What is the sum of all of the integers in the chart above?

Reorder the numbers in columns in ascending absolute value:



Now, it's easy to spot that the numbers in each column represent an equally spaced set (AP) and the sum in each can be found by mean*# of terms:

\(Sum=5*100+5*(-200)+5*(300)+5*(-400)+5*(500)=500-1,000+1,500-2,000+2,500=1,500\).

Answer: D.

Hope it's clear.


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Absolutely and many thanks. As always, thorough and a very helpful answer.
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Sum of the first column = 5*100 = 500.
Sum of the last column = 500+5*400 = 2500[As all the corresponding terms are increased by 400]
Sum of the second column = 5*-200 = -1000
Sum of the fourth column = -1000-5*200 = -2000
The total sum = 500+2500-1000-2000 = 0
Thus everything when added gets cancelled. Except the inner column. That is an AP and the sum = 5*300 = 1500.

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It is easy once you spot that in the same row, the 1st outer most and the second outer most cancel one another. You are then left with adding the column in the middle.

Always check the answer choices BEFORE solving the question!!
Since the answer choices are nicely spread apart, we can be quite aggressive in our estimation

Let's add the 5 numbers in each COLUMN

COLUMN 1: Sum ≈500
COLUMN 2: Sum ≈-1000
COLUMN 3: Sum ≈1500
COLUMN 4: Sum ≈-2000
COLUMN 5: Sum ≈2500

TOTAL SUM ≈500 + (-1000) + 1500 + (-2000) + 2500
≈1500

Answer: D

RELATED VIDEO FROM OUR COURSE

We don’t have actually add up all the numbers; we can just approximate the sum. Let’s add the numbers column by column.

We see that the 5 numbers in the first column are each about 100, so the sum is approximately 500. The 5 numbers in the second column are each about -200, so the sum is approximately -1000. The 5 numbers in the third column are each about 300, so the sum is approximately 1500. The 5 numbers in the fourth column are each about -400, so the sum is approximately -2000. Lastly, the 5 numbers in the fifth column are each about 500, so the sum is approximately 2500.

Therefore, the sum of all 25 numbers is approximately 500 + (-1000) + 1500 + (-2000) + 2500 = 4500 - 3000 = 1500

Answer: D.
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Hi,
your replies are really helpful in learning tricks and shortcuts. Can you please tell why did you take the middle line only

The complete property is:
In any evenly spaced set the arithmetic mean (average) is equal to the median and the sum of the terms in any evenly spaced set is the mean (average) multiplied by the number of terms.

So, while you are arranging the columns you are getting the middle as average and multiplying it to get the sum.

Now, (100x5) + (-200x5) + (300x 5)+(-400x5)+(500x5)
you don't need to multy you take 5 Common
5[100-200+300-400+500]
5[900-600] = 1500

Birds just don't fly, they fall down and get up.
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Average of first column = 100
Average of first column = -200
Average of first column = 300
Average of first column = -400
Average of first column = 500

(100+500)+(-400)+(-200) cancel out. Only left is 5x300=1,500