Many problems require the student to use common sense to solve them—that is, mathematical reasoning. For instance, when a problem refers to consecutive integers, the student is expected to realize that any two consecutive integers differ by one. If two numbers are consecutive, normally x is set equal to the first and x + 1, the second.
Examples
The sum of
two consecutive integers is 25. What are the numbers?
Let x = first number.
x + 1 = second number
Their sum is
25, so x + (x + 1) = 25.
The first
number is 12 and the second number is x + 1 = 12 + 1 = 13.
The sum of
three consecutive integers is 27. What are the numbers?
Let x = first number.
x + 1 = second number
x + 2 = third number
Their sum is
27, so x + (x + 1) + (x + 2) = 27.
The first
number is 8; the second is x + 1 = 8 + 1 = 9; the third is x + 2 = 8 + 2 = 10.
Practice
1.
Find two consecutive numbers whose sum is 57.
2.
Find three consecutive numbers whose sum is 48.
3.
Find four consecutive numbers whose sum is 90.
Solutions
1. Let
x = first number.
x
+ 1 = second number
Their
sum is 57, so x + (x + 1) = 57.
The
first number is 28 and the second is x + 1 = 28 + 1 = 29.
2. Let
x = first number.
x
+ 1 = second number
x
+ 2 = third number
Their
sum is 48, so x + (x + 1) + (x + 2) = 48.
The
first number is 15; the second, x + 1 = 15 + 1 = 16; and the third, x + 2 = 15 +
2 = 17.
3. Let
x ¼ first number.
x
+ 1 = second number
x
+ 2 = third number
x
+ 3 = fourth number
Their
sum is 90, so x + (x + 1) + (x + 2) + (x + 3) = 90.
The
first number is 21; the second, x + 1 = 21 + 1 = 22; the third, x + 2 = 21 + 2 =
23; and the fourth, x + 3 = 21 + 3 = 24.
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