UPSC Prelims 2021 · CSAT
Question 76: Basic Numeracy
The difference between a 2-digit number and the number obtained by interchanging the positions of the digits is 54. Consider the following statements :
- 1.The sum of the two digits of the number can be determined only if the product of the two digits is known.
- 2.The difference between the two digits of the number can be determined.
Which of the above statements is/are correct ?
AnswerUPSC’s answer is (b).
Write the number as 10a + b. Its difference from the reversed number is 9(a − b) = 54, so a − b = 6. Statement 2 is correct: the digits always differ by 6. The digit pairs are {6, 0}, {7, 1}, {8, 2} and {9, 3}, with sums 6, 8, 10 and 12. Statement 1 says the sum can be found only if the product is known; knowing the product does pick the pair, but so would knowing one of the digits, so 'only if' is wrong and Statement 1 is incorrect. Remember: 'only if' claims fail when a second route exists.
- 1. Knowing the product fixes the digit pair, but so would knowing either digit, so the product is not the only way.
- 2. 9(a − b) = 54, so the digits always differ by 6.
How you could have got this
- If you were sure statement 1 is not correct, (a) and (c) go: each of them includes 1.
What you needed to know: Knowing the product fixes the digit pair, but so would knowing either digit, so the product is not the only way.
- Then, if you were sure statement 2 is correct, (d) goes: it leaves 2 out.
What you needed to know: 9(a − b) = 54, so the digits always differ by 6.
Only (b) is left. This one needs every statement.
Which options go is worked out from UPSC’s key and the options, not by a model.
Why each option is right or wrong
- (a) Statement 1 is wrong: the product is not the only way to pin down the sum; a known digit also does.
- (b) Right: 9(a − b) = 54 gives a digit difference of 6.
- (c) Statement 1 is false because 'only if' is too strong, so both cannot be correct.
- (d) Statement 2 is correct, so this fails.
Answer: UPSC’s key. Explanation: written by UPSC Grade, not by UPSC.
Source: UPSC, Civil Services (Preliminary) Examination 2021, General Studies Paper II (CSAT), Series A. The paper on upsc.gov.in.