Exponent shortcut clarification

[RyanC896]

SAMPLE QUESTION:
A certain country had a total annual expenditure of $1.2 x 10^12 last year. If the population of the country was 240 million last year, what was the per capita expenditure?

MY QUESTION:
The suggested solution simplifies this problem in multiple steps (as shown below). And I'm having a problem getting from step 1 to step 2. Specifically, around how the exponent on 10 in the numerator is reduced from 12 to 11 causing 1.2 to increase to 12. Is there a property/short cut that allows you to do this? If so, could someone elaborate on the property/shortcut so I could tell when and how to apply it? Thanks in advance for any insight!

Step 1
1.2 x 10^12
240 million

Step 2
12 x 10^11
24 x 10^7

[RonPurewal]

here, how about this: the number 1,200 is "1.2 thousand", which is the same thing as "12 hundred". (i'm betting this will make perfect sense, because "12 hundred" is the way most people pronounce this number in conversation.)

if you understand that example, then you're good to go here, because "thousand" (= 10^3) and "hundred" (= 10^2) have exactly the same relationship as do 10^12 and 10^11.

1.2 thousand = 12 hundred
1.2(1 thousand) = 12(1 hundred)
1.2(10^3) = 12(10^2)
just apply the same intuitive understanding to the bigger powers.

[RonPurewal]

if you're looking for a more "mathy" justification, you can also realize that
1.2 x 10^12
= 1.2 x (10 x 10^11)
= (1.2 x 10) x 10^11
= 12 x 10^11.

[RonPurewal]

or, you know what's a lot easier: just write out the numbers.

(1.2 x 10^12) / (240 million)
= (1,200,000,000,000) / (240,000,000)
= 1,200,000 / 240
= 120,000 / 24
= 10,000 / 2
= 5000

yes, there are some steps that can be condensed there, but, for me at least, it's more intuitive to eliminate the ",000"s before eliminating single zeroes.

incidentally, this is how i would do the problem, because i am like a giant 10-year-old (and i'm much more likely to catch mistakes with numbers that are actually numbers, rather than weirdo algebraic expressions).

[RonPurewal]

also—IF your first reaction to the above "grind" method ^^ is "oh, but that might take a long time"...

...NO.
no.
absolutely not.

people are rotten at guessing how long these processes will take. (that's "people" as in ALL people, myself and yourself included.) seriously, do not try to guess how long processes will take; rather, just pick up the proverbial shovel, and start digging.

for me personally, in fact, the last method is FASTER than tinkering algebraically with the powers. and, as i implied above, i can be more confident in the answer i get that way, too (since, by lining up the numbers, i can get an intuitive sense of how big the answer should be).

[RyanC896]

Ron, that is perfect. Thank you!

[RonPurewal]

no problem!