The lowest number among these is $175$,and its middle digit is $7$.
The sum of these middle digits is $1 + 7 = 8$.

(C) The given numbers are: $519, 368, 437, 246, 175$.
Arranging these numbers in descending order: $519 > 437 > 368 > 246 > 175$.
The highest number is $519$.
The second highest number is $437$.
The third digit of the second highest number $(437)$ is $7$.

(B) Original number: $7, 5, 2, 4, 6, 9, 3$
Positions: $1, 2, 3, 4, 5, 6, 7$
Arranging the digits in ascending order:
Sorted digits: $2, 3, 4, 5, 6, 7, 9$
Positions: $1, 2, 3, 4, 5, 6, 7$
Comparing the positions:
Original: $7, 5, 2, 4, 6, 9, 3$
Sorted: $2, 3, 4, 5, 6, 7, 9$
Comparing digit by digit:
Position $1: 7 \neq 2$
Position $2: 5 \neq 3$
Position $3: 2 \neq 4$
Position $4: 4 \neq 5$
Position $5: 6 = 6$
Position $6: 9 \neq 7$
Position $7: 3 \neq 9$
Only the digit $6$ remains in the same position. Therefore,there is only one such digit.

(NONE) Original number: $26839514$
Arranging the digits in descending order: $98654321$
Comparing the original positions with the new positions:
Original: $2, 6, 8, 3, 9, 5, 1, 4$
New: $9, 8, 6, 5, 4, 3, 2, 1$
By comparing each position:
$1^{st}$ digit: $2 \neq 9$
$2^{nd}$ digit: $6 \neq 8$
$3^{rd}$ digit: $8 \neq 6$
$4^{th}$ digit: $3 \neq 5$
$5^{th}$ digit: $9 \neq 4$
$6^{th}$ digit: $5 \neq 3$
$7^{th}$ digit: $1 \neq 2$
$8^{th}$ digit: $4 \neq 1$
Since no digit remains in its original position,the number of digits that remain unchanged is $0$.

(C) The original number is $3675249$.
For each odd digit,subtract $2$:
$3 - 2 = 1$
$7 - 2 = 5$
$5 - 2 = 3$
$9 - 2 = 7$
For each even digit,add $3$:
$6 + 3 = 9$
$2 + 3 = 5$
$4 + 3 = 7$
Replacing these digits in their original positions,the new number is $1953577$.
In the new number $1953577$,the digits $5$ and $7$ appear twice each.
Therefore,there are $2$ such digits.

(C) Original number: $5831649$
Arranging the digits in ascending order: $1345689$
Comparing the positions:
Original: $5, 8, 3, 1, 6, 4, 9$
Ascending: $1, 3, 4, 5, 6, 8, 9$
Comparing index by index:
$1^{st}$ digit: $5 \neq 1$
$2^{nd}$ digit: $8 \neq 3$
$3^{rd}$ digit: $3 \neq 4$
$4^{th}$ digit: $1 \neq 5$
$5^{th}$ digit: $6 = 6$ (Match)
$6^{th}$ digit: $4 \neq 8$
$7^{th}$ digit: $9 = 9$ (Match)
Thus,there are $2$ such digits ($6$ and $9$) that remain in the same position.

(D) The numbers from $1$ to $45$ that are exactly divisible by $3$ form an arithmetic progression: $3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45$.
Arranging these in ascending order,the sequence is $3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45$.
The $1^{st}$ number is $3$,the $2^{nd}$ is $6$,the $3^{rd}$ is $9$,and so on.
The $n^{th}$ term is given by $3 \times n$.
For the $9^{th}$ place,the number is $3 \times 9 = 27$.

(D) Analyze the pattern of each letter in the series:
$1$. The first letter follows the sequence: $n \xrightarrow{+1} o \xrightarrow{+1} p \xrightarrow{+1} q \xrightarrow{+1} r$.
$2$. The second letter follows the sequence: $s \xrightarrow{-4} o \xrightarrow{+2} q \xrightarrow{-1} p \xrightarrow{-1} o$.
Wait,let's re-evaluate the pattern:
$n, o, p, q \rightarrow r$ (First letter +$1$).
$s, r, q, p \rightarrow o$ (Second letter -$1$).
$i, g, e, c \rightarrow a$ (Third letter -$2$).
Therefore,the next group is $roa$.

(B) The original number is $53216894$.
There are $8$ digits in total. The rule is to interchange the $1^{st}$ with $5^{th}$,$2^{nd}$ with $6^{th}$,$3^{rd}$ with $7^{th}$,and $4^{th}$ with $8^{th}$ digits.
Original positions: $1(5), 2(3), 3(2), 4(1), 5(6), 6(8), 7(9), 8(4)$.
After interchanging:
$1^{st}$ becomes $6$,$5^{th}$ becomes $5$.
$2^{nd}$ becomes $8$,$6^{th}$ becomes $3$.
$3^{rd}$ becomes $9$,$7^{th}$ becomes $2$.
$4^{th}$ becomes $4$,$8^{th}$ becomes $1$.
The new number is $68945321$.
The digits from the right end are $1, 2, 3, 5, 4, 9, 8, 6$.
The second digit from the right end is $2$.

(B) Original number: $53147926$
Arrange the digits in descending order: $9, 7, 6, 5, 4, 3, 2, 1$.
Comparing the original and the rearranged number:
Original: $5, 3, 1, 4, 7, 9, 2, 6$
Rearranged: $9, 7, 6, 5, 4, 3, 2, 1$
Comparing positions:
$1^{st}$ digit: $5 \neq 9$
$2^{nd}$ digit: $3 \neq 7$
$3^{rd}$ digit: $1 \neq 6$
$4^{th}$ digit: $4 \neq 5$
$5^{th}$ digit: $7 \neq 4$
$6^{th}$ digit: $9 \neq 3$
$7^{th}$ digit: $2 = 2$
$8^{th}$ digit: $6 \neq 1$
The only digit whose position remains unchanged is $2$. Therefore,there is only one such digit.