# Which is greater: 355*356 or 354*357?

• Aug 6th 2009, 03:01 PM
Voluntarius Disco
Which is greater: 355*356 or 354*357?
Well, I'm looking through a means of solving problems and I just want a little more explanation to this method of solving that question.
Represent 354=a, 357=b, 355=c, and 356=d
We now have that 1) a+b=c+d; 2) |b-a|=|d-c|
We want to prove that ab < dc
This is an elementary question maybe but...how did the problem solving leap from multiplication to addition for the first step?
Can anybody clear that up for me - on why it's a+b=c+d because...to me...it doesn't. And I'm not seeing it. D:
• Aug 6th 2009, 03:13 PM
CaptainBlack
Quote:

Originally Posted by Voluntarius Disco
Well, I'm looking through a means of solving problems and I just want a little more explanation to this method of solving that question.
Represent 354=a, 357=b, 355=c, and 356=d
We now have that 1) a+b=c+d; 2) |b-a|=|d-c|
We want to prove that ab < dc
This is an elementary question maybe but...how did the problem solving leap from multiplication to addition for the first step?
Can anybody clear that up for me - on why it's a+b=c+d because...to me...it doesn't. And I'm not seeing it. D:

$355\times 356= 355 \times (355+1)=355^2+355$

$354\times 357=(355-1)(355+2)=355^2+355-2$

so we conclude $355\times 356>354 \times 357$

CB
• Aug 6th 2009, 04:22 PM
colby2152
You could always solve for when this identity works too by setting up an inequality.

$n(n+1)>(n-1)(n+2)$

In our case, n is equal to 355, but let's expand the inequality to solve for n.

$n^2+n>n^2+2n-n-2$

$0>-2$

Now, unfortunately, we cannot solve for any n since that variable falls out of the equation. However, what this result does tell us is that the left side is greater by two which is exactly what CaptainBlack showed with numbers.