Transcript Consecutive Numbers

Translating & Solving
Word Problems
Some Hints:
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We are going to cover several different types
of word problems in this unit.
One of the keys to success will be the ability to
recognize word problems by type and apply an
appropriate problem-solving strategy to them.
Motion problems lend themselves to the use of
one strategy while mixture problems require a
slightly different approach.
There are several basic principles that can be
applied to most any word problem and we will use
them as we move from one type to another.
How To Solve Word Problems
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Read the problem several times carefully. Looking
for key ideas. Guess the answer. Write down
your guess on scratch paper and see if it fits
what the problem is asking. Guessing brings
common sense into play immediately.
Define “x”, as the answer to the question. For
example, if the question is: “How many pizzas did
Dawn order?” then let x equal the number of
pizzas Dawn ordered.
Label other unknown quantities in terms of x. For
example, if you know from reading the problem
that Barry ordered 3 more pizzas than Dawn, let
x+3 equal the number of pizzas Barry ordered.
How To Solve Word Problems
cont.
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Form an equation. This is a good time to read
the problem again to be sure that you are
making use of all important information.
Solve the problem. This may require solving an
equation or making a table or counting
something.
Answer the question. Compare your answer to
your original guess. This may keep you from
entering an answer that may not be correct due
to a mistake in setting up the problem. You may
then be able to trace your mistake and come up
with an answer that seems reasonable.
* Lets work some problems to see how these strategies work.
MYSTERY NUMBERS
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MYSTERY NUMBERS
1) The sum of three consecutive integers is 39.
Find the three integers.
What are the Integers?
{…-4, -3, -2, -1, 0, 1, 2, 3, 4, …}
What does consecutive mean?
Next to, such as 1 next to 2, -2 next to -1
What is the difference between two consecutive
integers?
1 unit
1) The sum of three consecutive integers is 39.
Find the three integers.
How do we do this problem algebraically?
Let n = first number
n = 12,
then
n + n+1 + n+2 =39
3n + 3 = 39
3n = 36
n = 12,
n + 1 = second number
n+1 = 13,
n + 2 = third number
n + 2 = 14
Consecutive Odd or Even
Odd Integers are {…-3, -1,1,3,5,…}
The difference between each consecutive pair is
2 units. Thus we would set up three consecutive
odd integers as n, n+2, and n + 4.
Even Integers are { …-6, -4, -2, 0, 2,4, …}
The difference between each consecutive pair is
2 units also. Thus we would set up three
consecutive even integers exactly like we did the
odd: n , n + 2, n + 4
Finding mystery numbers
A mystery number is a number that has
been changed to a new number by using
one of the four operations: addition,
subtraction, multiplication, and division.
For example:
If twice our number is 10 what is
our mystery number?
In other words 2 * ? = 10
To find the answer divide
We use division, the operation that
undoes multiplication to solve this
problem.
So
Likewise
if the sum of a number and 10 is 25.
We would use subtraction to undo
addition.
Undoing operations
So to guess a mystery number we
would use the opposite operation.
Operation
+ Addition
- Subtraction
 Multiplication
 Division
Opposite
operation
- Subtraction
+ Addition
 Division
 Multiplication
More mystery numbers
Sometimes it is harder to find a
mystery number because more than
one operation is used on the number.
In this case we need to use our
problem-solving process. Let’s see
how it works on an example.
Fifty-two is 2 less than 6 times a number.
Find the number.
See if you can guess the answer before you do it
algebraically. Write your guess down.
Define x.
x = the number we are looking for.
Label the other unknown quantities in terms of x.
6 times a number is 6x
2 less than 6 times a number is 6x - 2
Most of the words in a sentence will translate
directly into an equation. But the words “less
than” reverse things.
Next form an equation.
Fifty-two is 2 less than 6 times a number.
52
=
6 x
-
2
Verbs like “is” are where you place the “=” sign
Solve the equation: 52 = 6x - 2 add 2 to both sides
52 + 2 = 6x -2 +2 -2 + 2 = 0
54 = 6x Divide by 6 on both sides
9=x
HANDOUT
Click the back button when you finish.
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Problem 1: Find three consecutive
integers that add to 180.
(You may want to go to the end of this presentation to use
the LiveMath program to guess the answer. But first read on.)
Define x
Let x = the first integer.
2. Label other unknown quantities in terms of x
Let x + 1= the second integer
Let x + 2 = the third integer
3. Form an equation
First Integer +Second Integer + Third Integer = 180
x + (x+1) + ( x+2) = 180
Go to next slide.
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4.
Solve the equation.
x + (x+1) + (x+2)= 180
3x + 3= 180 Add Like Terms
3x = 177 Subtract 3 from both sides.
x = 59 Divide by 3 on both sides.
5. Answer the question.
x = 59
x + 1= 60
x + 2= 61
Check to see if your answer has a sum of 180.
59 + 60 + 61 = 180
180 = 180
How does this compare to your original guess?
2) The larger of two integers is 8 less than twice the
smaller. When the smaller number is subtracted from the
larger, the difference is 17. Find the two numbers.
Let x = smaller
8 less than twice the smaller is written
2x - 8
Note: when you see “less than” reverse
The order when you write them down.
2x – 8 is the larger number
Now subtract the smaller from the larger
2x - 8 - x = 17 then solve
x - 8 = 17
x = 25 Next, find the larger by substituting the answer
into 2x – 8
2(25) - 8 = 50 - 8 = 42
The two numbers are 25 and 42.
Try these problems:
1. Five times an unknown number decreased by 7 is 43. Find the
number.
2. When 4 is subtracted from half of an unknown number the result
is 17. Find the number.
3. The sum of 6 and twice an unknown number is 32. Find the
number.
4. Fifty-two is 2 less than 6 times a number. Find the number.
5. Eight less than 5 times a number is 57. Find the number.
6. Eight less than 5 times a number is four more than 8 times the
number. Find the number.
7. The sum of two numbers is 30. Three times the first plus twice
the second number is 72. Find the numbers.
8. The sum of two numbers is 40. One number is one more than
twice the other. Find the numbers.
9. The sum of two consecutive even integers is 106. Find the
integers.
10. The sum of two consecutive odd integers is –40. Find the
integers.
11. Find three consecutive integers if twice the largest is 16 less
than three times the smallest.
Complete solutions follow this slide, so work these first before
moving to the next slide.
Complete solutions to Mystery Numbers: 1 – 11.
1. Five times an unknown number decreased by 7 is 43. Find the number.
Read the problem several times. Guess the answer.
You might be able to get this one with just a couple of guesses.
Define x, usually to answer the question.
Let x = the number.
Label other unknown quantities in terms of x.
This step is not necessary on this problem.
Form the equation.
5x – 7 = 43
Solve the equation.
5x – 7 = 43
5x = 50
x = 10
Answer the question.
The number is 10. Check to see that 5 times this number decreased by 7 is 43.
2. When 4 is subtracted from half of an unknown number the result is 17.
Find the number.
a.
Read the problem several times. Guess.
If we guess 40 then half of 40 is 20 and subtracting 4 from 20 leaves us with
16…pretty close.
b.
Define x, usually to answer the question.
Let x = the number.
c.
Label other unknown quantities in terms of x.
There are no other unknowns in this problem.
d.
Form the equation.
(1/2)x – 4 = 17
e.
Solve the equation.
(1/2)x – 4 + 4 = 17 + 4
(1/2)x = 21
(2)(1/2)x = 21(2)
x = 42
f.
Answer the question.
Check to see that 42 behaves as advertised.
3. The sum of 6 and twice an unknown number is 32. Find the number.
Let x = the number.
Form an equation.
2x + 6 = 32
Solve the equation.
2x + 6 = 32
2x = 26
x = 13
Answer the question.
Check to see that the number 13 is correct.
4. Fifty-two is 2 less than 6 times a number. Find the number.
Let x = the number.
Form an equation.
52 = 6x – 2. Some people are tempted to try 52 = 2 – 6x. Be especially
careful when translating subtraction.
Solve the equation.
52 = 6x – 2
54 = 6x
9 = x Answer the question. The mystery number is 9.
5. Eight less than 5 times a number is 57. Find the number.
Let x = the number
5x – 8 = 57
5x – 8 = 57
5x = 65
x = 13
Answer the question.
To check, 5 times 13 is 65 and 8 less than 65 is 57.
6.
Eight less than 5 times a number is four more than 8 times the number. Find the number.
Let x = the number.
5x – 8 = 8x + 4
Solve the equation.
5x – 8 = 8x + 4
-3x = 12
x = -4
Answer the question.
Is 5*(-4) – 8 = 4 + 8*(-4 )
-20 – 8 ? 4 – 32
-28 = -28 yes it checks
7. The sum of two numbers is 30. Three times the first plus twice the
second number is 72. Find the numbers.
Let x = the first number.
Label other unknown quantities in terms of x.
Let 30 – x = the other number. Since the numbers have to add to 30, if we
subtract one number (x) from 30 we will get an algebraic name for the other.
Form an equation.
3x + 2(30 – x) = 72
Solve the equation.
3x + 60 – 2x = 72
x = 12
Answer the question.
If x = 12 then 30 – x = 30 – 12 = 18. The two numbers are 18 and 12.
8.
The sum of two numbers is 40. One number is one more than
twice the other. Find the numbers.
Let x = one of the numbers
Label other unknown quantities in terms of x.
Let 40 – x = the other number.
Form an equation.
x = 2(40 – x) + 1
Solve the equation.
x = 2(40 – x) +1
x = 80 – 2x + 1
3x = 81
x = 27
Answer the question.
If x = 27 then 40 – x = 40 – 27 = 13.
9. The sum of two consecutive even integers is 106. Find the integers.
Let x = the number
Label other unknown quantities in terms of x.
Let x + 2 = the second even integer.
Form an equation.
x + (x + 2) = 106
Solve the equation.
x + (x + 2) = 106
2x + 2 = 106
2x = 104
x = 52
Answer the question.
If x = 52, then x + 2 = 54.
10. The sum of two consecutive odd integers is – 40. Find the integers.
Let x = the first odd integer.
Label other unknown quantities in terms of x.
Let x + 2 = the second odd integer.
Form an equation.
x + (x + 2) = - 40
Solve the equation.
2x + 2 = - 40
2x = - 42
x = -21
Answer the question.
If x = -21, then x + 2 = -19. Note that the two integers do add to – 40.
11. Find three consecutive integers if twice the largest is 16 less
than three times the smallest.
Let x = the first integer
Let x + 1 = the second integer.
Let x + 2 = the third integer.
Form an equation.
2(x + 2) = 3x - 16
Solve the equation.
2(x + 2) = 3x – 16
2x + 4 = 3x – 16
4 = x – 16
20 = x
Answer the question.
If the first integer is 20 then the next two will be 21 and 22.