Multiplying numbers from One set with another set

[brisk11228]

Homework Statement

Bob makes two sets: one with all the even integers between 1 and 30 inclusive, and another with all the odd integers inclusive. He called the sets Q and R. He multiplied each number from Q with each number in R. Then he added the 225 products together and called the result "S". What number does S represent?

(A) 8,000 (B) 18,000 (C) 36,000 (D) 54,000 (E) 86,000

Homework Equations

The Attempt at a Solution

Set Q = {1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29}
Set R = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30}

I noticed if I took 1 from Q and multiplied with every number in R, I get multiples of 2 from 2 to 30
Likewise, if I took 3 from Q and multiplied with every number in R, I get multiples of 6 from 6 to 90 and so on.

# from Q Range Sum (Added first term and last term and multiplied by 7 and then added the middle #)
1 2 to 30 32 * 7 + 16 = 240
3 6 to 90 96 * 7 + 48 = 720
5 10 to 150 160 * 7 + 80 = 1200
7 14 to 210 224 * 7 + 112 = 1680
9 18 to 270 288 * 7 + 144 = 2160

I stopped here because I noticed the sums were odd multiples of 240.

So then I found the first term which is 240 and the last term which would be 240 * 29 = 6960. The middle # here would be 3600.

6960 + 240 = 7200. There are 7 pairs of 7200 and the left out middle number 3600, thus the sum would be 54,000.

I was wondering if there is a shorter or easier or alternative method of solving this problem.

 

[LCKurtz]

Yes. Multiply the sum of the odds by the sum of the evens. 225x240 = 54000.

 

[mfb]

And there is a formula you can use for those sums to make it even faster.