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The greatest integer of a set of consecutive even integers is 12. If the sum of these integers is 40, how many integers are in this set?
- a)5
- b)6
- c)12
- d)20
- e)40
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
The greatest integer of a set of consecutive even integers is 12. If t...
One quick way to solve this problem is to start with 12 and add each preceding even integer until you get to a sum of 40, and then count the terms:
12 + 10 + 8 + 6 + 4 = 40.
There are 5 terms in the set.
12 + 10 + 8 + 6 + 4 = 40.
There are 5 terms in the set.
Community Answer
The greatest integer of a set of consecutive even integers is 12. If t...
Given Information:
- The greatest integer in the set of consecutive even integers is 12.
- The sum of these integers is 40.
Solution:
Let's assume the set of consecutive even integers starts from x.
- Since the greatest integer in the set is 12, we can write the set as {x, x+2, x+4, ..., 12}.
- The number of integers in this set can be calculated using the formula:
Number of Integers = (Largest Integer - Smallest Integer)/2 + 1
= (12 - x)/2 + 1
- The sum of the set can be calculated by adding all the integers in the set:
Sum = x + (x+2) + (x+4) + ... + 12
= Number of Integers * (x + 12)/2
= ((12 - x)/2 + 1) * (x + 12)/2
= (12 - x)(x + 12)/4 + (x + 12)/2
- Given that the sum of the set is 40, we can substitute this value into the equation and solve for x:
(12 - x)(x + 12)/4 + (x + 12)/2 = 40
(12 - x)(x + 12) + 2(x + 12) = 160
12x + 144 - x^2 - 12x + 2x + 24 = 160
-x^2 + 2x + 168 = 160
-x^2 + 2x + 8 = 0
x^2 - 2x - 8 = 0
(x - 4)(x + 2) = 0
x = 4 (since x is even)
- Therefore, the set of consecutive even integers is {4, 6, 8, 10, 12} with 5 integers in the set. Hence, the answer is option A.
- The greatest integer in the set of consecutive even integers is 12.
- The sum of these integers is 40.
Solution:
Let's assume the set of consecutive even integers starts from x.
- Since the greatest integer in the set is 12, we can write the set as {x, x+2, x+4, ..., 12}.
- The number of integers in this set can be calculated using the formula:
Number of Integers = (Largest Integer - Smallest Integer)/2 + 1
= (12 - x)/2 + 1
- The sum of the set can be calculated by adding all the integers in the set:
Sum = x + (x+2) + (x+4) + ... + 12
= Number of Integers * (x + 12)/2
= ((12 - x)/2 + 1) * (x + 12)/2
= (12 - x)(x + 12)/4 + (x + 12)/2
- Given that the sum of the set is 40, we can substitute this value into the equation and solve for x:
(12 - x)(x + 12)/4 + (x + 12)/2 = 40
(12 - x)(x + 12) + 2(x + 12) = 160
12x + 144 - x^2 - 12x + 2x + 24 = 160
-x^2 + 2x + 168 = 160
-x^2 + 2x + 8 = 0
x^2 - 2x - 8 = 0
(x - 4)(x + 2) = 0
x = 4 (since x is even)
- Therefore, the set of consecutive even integers is {4, 6, 8, 10, 12} with 5 integers in the set. Hence, the answer is option A.
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The greatest integer of a set of consecutive even integers is 12. If the sum of these integers is 40, how many integers are in this set?a)5b)6c)12d)20e)40Correct answer is option 'A'. Can you explain this answer?
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