Transcript consecutive integers - Algebra 1 -

– ALGEBRA I –
Unit 1 – Section 2
Consecutive Integer Problems
In order to work with a
“consecutive integer” problems,
we need to start by
understanding the terminology:
Consecutive
Consecutive means: “In a
row” or “In order.”
– ALGEBRA I –
Unit 1 – Section 2
Consecutive Integer Problems
In order to work with a
“consecutive integer” problems,
we need to start by
understanding the terminology:
Integer
An integer is: a nice, round,
positive/negative number.
– ALGEBRA I –
Unit 1 – Section 2
Consecutive Integer Problems
Let’s go
through
some
examples.
The key thing to remember is
that your answers will be
consecutive integers. In other
words, the numbers you get
should be “nice” (a.k.a. no
fractions or decimals) and they
should be in a row.
Example Problems
Consecutive Integer Problem
The sum of three consecutive integers
is 51. Find the numbers
• The first thing to note is that we are dealing with
consecutive integers.
• An example (that is not necessarily the solution) of
consecutive integers could be:
20 21 22
+1
Notice that to get from the first number in the list
to the second, we need to add 1.
Example Problems
Consecutive Integer Problem
The sum of three consecutive integers
is 51. Find the numbers
• The first thing to note is that we are dealing with
consecutive integers.
• An example (that is not necessarily the solution) of
consecutive integers could be:
20 21 22
+1
+2
To get from the first number in the list to the third,
we need to add 2.
Example Problems
Consecutive Integer Problem
The sum of three consecutive integers
is 51. Find the numbers
• Instead of using numbers, we need to switch to
variables.
20 21 22
+1
+1
+2
+2
N N+1 N+2
• Note that we follow the same “addition” procedure.
Example Problems
Consecutive Integer Problem
The sum of three consecutive integers
is 51. Find the numbers
• Now to start the problem, we begin by writing the
expressions for the THREE integers:
N
N+1
N+2
• Since we are looking for the sum, the equation is:
N + N + 1 + N + 2 = 51
Example Problems
Consecutive Integer Problem
The sum of three consecutive integers
is 51. Find the numbers
• Suppose that the possible solutions for the first
number are 15, 16, or 17. Guess and test to solve:
(15) + (15) + 1 + (15) + 2 = 15 + 16 + 17 = 48  51
(16) + (16) + 1 + (16) + 2 = 16 + 17 + 18 = 51 = 51
(17) + (17) + 1 + (17) + 2 = 17 + 18 + 19 = 54  51
Example Problems
Consecutive Integer Problem
The sum of three consecutive integers
is 51. Find the numbers
• Since 16 is the solution for the FIRST number in the
list, the complete solution is:
N = 16 Notice that the 3
N + 1 = 16 + 1 = 17
N + 2 = 16 + 2 = 18
solutions are integers,
add up to 51, and are
consecutive.
Example Problems
Consecutive EVEN Integer Problem
The sum of three consecutive EVEN
integers is 84. Find the numbers
• The first thing to note is that we are dealing with
consecutive EVEN integers.
• An example (that is not necessarily the solution) of
consecutive integers could be:
20 22 24
+2
Notice that to get from the first number in the list
to the second, we need to add 2.
Example Problems
Consecutive EVEN Integer Problem
The sum of three consecutive EVEN
integers is 84. Find the numbers
• The first thing to note is that we are dealing with
consecutive integers.
• An example (that is not necessarily the solution) of
consecutive integers could be:
20 22 24
+2
+4
To get from the first number in the list to the third,
we need to add 4.
Example Problems
Consecutive EVEN Integer Problem
The sum of three consecutive EVEN
integers is 84. Find the numbers
• Instead of using numbers, we need to switch to
variables.
20 22 24
+2
+2
+4
+4
N N+2 N+4
• Note that we follow the same “addition” procedure.
Example Problems
Consecutive EVEN Integer Problem
The sum of three consecutive EVEN
integers is 84. Find the numbers
• Now to start the problem, we begin by writing the
expressions for the THREE integers:
N
N+2
N+4
• Since we are looking for the sum, the equation is:
N + N + 2 + N + 4 = 84
Example Problems
Consecutive EVEN Integer Problem
The sum of three consecutive EVEN
integers is 84. Find the numbers
• Suppose that the possible solutions for the first
number are 22, 24, or 26. Guess and test to solve:
(22) + (22) + 1 + (22) + 2 = 22 + 24 + 26 = 72  84
(24) + (24) + 1 + (24) + 2 = 24 + 26 + 28 = 78  84
(26) + (26) + 1 + (26) + 2 = 26 + 28 + 30 = 84 = 84
Example Problems
Consecutive EVEN Integer Problem
The sum of three consecutive EVEN
integers is 84. Find the numbers
• Since 26 is the solution for the FIRST number in the
list, the complete solution is:
N = 26 Notice that the 3
N + 2 = 26 + 2 = 28
N + 4 = 26 + 4 = 30
solutions are integers,
even, add up to 84,
and are consecutive.
Example Problems
Consecutive ODD Integer Problem
The sum of three consecutive ODD
integers is 57. Find the numbers
• The first thing to note is that we are dealing with
consecutive ODD integers.
• An example (that is not necessarily the solution) of
consecutive integers could be:
21 23 25
+2
Notice that to get from the first number in the list
to the second, we need to add 2.
Example Problems
Consecutive ODD Integer Problem
The sum of three consecutive ODD
integers is 57. Find the numbers
• The first thing to note is that we are dealing with
consecutive integers.
• An example (that is not necessarily the solution) of
consecutive integers could be:
21 23 25
+2
+4
To get from the first number in the list to the third,
we need to add 4.
Example Problems
Consecutive ODD Integer Problem
The sum of three consecutive ODD
integers is 57. Find the numbers
• Instead of using numbers, we need to switch to
variables.
21 23 25
+2
+2
+4
+4
N N+2 N+4
• Note that we follow the same “addition” procedure.
Example Problems
Consecutive ODD Integer Problem
The sum of three consecutive ODD
integers is 57. Find the numbers
• Now to start the problem, we begin by writing the
expressions for the THREE integers:
N
N+2
N+4
• Since we are looking for the sum, the equation is:
N + N + 2 + N + 4 = 57
Example Problems
Consecutive ODD Integer Problem
The sum of three consecutive ODD
integers is 57. Find the numbers
• Suppose that the possible solutions for the first
number are 17, 19, or 21. Guess and test to solve:
(17) + (17) + 1 + (17) + 2 = 17 + 19 + 21 = 57 = 57
(19) + (19) + 1 + (19) + 2 = 19 + 21 + 23 = 63  57
(21) + (21) + 1 + (21) + 2 = 21 + 23 + 25 = 69  57
Example Problems
Consecutive ODD Integer Problem
The sum of three consecutive ODD
integers is 57. Find the numbers
• Since 17 is the solution for the FIRST number in the
list, the complete solution is:
N = 17 Notice that the 3
N + 2 = 17 + 2 = 19
N + 4 = 17 + 4 = 21
solutions are integers,
odd, add up to 57,
and are consecutive.
Try This Problem…
Use the procedures that we just went
through to the solve the following
problems.
1. The sum of four consecutive even integers is
268. Find the numbers. (Possible solutions
for the first number: 60, 64, or 70)
2. The product of two consecutive odd integers is
143. Find the numbers. (Possible solutions
for the first number: 9, 11, or 13)
**The answers can be found at the end of the PowerPoint.
ALGEBRA
IS FUN
AND EASY!
**Answers: 1) 64, 66, 68, and 70 2) 11 and 13