Hi, there--

QUESTION:
I am subtracting two three digit numbers. 771-252. The 1 becomes an 11, 11-2=9 bring the 
7 in the tens column to a 6, 6-5=1. Now I don't bring the 7 in the hundreds column down to 
a 6. Why is that? In 916-618, the 6 becomes a 16, the 1 is brought down to a zero and 
becomes a 10, and the 9 is brought down to an 8. Why must all the numbers in 916 be 
regrouped but not all in 771?

SOLUTION:
This is a great question. It shows that you won't accept the mathematics when it doesn't make 
sense. Excellent! Here is one way to think about this:

Sometimes people will expand the numbers (into ones, tens, and hundreds) so they see what's 
happening on at a place value level:

771 is equivalent to 700 + 70 + 1
252 is equivalent to 200 + 50 + 2

Now let's subtract the ones. We want to subtract 2 from 1. Here we need to regroup (borrow) 
because we cannot take 2 things from 1 thing without going in the hole. (For example, I can't 
borrow $2 from you if you only have $1.)

771 = 700 + 60 + 11
252 = 200 + 50 +  2

ONES: Subtract 11 - 2 = 9 in the ones column.
TENS: Subtract 60 - 50 = 10 in the tens column. We don't need to regroup here because we 
can take 50 form 60 with out going in the hole. (For example, I can borrow $50 from you if 
you have $60.)
HUNDREDS: Subtract 700 - 200 = 500 in the hundreds column.

Putting this back together, we have 500 + 10 + 9 = 519.

Here's how it looks vertically,

771 = 700 + 70 + 1 = 700 + 60 + 10
252 = 200 + 50 + 2
-----------------
(regroup from tens column to ones column. We take 1 group of ten from the tens column 
and put in the ones column.)

Subtract:
771 = 700 + 60 + 11
252 = 200 + 50 + 2
-----------------
519 = 500 + 10 + 9

In your second example, 916-618, you do have to regroup (borrow) in each column. Let's 
break the numbers down to show their place value. 

916 = 900 + 10 + 6
618 = 600 + 10 + 8

ONES: We cannot take 8 things from 6 things without going in the hole, so we need to regroup. 
When we take 10 from the tens column we have 0 left because 10-10=0.

916 = 900 + 0 + 16
618 = 600 + 10 + 8

TENS: We cannot take 10 things from 0 things without going in the hole, so we regroup again 
from the hundreds column to the tens column.

916 = 800 + 100 + 16
618 = 600 + 10 + 8

Now we are ready to subtract.
ONES: 16-8 is 8.
TENS: 100-10 is 90.
HUNDREDS: 800-600 is 200.

Vertically, we have 
916 = 800 + 100 + 16
618 = 600 + 10 + 8
--------------------
298 = 200 + 90 + 8

I hope this helps. You can email me if you still have questions about this,

Mrs.Figgy
math.in.the.vortex@gmail.com