SOLUTION: The sum of two whole numbers is 45 and their difference is less than 10. The number of all possible pairs is: WAN.Algebra.com A. 4 B. 5 C. 6 D. 7

Question 1203135: The sum of two whole numbers is 45 and their difference is less than 10. The number of all possible pairs is: WAN.Algebra.com
A. 4
B. 5
C. 6
D. 7
Found 3 solutions by Theo, greenestamps, ikleyn:
Answer by Theo(13342)   (Show Source): You can put this solution on YOUR website!
the set of whole numbers is all positive integers plus 0.
you are given that x + y = 45 and x - y < 10
from the first equation, solve for y to get y = 45 - x
in the second equation, replace y with 45 - x to get x - (45 - x) < 10 which becomes 2x - 45 < 10 which becomes 2x < 55 which becomes x < 27.5.
since x has to be a whole number, you get x <= 27.
when x = 27, y = 45 - 27 = 18 and x - y becomes 27 - 18 = 9
when x = 26, y = 45 - 26 = 19 and x - y becomes 27 - 19 = 7
working down the line, you get:
when x = 25, x - y = 5
when x = 24, x - y = 3
when x = 23, x - y = 1
when x = 22, x - y = -1 ***** this is not a whole number.
you run out of x - y being a whole number after x = 23, so you have to stop there.
your total of possible values of x - y < 10 is equal to 5.
that's your solution.

Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!

Solving the problem using formal algebra is a bit awkward; and finding a solution using logical reasoning and simple arithmetic is possibly of more benefit.

You need to find how many pairs of whole numbers have a sum of 45 and a difference of less than 10.

So start with the pair of whole numbers with a sum of 45 that has the smallest possible difference. Simple analysis and simple arithmetic shows that pair to be 23 and 22.

Then find other pairs of whole numbers with a sum of 45 and larger differences by adding 1 to the larger number and subtracting 1 from the smaller number; continue until the difference is greater than 10.

23 and 22 (difference 1)
24 and 21 (difference 3)
25 and 20 (difference 5)
26 and 19 (difference 7)
27 and 18 (difference 9)

ANSWER: 5 pairs

An observant student will realize that the difference between the two whole numbers will change by 2 from one pair to the next, because you are adding 1 to one of the numbers and subtracting 1 from the other. So that student will be able to reach the answer of 5 without having to write out all the pairs.

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Final note....

The problem is, technically, not posed correctly. The number of pairs of whole numbers with a sum of 45 and a difference of less than 10 is much greater than 5.

For example, one such additional pair is 6 and 39. Their sum is 45, and their difference is 6-39 = -33, which is less than 10.

For the problem to be written correctly, it has to say that the difference of the two numbers is A WHOLE NUMBER less than 10.

Note that the other tutor, in making his list of pairs, stopped when the difference between the two whole numbers became negative, saying that he stopped because negative integers are not whole numbers.

But the statement of the problem didn't say that the difference had to be a whole number....

Answer by ikleyn(52781)   (Show Source): You can put this solution on YOUR website!
.

This problem is posed incorrectly and is SELF-CONTRADICTING.

If to understand it precisely as it is written, then the set of all possible solutions is in this table below

n First Second The The The number number sum difference pairs --------------------------------------------------------- 1 23 22 45 1 (23,22) 2 24 21 45 3 (24,21) 3 25 20 45 5 (25,20) 4 26 19 45 7 (26,19) 5 27 18 45 9 (27,18) 6 22 23 45 -1 (22,23) 7 21 24 45 -3 (21,24) 8 20 25 45 -5 (20,25) 9 19 26 45 -7 (19,26) 10 18 27 45 -9 (18,27) 11 17 28 45 -11 (17,28) 12 16 29 45 -13 (16,29) 13 15 30 45 -15 (15,30) 14 14 31 45 -17 (14,31) 15 13 32 45 -19 (13,32) 16 12 33 45 -21 (12,33) 17 11 34 45 -23 (11,34) 18 10 35 45 -25 (10,35) 19 9 36 45 -27 ( 9,36) 20 8 37 45 -29 ( 8,37) 21 7 38 45 -31 ( 7,38) 22 6 39 45 -33 ( 6,39) 23 5 40 45 -35 ( 5,40) 24 4 41 45 -37 ( 4,41) 25 3 42 45 -39 ( 3,42) 26 2 43 45 -41 ( 2,43) 27 1 44 45 -43 ( 1,44)

As you see, there are 27 solutions, which number 27 is not in your list of possible answers.

All 27 pairs in the last column are different and satisfy the problem's requirement.

Therefore, the conclusion is: the problem formulation is FATALLY and TOTALLY wrong.

I clearly understand that this problem, as presented in this post, migrates in the Internet
from site to site during the years, while its right place is in GARBAGE BIN.

I saw this problem (or similar, or in similar formulations) at this forum years ago,
and saw discussions of it by different tutors.

My point of view is that such GIBBERISH does not deserve any discussion.
What it really deserves, is to be removed/deleted from the Internet.

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