The sum of two consecutive integers is less than 65. Find the pair of integers with the greatest sum.

Answer 1

Learn to solve problems using the rules of inequalities.

Let the consecutive integers be: x and (x+1)

"The sum of two consecutive integers is less than 65"

x + (x+1) < 65

2x + 1 < 65

2x < 64 [positive 1 may be subtracted from both sides without changing inequality]

x < 32 [divide both sides by positive 2 and retain sense of ineuality]

The "greatest sum" uses the greatest integer value of x that is less than 32; that is, 31.

Check:

Is 31 + 32 < 64 ?

31 + 32 < 64 ?

63 < 64 ?yes

Answer 2

The pair is 31, 32

If you let x be the largest number in the two consecutive number and x - 1 be the small number in the two consecutive numbers then you can solve for x as follows

x = larger consecutive number

x - 1 = smaller consecutive number (the integer just before x)

x - 1 + x < 65

2x -1 < 65

Set 2x - 1 equal to 65 to find the limit on the larger consecutive number

2x - 1 = 65

Add 1 to both sides

2x = 65 +1

2x = 66

Divide both side of the equation by 2

x = 33

Going back to the original problem, x cannot be 33 or larger. 33 and the preceding consecutive integer (x -1) is 32 these sum to 65. X must be an integer less than 33. X can be less than or equal to 32.

x = 32

x - 1 = 32 -1 = 31

31 + 32 = 63

The pair of consecutive integers with the greatest sum is 31, 32.

32 + 33 is too much

32 + 32 does not represent consecutive integers.

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