of these numbers which ones total $1,560.29

Question

Of these # which ones added together total $1560.29? 35.61 28.50 11.79 19.90 867.86 45.67 23.36 1095.00 480.58 102.00 527.60 101.77 223.89 302.38 41.09 392.48 978.10 63.15 27.59 25.00 118.69 198.86 58.61

Answer 1

Solution methods:

(1) Random – Select a number then compute the amount still needed to reach $1,560.29.  If more is needed, compare that to the remaining numbers and continue this process.  However, if no history records are kept, there will likely be rework and options left unconsidered.  This is not a “best practice” method.

 

(2) Table – Much like the elementary addition table, construct a two-dimensional matrix of all the sums.  Note that, like a simple additional table, only a diagonal half must be completed.  This is exhaustive for all of pairs of numbers that add to $1,560.29.  The rules did not specify pairs, so let’s think of more solutions.

 

(3) Math knowledge – Considering the numbers, realize that, in order to get a sum of $nnn.n9, the numbers must have cents that add to 9.  When a number is selected, this method eliminates many of the remaining numbers from being the very last number chosen.  If a number can still be chosen (and not exceed $1,560.29), then there is a different set of possible/eliminated remaining numbers.  Now, we are beginning to have enough procedures to need a computer.

 

(4) Sort the numbers high to low.  Add alternatives in a pattern that considers all possibilities.  This algorithm may abort when the “residual sum” (the sum of all the remaining numbers) is not enough to reach $1,560.29.  This is best done with a second array that keeps the sum of the remaining numbers (since they are sorted high to low).  This turns out to be a very easy computer program (with 23 numbers [0=omit;1=include], 2^23 is only 8388608 and is still sub-second for most PCs).  It is also a method that the student will see again in calculus.

 

The possibilities are:

48058  30238  19886  11869  10200  10177  6315  5861  4567  4109  2759  1990 

48058  39248  22389  19886  10177  4567  4109  2759  2500  2336 

48058  39248  22389  19886  10200  5861  3561  2500  2336  1990 

48058  39248  22389  19886  11869  6315  2759  2336  1990  1179 

48058  39248  30238  11869  10177  6315  4109  2500  2336  1179 

48058  39248  30238  19886  11869  3561  1990  1179 

52760  30238  19886  10200  10177  6315  5861  4567  4109  3561  2850  2336  1990  1179 

52760  30238  22389  19886  10200  5861  4567  4109  2850  1990  1179 

52760  30238  22389  19886  11869  6315  4567  2500  2336  1990  1179 

52760  39248  22389  11869  6315  5861  4567  4109  3561  2850  2500 

52760  48058  11869  10200  10177  5861  4567  4109  2759  2500  1990  1179 

52760  48058  19886  11869  10177  4109  3561  2850  2759 

52760  48058  22389  19886  10177  2759 

86786  22389  11869  10200  10177  5861  3561  2850  2336 

86786  48058  11869  4567  2759  1990 

86786  52760  4567  3561  2850  2336  1990  1179 

97810  11869  10200  10177  6315  5861  4567  3561  2500  1990  1179 

97810  30238  10177  4567  3561  2850  2500  2336  1990 

109500  19886  10200  5861  4567  2500  2336  1179 

109500  19886  10200  6315  4109  2850  1990  1179