SOLVING WORD PROBLEMS INVOLVING CONSECUTIVE INTEGERS

Problem 1 :

If the average of three consecutive integers is 26, find the integers.

Solution :

Let x, (x + 1) and (x + 2) be the three consecutive integers.

Average of three consecutive integers = 26

Multiply both sides by 3.

3x + 3 = 78

Subtract 3 from both sides.

3x = 75

Divide both sides by 3.

x = 25

x + 1 = 26

x + 2 = 27

Therefore, the three consecutive integers are 25, 26 and 27.

Problem 2 :

If the average of three consecutive even integers is 14, find the integers.

Solution :

Let x, (x + 2) and (x + 4) be the three consecutive even integers.

Average of three consecutive even integers = 14

Multiply both sides by 3.

3x + 6 = 42

Subtract 6 from both sides.

3x = 36

Divide both sides by 3.

x = 12

x + 2 = 14

x + 4 = 16

Therefore, the three consecutive even integers are 12, 14 and 16.

Problem 3 :

Among the three consecutive odd integers, 26 less than thrice the largest integer is equal to twice the smallest integer. Find the integers.

Solution :

Let x, (x + 2) and (x + 4) be the three consecutive odd integers.

3(x + 4) - 25 = 2x

3x + 12 - 25 = 2x

3x - 13 = 2x

x = 13

x + 2 = 15

x + 4 = 17

Therefore, the three consecutive odd integers are 12, 14 and 16.

Problem 4 :

Find two consecutive natural numbers whose product is 30.

Solution :

Let x, (x + 1) be the two consecutive integers.

Product of two consecutive integers = 30

x(x + 1) = 30

x² + x = 30

x² + x - 30 = 0

x² - 5x + 6x - 30 = 0

x(x - 5) + 6(x - 5) = 0

(x - 5)(x - 6) = 0

x - 5 = 0  or  x + 6 = 0

x = 5  or  x = -6

Since the natural numbers are always positive, x can not be -6.

So,

x = 5

x + 1 = 6

Therefore, the required two consecutive natural numbers are 5 and 6.

Verification :

Product of two positive integer is 30.

5(6) = 30 

 30 = 30

Problem 5 :

There are three consecutive positive integers such that the sum of the square of first and the product of the other two is 154. Find the integers.

Solution :

Let x, (x + 1) and (x + 2) be the required three consecutive integers

The sum of the squares of first and the product of the other two is 154.

x² + (x + 1)(x + 2) = 154

x² + x² + 2x + x + 2 = 154

2x² + 3x + 2 = 154

2x² + 3x + 2 - 154 = 0

2x² + 3x - 152 = 0

2x² + 3x - 152 = 0

2x² - 16x + 19x - 152 = 0

2x(x - 8) + 19(x - 8) = 0

(x - 8)(2x + 19) = 0

x - 8 = 0  or  2x + 19 = 0

x = 8  or  x = -9.5

Since the numbers are positive integers, x can not be -9.5.

So,

x = 8

x + 1 = 9

x + 2 = 10

Therefore, the required three consecutive positive integers are 8, 9 and 10.

Verification :

The sum of the square of first and the product of the other two is 154.

8² + (9) (10) = 154

 64 + 90 = 154

 154 = 154

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