Welcome to two examples of decomposing numbers additively.
Decomposing number additively means to break a number into two or more groups so that the
combined amount in the groups is equivalent to the original amount.
In our first example, Jen and Dan pooled their money together and had a total of 16 1 dollar
bills as shown here below.
Determine three of the possible amounts they may have originally had if each contributed
at least two dollars and each contributed a different amount.
So based upon these restrictions, we will not have a contribution of one dollar or two
contributions of eight dollars.
So as one possibility, let's say Jen contributes 4 dollars.
If Jen contributed 4 dollars, because we know the total must be sixteen dollars, Dan contributes
twelve dollars.
Another way to approach this would be to find two whole numbers that meet these restrictions
and have a sum of 16.
Another possibility would be Jen contributes 11 dollars.
Because 11 plus 5 is equal to 16, Dan contributes 5 dollars.
To model this, Jen's contribution would be 11 dollars.
And Dan's contribution would be 5 dollars.
And for the last row, let's just say Jen contributes 3 dollars, because 3 plus 13 is equal to 16,
we know Dan contributes 13 dollars.
Let's look at a second example.
For this example there are three people.
Jen, Dan, and Paul pooled their money together and had a total of 21 $1 bills, again as shown
below.
Determine three of the possible amounts they may have originally had in each contributed
at least $2 and each contributed a different amount.
So one possibility would be that Jen contributes let's say $5.
Dan contributes $6.
And therefore Paul contributes $10.
Notice $5 plus $6 plus $10 is $21.
Another way to approach this, would be to find 3 whole numbers that satisfy these two
conditions and have a sum of 21.
For example, if Jen contributes $3 and Dan contributes $5, 3 plus 5 is equal to 8.
Because 8 plus 13 is equal to 21, Paul contributes $13.
If we wanted to model this, here are the $3 that Jen contributes.
Here are the $5 that Dan contributes.
And here are the $13 that Paul contributes.
And finally for the third row, let's say Jen contributes $15 and Dan contributes $3, 15
plus 3 is equal to 18, 18 plus 3 is equal to 21.
But we can't have a contribution of $3 from Dan and Paul because we are told the contributions
are of different amount.
Let's go back and change Jen's contribution to $14.
Well 14 plus 3 is equal to 17, 17 plus 4 is equal to 21.
And therefore, Paul contributes $4.
14 plus 3 plus 4 equals 21.
Each contribution is at least $2 and all the contributions are a different amount.
I hope you found this helpful.