Solved Examples: Consecutive Integers Problems
Example 1: John has a wooden board that is 5 feet long. He plans to make 4 shelves whose lengths are to be a series of consecutive even numbers. Find the length of each shelf in inches?
Solution:
Step 1: We know that consecutive even integers are even integers that follow each other by a difference of 2.
Let x = length of first shelf
x + 2 = length of second shelf
x + 4 = length of third shelf
x + 6 = length of fourth shelf
Step 2: Converting feet to inches
1 feet = 12 inches
Hence, 5 × 12 = 60 inches
Step 3: adding the 4 consecutive integers equal to 60
x + x + 2 + x + 4 + x + 6 = 60
Combine like terms
4x + 12 = 60
Isolate variable x
4x = 60 – 12
4x = 48
x = 12
Step 4: substitute the values of x
length of first shelf = x = 12 inches
length of second shelf = x + 2 = 14 inches
length of third shelf = x + 4 = 16 inches
length of fourth shelf = x + 6 = 18 inches
Therefore, the lengths of the shelves should be 12inch, 14 inch, 16 inch and 18 inch.
Let’s check our answer
12 + 12 + 2 + 12 + 4 + 12 + 6 = 60
Example 2: If the sum of three consecutive integers is 81, Find the three integers and then find what is the product of the first and the third integer?
Solution:
Step 1: Let us assume the three consecutive integers: x, x + 1 and x + 2
Step 2 : Now, as given
x + x + 1 + x + 2 = 81
3x + 3 = 81
3x = 81 – 3
3x = 78
x = 78/3
x = 26
x + 1 = 27
x + 2 = 28
Product of the first and third integer = 26 × 28 = 728.