SOLUTION: If the sum of six consecutive odd integers is 468, what is the smallest of the six integers?

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Question 1201138: If the sum of six consecutive odd integers is 468, what is the smallest of the six integers?
Found 7 solutions by mananth, Theo, math_tutor2020, greenestamps, ikleyn, josgarithmetic, MathTherapy:
Answer by mananth(16946) About Me  (Show Source):
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Let the odd integers be x, x+2, x+4, x+6, x+8,x+10
sum =468
x+ x+2+ x+4+ x+6+ x+8+x+10 =468
6x+30 =468
6x =438
x=73 smallest
b

Answer by Theo(13342) About Me  (Show Source):
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the 6 consecutive odd integers are x, x + 2, x = 4, x + 6, x + 8, x + 10
the sum of all of them is 6x + 30
you get 6x + 30 = 468.
subtract 30 from both sides of the equation to get 6x = 438.
solve for x to get x = 438 / 6 = 73.
the consecutive odd integers are 73, 75, 77, 79, 81, 83.
that's your solution.
the sum of those 6 odd integers is 468.
solution is confirmed to be good.

Answer by math_tutor2020(3817) About Me  (Show Source):
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Answer: 73


Work Shown:
x = first odd integer
x+2 = second odd integer
(x+2)+2 = x+4 = third odd integer
(x+4)+2 = x+6 = fourth odd integer
(x+6)+2 = x+8 = fifth odd integer
(x+8)+2 = x+10 = sixth odd integer
We add 2 to each odd integer to get the next consecutive odd integer.

first+second+third+fourth+fifth+sixth = 468
x+(x+2)+(x+4)+(x+6)+(x+8)+(x+10) = 468
6x+30 = 468
6x = 468-30
6x = 438
x = 438/6
x = 73

x = 73
x+2 = 73+2 = 75
x+4 = 73+4 = 77
x+6 = 73+6 = 79
x+8 = 73+8 = 81
x+10 = 73+10 = 83
The six consecutive odd integers are 73, 75, 77, 79, 81, 83

Check:
73+75+77+79+81+83 = 468
The answer is confirmed.

Answer by greenestamps(13214) About Me  (Show Source):
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The sequence of consecutive odd integers forms an arithmetic sequence with common difference 2.

The average of the 6 consecutive odd integers is 468/6 = 78.

That means the two odd integers "in the middle" (the 3rd and 4th of them) are 77 and 79.

Simple arithmetic then shows the sequence to be 73, 75, 77, 79, 81, 83.

ANSWER: 73

Answer by ikleyn(52898) About Me  (Show Source):
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.

See the lesson
    - Problems with consecutive integer numbers; odd/even consecutive integer numbers
in this site.

Find there many similar solved problems - your  TEMPLATES.


        Learn the subject from there.



Answer by josgarithmetic(39630) About Me  (Show Source):
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The six consecutive odd integers
system%28n-5%2Cn-3%2Cn-1%2Cn%2B1%2Cn%2B3%2Cn%2B5%29

Sum
%28n-5%29%2B%28n-3%29%2B%28n-1%29%2B%28n%2B1%29%2B%28n%2B3%29%2B%28n%2B5%29
.
.
6n

That sum is 468.
6n=468
n=468%2F6
n=78--------------

The smallest of the asked-for integers,
n-5
78-5
highlight%2873%29

Answer by MathTherapy(10557) About Me  (Show Source):
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If the sum of six consecutive odd integers is 468, what is the smallest of the six integers?

You can use the formula for the SUM of an A.P., which is: matrix%281%2C3%2C+S%5Bn%5D%2C+%22=%22%2C+%28n%2F2%29%282a%5B1%5D+%2B+%28n+-+1%29d%29%29%29, where:
                                    n = The number of elements in the set
                                   S%5Bn%5D = SUM of the series
                                   a%5B1%5D = the 1st number (unknown, in this case)
                                    d = the COMMON DIFFERENCE in the series (ODD, so 2, in this case)
                                  
                                   156 = 2a1 + 10 ---- Dividing by 3
                              156 - 10 = 2a1
                                   146 = 2a1
First/Smallest of the INTEGERS, or