Need assistance in Deciphering this solution

[bjuveges]

The average of the numbers and is . What is ?

[h=2]Solution[/h]We first sum the first numbers: . Then, we know that the sum of the series is . Since the average is , and there are terms, we also find the sum to equal . Setting equal - . Thus, the answer is .

Ok I know that to find the sum of the terms you need to do some arithmetic, 99(100)/2=4950. but after the solution states that it needs to satisfy a sum =10,000x I get lost. How did the solution get a 9,999x? Basically I don't understand anything after the phrase "Setting equal - 10,000x" on the second line down of the solution. ( and the - is not a negative just to eradicate any confusion)

 

[Quaid]

after the solution states that it needs to satisfy a sum =10,000x I get lost. How did the solution get a 9,999x?

Basically I don't understand anything after the phrase "Setting equal - 10,000x" on the second line down of the solution.

We know the following.

"Since the average is , and there are terms …"

In other words, if you take 100 instances of the number 100x and add them all together, you get 100 times 100x, which is 10000x.

(It is dividing this total by 100 that gives the average 100x.)

So, we know two different ways to write the given sum: 10000x and 4950+x

As these two expressions are equal, we solve the following equation for x, in the usual way.

10000x = 4950 + x

Subtract x from each side

10000x - x = 4950 + x - x

If you have 10,000 identical objects, and you take one away, you're left with 9,999 objects.

9999x = 4950 + x - x

(You already know that x minus x is zero.)

9999x = 4950

Divide each side by 9999, to solve for x.

the - is not a negative just to eradicate any confusion

A dash is always typed (using keyboard characters) as two hyphens -- like that.

Or, one could simply omit the dash in your post, as it's unnecessary in the phrase where it appears. (We see the equation, after all.)

Cheers

 

[bjuveges]

We know the following.

"Since the average is , and there are terms …"

In other words, if you take 100 instances of the number 100x and add them all together, you get 100 times 100x, which is 10000x.

(It is dividing this total by 100 that gives the average 100x.)

So, we know two different ways to write the given sum: 10000x and 4950+x

As these two expressions are equal, we solve the following equation for x, in the usual way.

10000x = 4950 + x

Subtract x from each side

10000x - x = 4950 + x - x

If you have 10,000 identical objects, and you take one away, you're left with 9,999 objects.

9999x = 4950 + x - x

(You already know that x minus x is zero.)

9999x = 4950

Divide each side by 9999, to solve for x.




A dash is always typed (using keyboard characters) as two hyphens -- like that.

Or, one could simply omit the dash in your post, as it's unnecessary in the phrase where it appears. (We see the equation, after all.)

Cheers

Haah true, I'll remember for next time! And thanks for clearing up my confusion

 

[guitarguy]

Maybe this is helpful. The average of the numbers 1 through 99 is 50. 1 and 99 average to 50, 2 and 98 average to 50, 3 and 97 average to fifty and so forth plus the remaining 50.

So this number, 50, and x equal 100x.

Therefore, 50 + x = 100x

solving to 50/101.

 

[lookagain]

Maybe this is helpful. The average of the numbers 1 through 99 is 50. 1 and 99 average to 50, 2 and 98 average to 50,
3 and 97 average to fifty and so forth plus the remaining 50.

So this number, 50, and x equal 100x.

Therefore, 50 + x = 100x \(\displaystyle \ \ \ \) <-----

solving to 50/101.

Your last line doesn't follow from the line just above it.

It would lead to

50 = 99x ==>

50/99 = x \(\displaystyle \ \ \ \)if the indicated line in the quote box were true.