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Amol was asked to calculate the arithmetic mean of ten positive integers each of which had two digits. By mistake, he interchanged the two digits, say a and b, in one of these ten integers. As a result, his answer for the arithmetic mean was 1.8 more than what it should have been. Then b –a equals
- a)1
- b)2
- c)3
- d)4
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
Amol was asked to calculate the arithmetic mean of ten positive intege...
► (10b + a) - (10a + b) = 1.8 × 10. So b – a = 2.
Most Upvoted Answer
Amol was asked to calculate the arithmetic mean of ten positive intege...
Let the original number be $10a+b$ and the new number be $10b+a$. The sum of the ten original numbers is $10(10a_1+b_1)+10(10a_2+b_2)+\cdots+10(10a_{10}+b_{10}) = 100(a_1+a_2+\cdots+a_{10}) + (b_1+b_2+\cdots+b_{10})$, since each digit is counted ten times in the units place and once in the tens place. Similarly, the sum of the ten new numbers is $100(b_1+b_2+\cdots+b_{10}) + (a_1+a_2+\cdots+a_{10})$. Therefore, we have \begin{align*}
\frac{100(a_1+a_2+\cdots+a_{10}) + (b_1+b_2+\cdots+b_{10}) - (10a+b) + 10b+a}{10} &= \frac{100(b_1+b_2+\cdots+b_{10}) + (a_1+a_2+\cdots+a_{10}) - (10b+a) + 10a+b}{10} + 1.8 \\
\Rightarrow \qquad 20(b-a) &= 18 \\
\Rightarrow \qquad b-a &= \frac{9}{10}.
\end{align*}Since $a$ and $b$ are digits, we must have $b-a=1$, so $b=a+\frac{9}{10}$. Since $a$ and $b$ are both digits, we must have $a=1$ and $b=2$, so the original number is $10a+b=\boxed{12}$.
\frac{100(a_1+a_2+\cdots+a_{10}) + (b_1+b_2+\cdots+b_{10}) - (10a+b) + 10b+a}{10} &= \frac{100(b_1+b_2+\cdots+b_{10}) + (a_1+a_2+\cdots+a_{10}) - (10b+a) + 10a+b}{10} + 1.8 \\
\Rightarrow \qquad 20(b-a) &= 18 \\
\Rightarrow \qquad b-a &= \frac{9}{10}.
\end{align*}Since $a$ and $b$ are digits, we must have $b-a=1$, so $b=a+\frac{9}{10}$. Since $a$ and $b$ are both digits, we must have $a=1$ and $b=2$, so the original number is $10a+b=\boxed{12}$.
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Amol was asked to calculate the arithmetic mean of ten positive intege...
supposed the number which was interchanged is ba, if we interchange the digits then the number will become ab
Average before interchanging = (ba + x)/10 , where x is sum of other 9 numbers
Average after interchanging = (ab + x)/10
Given that average before interchanging = average after interchanging + 1.8
(ba + x)/10 = (ab+x)/10 + 1.8
or
ba + x = ab + x + 18
ba can be written as 10b + a and ab can be written as 10a + b
10b + a + x = 10a + b + x + 18
or
b-a = 2
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Amol was asked to calculate the arithmetic mean of ten positive integers each of which had two digits. By mistake, he interchanged the two digits, say a and b, in one of these ten integers. As a result, his answer for the arithmetic mean was 1.8 more than what it should have been. Then b –a equalsa) 1b) 2c) 3d) 4Correct answer is option 'B'. Can you explain this answer?
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Amol was asked to calculate the arithmetic mean of ten positive integers each of which had two digits. By mistake, he interchanged the two digits, say a and b, in one of these ten integers. As a result, his answer for the arithmetic mean was 1.8 more than what it should have been. Then b –a equalsa) 1b) 2c) 3d) 4Correct answer is option 'B'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about Amol was asked to calculate the arithmetic mean of ten positive integers each of which had two digits. By mistake, he interchanged the two digits, say a and b, in one of these ten integers. As a result, his answer for the arithmetic mean was 1.8 more than what it should have been. Then b –a equalsa) 1b) 2c) 3d) 4Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Amol was asked to calculate the arithmetic mean of ten positive integers each of which had two digits. By mistake, he interchanged the two digits, say a and b, in one of these ten integers. As a result, his answer for the arithmetic mean was 1.8 more than what it should have been. Then b –a equalsa) 1b) 2c) 3d) 4Correct answer is option 'B'. Can you explain this answer?.
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