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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, then the new average of the four numbers will become 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...
It is given that average of the four numbers is 27.
Let A, B, C and D be the four numbers.
Let A, B, C and D be the four numbers.
A + B + C + D = 108 … (1)
Also, A is increased by K; A + K.
B is increased by K + 7; B + K + 7.
C is increased by 2K + 3; C + 2K + 3.
D is decreased by 2; D – 2.
Also, A is increased by K; A + K.
B is increased by K + 7; B + K + 7.
C is increased by 2K + 3; C + 2K + 3.
D is decreased by 2; D – 2.
⇒ A + B + C + D + 4K = 128 – 8 = 120
⇒ 108 + 4K = 120 {From equation (1)}
⇒ 4K = 12
K = 3
⇒ 108 + 4K = 120 {From equation (1)}
⇒ 4K = 12
K = 3
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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, then the new average of the four numbers will become 32.
To find:
The value of K.
Solution:
Let's assume the four numbers as a, b, c, and d.
Step 1: Calculate the sum of the four numbers.
- The average of the four numbers is 27, so the sum of the four numbers is 27 * 4 = 108.
- Therefore, a + b + c + d = 108.
Step 2: Calculate the new sum of the four numbers.
- The new average of the four numbers is 32, so the new sum of the four numbers is 32 * 4 = 128.
Step 3: Express the new sum of the four numbers in terms of K.
- The first number increases by K, so a + K.
- The second number increases by (K + 7), so b + (K + 7).
- The third number increases by (2K - 3), so c + (2K - 3).
- The fourth number decreases by 2, so d - 2.
- Therefore, the new sum of the four numbers is (a + K) + (b + K + 7) + (c + 2K - 3) + (d - 2).
Step 4: Equate the two sums and solve for K.
- (a + K) + (b + K + 7) + (c + 2K - 3) + (d - 2) = 128.
- Simplify the equation: a + b + c + d + 3K + 2 = 128.
- From Step 1, we know that a + b + c + d = 108, so substitute the value: 108 + 3K + 2 = 128.
- Simplify further: 3K + 110 = 128.
- Subtract 110 from both sides: 3K = 18.
- Divide both sides by 3: K = 6.
Answer:
The value of K is 6.
Therefore, the correct option is (B) 6.
- 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, then the new average of the four numbers will become 32.
To find:
The value of K.
Solution:
Let's assume the four numbers as a, b, c, and d.
Step 1: Calculate the sum of the four numbers.
- The average of the four numbers is 27, so the sum of the four numbers is 27 * 4 = 108.
- Therefore, a + b + c + d = 108.
Step 2: Calculate the new sum of the four numbers.
- The new average of the four numbers is 32, so the new sum of the four numbers is 32 * 4 = 128.
Step 3: Express the new sum of the four numbers in terms of K.
- The first number increases by K, so a + K.
- The second number increases by (K + 7), so b + (K + 7).
- The third number increases by (2K - 3), so c + (2K - 3).
- The fourth number decreases by 2, so d - 2.
- Therefore, the new sum of the four numbers is (a + K) + (b + K + 7) + (c + 2K - 3) + (d - 2).
Step 4: Equate the two sums and solve for K.
- (a + K) + (b + K + 7) + (c + 2K - 3) + (d - 2) = 128.
- Simplify the equation: a + b + c + d + 3K + 2 = 128.
- From Step 1, we know that a + b + c + d = 108, so substitute the value: 108 + 3K + 2 = 128.
- Simplify further: 3K + 110 = 128.
- Subtract 110 from both sides: 3K = 18.
- Divide both sides by 3: K = 6.
Answer:
The value of K is 6.
Therefore, the correct option is (B) 6.
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