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The average of four numbers is 27. If the first number is increased by k, the second by (k + 7), the third by (2k + 3) and the fourth is decreased by 2, the new average of the four numbers becomes 32. Find the value of k.
- a)2
- b)3
- c)4
- d)5
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
The average of four numbers is 27. If the first number is increased by...
The given average of the four numbers is 27.
Let A, B, C and D be the four numbers.
∴
= 27
= 27A + B + C + D = 108 … (1)
Also,
A is increased by k, so it becomes A + k.
B is increased by k + 7, so it becomes B + k + 7
C is increased by 2k + 3, so it becomes C + 2k + 3.
D is decreased by 2, so it becomes D – 2.
New average = 
A + B + C + D + 4k = 128 – 8 = 120
108 + 4k = 120 {from equation (1)}
4k = 12∴ k = 3
Why Incorrect?
Solution: This could be eliminated.
The given average of the four numbers is 27.
Let A, B, C and D be the four numbers.
= 27A + B + C + D = 108 … (1)
Also,
A is increased by k, so it becomes A + k.
B is increased by k + 7, so it becomes B + k + 7
C is increased by 2k + 3, so it becomes C + 2k + 3.
D is decreased by 2, so it becomes D – 2.
New average =
A + B + C + D + 4k = 128 – 8 = 120
108 + 4k = 120 {from equation (1)}
4k = 12
∴ k = 3 ≠ 4
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Community Answer
The average of four numbers is 27. If the first number is increased by...
Given:
- The average of four numbers is 27.
- If the first number is increased by k, the second by (k + 7), the third by (2k + 3), and the fourth is decreased by 2, the new average of the four numbers becomes 32.
To find:
The value of k.
Solution:
Let's assume the four numbers as a, b, c, and d, respectively.
Step 1: Average of Four Numbers
The average of the four numbers is given as 27.
So, we can write the equation as:
(a + b + c + d)/4 = 27
Step 2: New Average of Four Numbers
When the first number is increased by k, the second by (k + 7), the third by (2k + 3), and the fourth is decreased by 2, the new average becomes 32.
So, we can write the equation as:
(a + k + b + (k + 7) + c + (2k + 3) + d - 2)/4 = 32
Simplifying the above equation, we get:
(a + b + c + d + 4k + 8)/4 = 32
(a + b + c + d + 4k + 8) = 32 * 4
(a + b + c + d + 4k + 8) = 128
(a + b + c + d) + 4k + 8 = 128
(a + b + c + d) + 4k = 120
Step 3: Substituting the Average equation
We can substitute the average equation (from Step 1) into the above equation to eliminate (a + b + c + d).
27 + 4k = 120
4k = 120 - 27
4k = 93
k = 93/4
k = 23.25
Since k is not an integer, we cannot have a fraction as the value of k. Therefore, the given answer options are incorrect.
Hence, the correct answer is not among the given options.
- The average of four numbers is 27.
- If the first number is increased by k, the second by (k + 7), the third by (2k + 3), and the fourth is decreased by 2, the new average of the four numbers becomes 32.
To find:
The value of k.
Solution:
Let's assume the four numbers as a, b, c, and d, respectively.
Step 1: Average of Four Numbers
The average of the four numbers is given as 27.
So, we can write the equation as:
(a + b + c + d)/4 = 27
Step 2: New Average of Four Numbers
When the first number is increased by k, the second by (k + 7), the third by (2k + 3), and the fourth is decreased by 2, the new average becomes 32.
So, we can write the equation as:
(a + k + b + (k + 7) + c + (2k + 3) + d - 2)/4 = 32
Simplifying the above equation, we get:
(a + b + c + d + 4k + 8)/4 = 32
(a + b + c + d + 4k + 8) = 32 * 4
(a + b + c + d + 4k + 8) = 128
(a + b + c + d) + 4k + 8 = 128
(a + b + c + d) + 4k = 120
Step 3: Substituting the Average equation
We can substitute the average equation (from Step 1) into the above equation to eliminate (a + b + c + d).
27 + 4k = 120
4k = 120 - 27
4k = 93
k = 93/4
k = 23.25
Since k is not an integer, we cannot have a fraction as the value of k. Therefore, the given answer options are incorrect.
Hence, the correct answer is not among the given options.
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