Alright this might sound stupid. Why do we expand by 10?

Question

Specifically the math problem 90.5b+21*92/21+b=91.2. The math calculator on the internet kept giving me errors when I put it in like that. So I changed it to 90.5b+1932/21+b=91.2, and it gave me an answer with step by step so I could follow along. Honesty first time hearing distributive property in a math problem, but easy enough to understand once I saw what it did for the problem. What I don't quite understand is the second to last step of the problem. It says expand by 10. So my question is why? Where does this 10 come from? Without this 10 the answer is 2.4, but with it it becomes 24. This math problem I got from a word problem involving students, but even just numbers it answers 24. Which brought up the questions in bold and italic. I'm not to bright so treat this like you're speaking to a 4th grader please? Incase anyone is wondering since I messed up even after seeing the steps (distributive property got me) the steps are

                         90.5b+(21*92)/21+b=91.2

                         90.5b+1932/21+b=91.2

                         90.5b+1932(/21+b*21+b)=91.2(21+b)

                 90.5b+1932=(91.2*21)+(91.2*b) (this is that distributive property part)

                         90.5b+1932=1915.2+91.2b

                        (90.5b-91.2b)+1932=1915.2+(91.2b-91.2b)

                         -0.7b+1932=1915.2

                         -0.7b+(1932-1932)=1915.2-1932

                         -0.7b=-16.8

(This is the part I don't get) -0.7x10=-16.8x10 (not sure why the * didn't work in final draft)

                         -7b=-168

                         -168/-7=b

                          b=24

I know a lot of people are smarter than me, and didn't need the step by step. But hey it's there so if it helps great. Thanks for your time on what is probably an easy answer.

Answer 1

Why you multiply by 10: to make the numbers easier to calculate.  It's as simple as that.

Dividing whole numbers is easier than dividing fractions; since both of the numbers involved has one decimal place, then multiplying by 10 will turn them into integers.  (If one of them had two decimal places, you'd need to multiply them by 10² = 100 to get integers; and so on.)

It's not strictly needed: you'd get the same end result without multiplying by 10.  But it's very easy to do, and makes the division easier.

(And why you can multiply by 10: because you're multiplying both sides by the same number, so the equality still holds.)

Answer 2

I'm not sure what I am looking at in that equation. Looking at the first equation in the post, I see $90.5b+21\times\frac{92}{21}+b=91.2$. This from your post "90.5b+21*92/21+b=91.2" And this yields something like -.008. I think you might have left out some parenthesis.