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For any positive number n, the function #n represents the value of the number n rounded to the nearest integer. If k is a positive number, what is the units digit of #k?
(1)  #(10k) = 10k
(2)  #(100k) is 10300.
  • a)
    Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
  • b)
    Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
  • c)
    BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
  • d)
    EACH statement ALONE is sufficient to answer the question asked.
  • e)
    Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
Correct answer is option 'B'. Can you explain this answer?
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For any positive number n, the function #n represents the value of the...
Steps 1 & 2: Understand Question and Draw Inferences
Given:
  • Let k = n + a.bcde. . . , where n is a multiple of 10 and a, b, c, d, e etc. denote the units, tenths, hundredths, thousandths, ten-thousandths digits of k respectively.
 
  • Example: Say k is equal to 123.789. We can also write this value as 120 + 3.789. This form is comparable to n + a.bcde . . .
 
  • So, n denotes the value of all the digits at the tens, hundreds, thousands and higher place values (if k = 123.789, n = 120; if k = 3456.987, n = 3450; if k = 43789.34, n = 43780 and so on)
 
  • We have split k into n and a.bcde. . . because the question doesn’t indicate whether the magnitude of k is in tens or hundreds or thousands etc. (that is, is k a number like 12.345 or like 123.789 or like 3456.987 etc.) This knowledge is not important either because all the action is happening at the units digit and the digits to the right of the decimal point. So, we’ve simply expressed the value of all the digits at the tens, hundreds, thousands and higher place values in a single term n
     
  • #k =
    • n + a, if b < 5
    • n + (a+1) if b ≥ 5
To find:
  • The units digit of #k
  • Since n is a multiple of 10, from the expression of #k, we can say that the units digit of #k =
    • a, if b < 5
    • a + 1, if b ≥ 5
Thus, to answer the question, we need to know the value of a and whether b < 5 or not
  
Step 3: Analyze Statement 1 independently
(1)  #(10k) = 10k
  • 10k = 10n + ab.cde. . .
    • Since a is now the tens digit and b is the units digit, we can write
10k = 10n + 10a + b.cde. . .
10k = 10(n+a) + b.cde. . .
 
  • Applying the definition of the function #n, we can write:
    • #(10k) =
      • 10(n+a) + b if c < 5
      • 10(n+a) + (b+1) if c ≥ 5
  • We are given that #(10k) = 10k.
    • #(10k) = 10(n+a) + b.cde . . .
 
  • If c < 5, then by substituting the value of #(10k) in the above equation, we get:
    • 10(n+a) + b = 10(n+a) + b.cde. . .
    • b = b + 0.cde. . .
    • This implies, c = d = e . . .= 0
    • Thus, k is a number of the form n + a.b, where b is the sole digit after the decimal point
 
  • If  c ≥ 5, then by substituting the value of #(10k) in the above equation, we get:
    • 10(n+a) + (b+1) = 10(n+a) + b.cde. . .
    • b  +1 = b + 0.cde . . .
    • 1 = 0.cde . .  .
    • For no values of digits c, d, e . . . will the above equation be satisfied.
    • Therefore, this case is not possible
 
  • Thus, from Statement 1, we conclude that k is a number of the form n + a.b, where b is the sole digit after the decimal point
  • However, we still don’t know the value of a or whether b is less than 5 or not.
So, Statement 1 alone is not sufficient.
 
Step 4: Analyze Statement 2 independently
(2) #(100k) is 10300.
  • 100k = 100n + abc.de. . .
    • 100k = (100n + 100a + 10b) + c.de. . .
 
  • Applying the definition of the function #n, we can write:
    • #(100k) =
      • (100n + 100a + 10b) + c if d < 5
      • (100n + 100a + 10b) + (c+1) if d ≥ 5
 
  • We are given that #(100k) = 10300
    • If d < 5, then we can write
      • (100n + 100a + 10b) + c = 10300
      • Remember that in the above expression, a, b and c are digits (therefore, lie between 0 and 9, inclusive) whereas n is a multiple of 10.
      • Comparing the units digits on both sides of the equation, we get: c = 0
      • Comparing the tens digits on both sides of the equation, we get: b = 0
      • Comparing the hundreds digits on both sides of the equation, we get: a = 3
      • Since b < 5 in this case, the units digit of #k = a = 3
 
  • If d ≥ 5, then we can write
    • (100n + 100a + 10b) + (c+1) = 10300
    • Comparing the units digits on both sides of the equation, we get: c + 1 = 0
      • So, c = 9 and 1 is carried over to tens place
    • Comparing the tens digits on both sides of the equation, we get: b + 1 = 0 (Note, we’re writing b+1 and not b due to the carry-over)
      • So, b = 9 and 1 is carried over to hundreds place
    • Comparing the hundreds digits on both sides of the equation, we get: a + 1 = 3
      • So, a = 2
    • Since b > 5 in this case, the units digit of #k = a  +1 = 3
 
  • Thus, we see that both possible values of #(100k) lead to the same value of the units digit of #k
  • Thus, Statement 2 is sufficient to find a unique answer to the question.
 
Step 5: Analyze Both Statements Together (if needed)
Since we’ve already arrived at a unique answer in Step 4, this step is not required
Answer: Option B
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Most Upvoted Answer
For any positive number n, the function #n represents the value of the...
Statement Analysis:

Statement 1: #(10k) = 10k
This statement tells us that the value of 10k rounded to the nearest integer is 10k. From this information, we can deduce that k must be a whole number. However, this statement alone does not provide enough information to determine the units digit of #k.

Statement 2: #(100k) is 10300
This statement gives us the rounded value of 100k. We can see that the units digit of this rounded value is 0. However, this information alone is not sufficient to determine the units digit of #k.

Combined Analysis:
When we combine both statements, we can see that k must be a whole number and that the units digit of the rounded value of 100k is 0. This means that the units digit of k must be 0 in order for the units digit of 100k to be 0. Therefore, statement 2 alone is sufficient to determine the units digit of #k.
Therefore, the correct answer is option B.
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For any positive number n, the function #n represents the value of the number n rounded to the nearest integer. If k is a positive number, what is the units digit of #k?(1) #(10k) = 10k(2) #(100k) is 10300.a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.b)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.c)BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.d)EACH statement ALONE is sufficient to answer the question asked.e)Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.Correct answer is option 'B'. Can you explain this answer?
Question Description
For any positive number n, the function #n represents the value of the number n rounded to the nearest integer. If k is a positive number, what is the units digit of #k?(1) #(10k) = 10k(2) #(100k) is 10300.a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.b)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.c)BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.d)EACH statement ALONE is sufficient to answer the question asked.e)Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.Correct answer is option 'B'. Can you explain this answer? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about For any positive number n, the function #n represents the value of the number n rounded to the nearest integer. If k is a positive number, what is the units digit of #k?(1) #(10k) = 10k(2) #(100k) is 10300.a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.b)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.c)BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.d)EACH statement ALONE is sufficient to answer the question asked.e)Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for For any positive number n, the function #n represents the value of the number n rounded to the nearest integer. If k is a positive number, what is the units digit of #k?(1) #(10k) = 10k(2) #(100k) is 10300.a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.b)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.c)BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.d)EACH statement ALONE is sufficient to answer the question asked.e)Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.Correct answer is option 'B'. Can you explain this answer?.
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