The sum of 8 consecutive integers is 180.

Question

The sum of 8 consecutive integers is 180. What is the sum of first 4 of these integers?

a. 80
b. 82
c. 85
d. 86
e. 90

Mohammadmo · Answer

n,n+1,...,n+7
sum of these 8 numbers:8n+28=180
8n=152,n=19
So,first four numbers are,19,20,21,22.
Which their sum equals 82.
Option B

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Kinshook · Answer

Given: The sum of 8 consecutive integers is 180.

Asked: What is the sum of first 4 of these integers?

Let the consecutive integers start with a

a+a+1+a+2+a+3= 4a+6=?
4a+6+a+4+a+5+a+6+a+7=8a+28=180

a=19
4a+6= 4*19+6=82

IMO B

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adkikani · Answer

Bunuel   VeritasKarishma   chetan2u   GMATPrepNow

I respect all above solutions, but in my view I still feel they are prone to calculation errors.
Any way I can use below facts to solve this problem:
1. For consecutive integers: average = (First term + last term) / 2
2. Even for using simple average equation (avg = sum / quantity) ; since 180 / 8 gives me a decimal,
I do not feel like proceeding since I make calculations mistakes in same.

May I know your approach please?

KarishmaB · Answer

Here, all the methods are almost equivalent in the amount of time and effort required.

\frac{(a + a + 7)}{2} = \frac{180}{8}

2a = 45 - 7 = 38

a = 19

Sum of first four numbers = 4a + 1 + 2 + 3 = 4*19 + 6 = 82

Answer (B)

Another method:

Each of the last 4 numbers will be 4 more than the previous 4 numbers hence the sum will have 16 extra from the last 4 numbers. 
180 - 16 = 164 is the sum of all 8 numbers if the last 4 numbers were the same as first 4 numbers. 
So sum of first 4 numbers = 164/2 = 82

Abhishek009 · Answer

a + (a + 1) + (a + 2) + (a + 3) + (a + 4) + (a + 5) + (a + 6) + (a + 7) = 180

Or,  8a + 28 = 180

Or,  a = \frac{152}{8} = 19

Now, sum of  a + (a + 1) + (a + 2) + (a + 3) = 4a + 6 = 4*19 + 6 => 82 , Answer must be (B)

The sum of 8 consecutive integers is 180.

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The sum of 8 consecutive integers is 180.

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