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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 i.e. ab. As a result, his answer for the arithmetic mean was 1.8 more than what it should have been. Then, b - a is equal to
- a)1
- b)2
- c)-2
- d)0
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
Amol was asked to calculate the arithmetic mean of ten positive intege...
The original number is ab.
Let the sum of the rest nine terms be x.
Average before the digits were interchanged = [(x/10) + (10a + b)/10]
Average after the digits were interchanged = [(x/10) + (10b + a)/10]
Given;
[(x/10) + (10b + a)/10] - [(x/10) + (10a + b)/10] = 1.8
(b - a) - (b - a)/10 = 1.8
9(b - a)/10 = 1.8
(b - a) = 2
Let the sum of the rest nine terms be x.
Average before the digits were interchanged = [(x/10) + (10a + b)/10]
Average after the digits were interchanged = [(x/10) + (10b + a)/10]
Given;
[(x/10) + (10b + a)/10] - [(x/10) + (10a + b)/10] = 1.8
(b - a) - (b - a)/10 = 1.8
9(b - a)/10 = 1.8
(b - a) = 2
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Amol was asked to calculate the arithmetic mean of ten positive intege...
Solution:
Let's assume that the original ten positive integers were a1, a2, a3, ..., a10.
Step 1: Calculation of the original arithmetic mean
The original arithmetic mean of the ten positive integers is given by:
(original sum) / 10
So, the original sum of the ten positive integers is:
a1 + a2 + a3 + ... + a10
And the original arithmetic mean is:
(original sum) / 10 = (a1 + a2 + a3 + ... + a10) / 10
Step 2: Calculation of the incorrect arithmetic mean
When Amol interchanged the two digits in one of the integers, the new integer became 10a + b (assuming a > b).
Thus, the new sum of the ten positive integers is:
(a1 + a2 + a3 + ... + a(n-1) + 10a + b + a(n+1) + ... + a10)
And the new arithmetic mean is:
(new sum) / 10 = (a1 + a2 + a3 + ... + a(n-1) + 10a + b + a(n+1) + ... + a10) / 10
Step 3: Calculation of the difference between the two arithmetic means
According to the given information, the new arithmetic mean is 1.8 more than what it should have been. Mathematically, we can express this as:
(new arithmetic mean) - (original arithmetic mean) = 1.8
Substituting the expressions for the new and original arithmetic means, we get:
[(a1 + a2 + a3 + ... + a(n-1) + 10a + b + a(n+1) + ... + a10) / 10] - [(a1 + a2 + a3 + ... + a10) / 10] = 1.8
Simplifying the above equation, we get:
(10a + b) - (a) = 18
9a + b = 18
Step 4: Calculation of b - a
We need to find the value of b - a. From Step 3, we know that:
9a + b = 18
We can rewrite this equation as:
b = 18 - 9a
Substituting this value of b in the expression b - a, we get:
b - a = (18 - 9a) - a
Simplifying the above equation, we get:
b - a = 18 - 10a
So, the value of b - a is 18 - 10a. To determine the value of b - a, we need the value of a. Unfortunately, the question does not provide any information about the value of a. Therefore, we cannot determine the exact value of b - a.
Hence, the correct answer is option 'd' (0).
Let's assume that the original ten positive integers were a1, a2, a3, ..., a10.
Step 1: Calculation of the original arithmetic mean
The original arithmetic mean of the ten positive integers is given by:
(original sum) / 10
So, the original sum of the ten positive integers is:
a1 + a2 + a3 + ... + a10
And the original arithmetic mean is:
(original sum) / 10 = (a1 + a2 + a3 + ... + a10) / 10
Step 2: Calculation of the incorrect arithmetic mean
When Amol interchanged the two digits in one of the integers, the new integer became 10a + b (assuming a > b).
Thus, the new sum of the ten positive integers is:
(a1 + a2 + a3 + ... + a(n-1) + 10a + b + a(n+1) + ... + a10)
And the new arithmetic mean is:
(new sum) / 10 = (a1 + a2 + a3 + ... + a(n-1) + 10a + b + a(n+1) + ... + a10) / 10
Step 3: Calculation of the difference between the two arithmetic means
According to the given information, the new arithmetic mean is 1.8 more than what it should have been. Mathematically, we can express this as:
(new arithmetic mean) - (original arithmetic mean) = 1.8
Substituting the expressions for the new and original arithmetic means, we get:
[(a1 + a2 + a3 + ... + a(n-1) + 10a + b + a(n+1) + ... + a10) / 10] - [(a1 + a2 + a3 + ... + a10) / 10] = 1.8
Simplifying the above equation, we get:
(10a + b) - (a) = 18
9a + b = 18
Step 4: Calculation of b - a
We need to find the value of b - a. From Step 3, we know that:
9a + b = 18
We can rewrite this equation as:
b = 18 - 9a
Substituting this value of b in the expression b - a, we get:
b - a = (18 - 9a) - a
Simplifying the above equation, we get:
b - a = 18 - 10a
So, the value of b - a is 18 - 10a. To determine the value of b - a, we need the value of a. Unfortunately, the question does not provide any information about the value of a. Therefore, we cannot determine the exact value of b - a.
Hence, the correct answer is option 'd' (0).
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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 i.e. ab. As a result, his answer for the arithmetic mean was 1.8 more than what it should have been. Then, b - a is equal toa)1b)2c)-2d)0Correct answer is option 'B'. Can you explain this answer?
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