Extra Hard SAT Math Question – Finding the Sum of Way Too Many Numbers

You'll never get out of this one!
You’ll never get out of this one!

At the end of every SAT Math section, the test makers try to come up with an extremely difficult problem that will leave even the cleverest students scratching their heads. The really evil part, though, is that even these problems can be solved in under a minute without a calculator – if you know what to do. This means that once you “figure out the trick,” these difficult problems become easy. So, while those test makers are busy cackling with sadistic glee, let’s see if we can’t beat them at their own game.

Consider the following problem:

If the sum of the following integers from 1 to 50 is s, which of the following is the sum of the integers from 1 to 100?

A) 2s

B) 2s + 2000

C) 2s + 2500

D) 2s + 2750

E) 4s

Now, you could try to solve this by punching in 1+2+3+4+…+100 into your calculator, but considering all the adding you’re going to do, that’s going to take way too long, especially when this problem can be solved in seconds. But how, you ask? Well, to find out, we have to ask a kid in elementary school.

Carl Friedrich Gauss, all grown up.
Carl Friedrich Gauss, all grown up.

Meet Carl Friedrich Gauss, great mathematician and one time child prodigy. Gauss was born in Brunswick in modern day Germany in 1777 into a poor family. No one expected him to be a genius, but as soon as he learned to talk he began to amaze everyone with his mathematical abilities. When it was time for him to begin school, his math teachers had no idea what to do with him. He was already better at math then they were and he was always bored in class and getting into trouble. One day his teacher, J.G. Büttner, decided to punish his misbehavior (or at least keep him busy for a few minutes) by telling him to add all of the numbers from 1 to 100 together. To his astonishment, little Gauss gave him the correct answer, 5050, after only a few seconds. Ever since, Gauss has been a hero to nerdy children everywhere, and he later went on to make many important mathematical discoveries which you can learn about if you study math in college.

But how did little Gauss do it? Could he just add crazy fast? Maybe, but he was smarter than that. Consider the following chart:

1

2

3

4

5

6

7

8

9

10

+10

+9

+8

+7

+6

+5

+4

+3

+2

+1

11

11

11

11

11

11

11

11

11

11

Gauss realized that if you take an arithmetic series (like the numbers 1 through 10) and add the smallest one to the largest one, the next smallest one to the next largest one, and so on, the sum will always be the same (in this case, 11). If you multiply this sum by the number of terms in the series (11*10), then you will get twice the sum of all the terms in the series. You can see this by examining the chart above: the first row of the chart is all the numbers from 1 to 10 and the second row is as well, so when you add them you’ve got all the numbers from 1 to 10 twice. Thus, you have to divide 11*10 by 2 in order to find just the sum of the numbers from 1 to 10:

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(11*10)/2 = 55

The same principle applies with larger series. The sum of the numbers from 1 to 50 is:

(51*50)/2 = 1275

and the sum of the number from 1 to 100 is:

(101*100)/2 = 5050

Seems pretty simple, right? Well, that’s the nature of genius – seeing the obvious things everyone else misses. The problem asks for the the sum of the integers from 1 to 100 in terms of s, so all we need to do is put this into a form that fits one of the answer choices. S is defined as the sum of the integers from 1 to 50, so

s = 1275

Most of the answers are in terms of 2s, so

2s = 2250

We know that the sum of the integers from 1 to 100 is 5050, so

5050 – 2s = 5050 – 2250 = 2500

Thus,

2s + 2500 = 5050

Et voila! The answer is choice C. If you know what to do, it takes only a few seconds to solve this problem. So you see, with practice, even the hardest problems on the SAT become easy. Check back here each week for more extra hard problems and the tricks you need to solve them! Also, remember that you can find out all the tricks from experts like me with a Test Masters course or private tutoring. Until  then, keep up the good work and happy studying!

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