SOLUTION: The sum of two whole numbers is 45 and their difference is less than 10. The numbers of all possible pairs is?

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Question 1163645: The sum of two whole numbers is 45 and their difference is less than 10. The numbers of all possible pairs is?
Found 2 solutions by greenestamps, ikleyn:
Answer by greenestamps(13203)   (Show Source): You can put this solution on YOUR website!


What do you need help with...?

Make a list of all pairs of whole numbers with a sum of 45 and identify the pairs in which the difference is less than 10.
  44  1
  43  2
  42  3
  41  4
 ...
  30 15
  29 16
  28 17
  27 18
  26 19
  25 20
  24 21
  23 22

The problem doesn't talk about ORDERED pairs, so I am assuming the pair 22 and 23 is the same as the pair 23 and 22; so you can stop there.

It is easy to see that there are 5 pairs of numbers that satisfy the given condition.

ANSWER: 5 pairs

Here is a solution using logical reasoning and a few simple mental calculations to find the answer, without writing out all those pairs.

(1) The sum of two whole numbers is 45; that means one of them is odd and one is even.
(2) That means the difference between the two numbers is odd.
(3) The difference has to be less than 10; there are 5 positive odd integers less than 10.

ANSWER: 5 pairs
Answer by ikleyn(52832)   (Show Source): You can put this solution on YOUR website!
.

The possible pairs are (x,45-x)  with  the conditions

    x is whole, 45 -x is whole and  | x - (45-x) | < 10.      (1)


I use the absolute value inequality, based on the context, saying "their difference", which means both possible differences.



In other words, the numbers are (x,45-x), where

    x is integer, 0 <= x <= 45  and  -10 < 2x -45 < 10.       (2)



The last compound inequality means

     -10 + 45 < 2x < 10 + 45.

     35       < 2x < 55

     17.5     < x < 27.5.                                     (3)


There are 10 integer solutions for the last compound inequality

     x = 18, 19, 20, 21, 22, 23, 24, 25, 26, 27.



They create 10 pairs

     (18,27), (19,26), (20,25), (21,24), (22,23), (23,22), (24,21), (25,20), (26,19), (27,18).



ANSWER.  There are 10 pairs satisfying given conditions, listed above.

Solved.



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