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The mean of three numbers is 10 more than the least of the numbers and
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18 Mar 2019, 02:19
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The mean of three numbers is 10 more than the least of the numbers and 15 less than the greatest. The median of the three numbers is 5. What is their sum?
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18 Mar 2019, 03:04
Bunuel wrote:
The mean of three numbers is 10 more than the least of the numbers and 15 less than the greatest. The median of the three numbers is 5. What is their sum?
(A) 5 (B) 20 (C) 25 (D) 30 (E) 36
a+b+c/3 = 10+a = c-15 b=5 a= m-10 c=m+15 so m-10+5+m+15 = 3m 10=m sum 30 IMO D
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The mean of three numbers is 10 more than the least of the numbers and
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Updated on: 18 Mar 2019, 19:57
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Bunuel wrote:
The mean of three numbers is 10 more than the least of the numbers and 15 less than the greatest. The median of the three numbers is 5. What is their sum?
(A) 5 (B) 20 (C) 25 (D) 30 (E) 36
Let x, y and z be the 3 numbers such that x ≤ y ≤ z
The median of the three numbers is 5. So, y = 5
The mean of three numbers is 10 more than the least of the numbers We can write: (x + 5 + z)/3 = x + 10 Multiply both sides by 3 to get: x + 5 + z = 3x + 30 Subtract 5 from both sides to get: x + z = 3x + 25 Subtract x from both sides to get: z = 2x + 25
The mean of three numbers is 15 less than the greatest. We can write: (x + 5 + z)/3 = z - 15 Multiply both sides by 3 to get: x + 5 + z = 3z - 45 Subtract 5 from both sides to get: x + z = 3z - 50 Subtract z from both sides to get: x = 2z - 50
What is the value of x + y + z? We get: x + y + z = (2z - 50) + 5 + 2x + 25 Simplify to get: x + y + z = 2z + 2x - 20 Subtract y and z from both sides to get: y = z + x - 20 Since y = 5, we get: 5 = z + x - 20 Add 20 to both sides: 25 = z + x Since y = 5, we get: 30 = x + y + z
Re: The mean of three numbers is 10 more than the least of the numbers and
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19 Mar 2019, 01:50
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Bunuel wrote:
The mean of three numbers is 10 more than the least of the numbers and 15 less than the greatest. The median of the three numbers is 5. What is their sum?
The least number is 10 less than mean while the greatest is 15 more. To balance out the shortfall with the excess, the third number should be 5 less than mean too. Since median is 5, it is the middle number. The middle number must be 5 less than mean so mean must be 10.
Hence the other two numbers must be 0 and 25.
Sum of all numbers = 0 + 5 + 25 = 30
Answer (D)
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Re: The mean of three numbers is 10 more than the least of the numbers and
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19 Mar 2019, 19:06
Bunuel wrote:
The mean of three numbers is 10 more than the least of the numbers and 15 less than the greatest. The median of the three numbers is 5. What is their sum?
(A) 5 (B) 20 (C) 25 (D) 30 (E) 36
We can let the least value = a, the middle value = 5, and the greatest value = c, thus:
(a + 5 + c)/3 = 10 + a
a + 5 + c = 30 + 3a
c = 25 + 2a
and
(a + 5 + c)/3 = c - 15
a + 5 + c = 3c - 45
a = 2c - 50
Substituting c = 25 + 2a into a = 2c - 50, we have:
a = 2(25 + 2a) - 50
a = 50 + 4a - 50
-3a = 0
a = 0
So c = 25 + 2(0) = 25, and their sum is 0 + 5 + 25 = 30.
Alternate Solution:
Let m denote the mean. Then, the smallest number is m -10 and the greatest number is m + 15. We are also given that the middle (median) number is 5; thus we can create the equation:
(m - 10 + 5 + m + 15)/3 = m
2m + 10 = 3m
m = 10
Since the mean of the three numbers is 10, the sum is 3 x 10 = 30.