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					3.1 Changing App Probs into Equations
How do different words and phrases relate to
mathematical operations?

What words do you know that indicate a
certain operation?

Refer to the chart on p 180


                                           1
3.1 Changing App Probs into Equations
One of the things you will be asked to do in
 this section is translate from words to
 symbols and from symbols to words:
Given a mathematical expression, provide
 an expression in words that would be
 equivalent:

 2x + 3

                                               2
Given a mathematical expression, provide an
expression in words that would be equivalent:

2x + 3                 • Three more than
                         twice a number
                       • The sum of twice a
                         number and three
                       • Twice a number,
                         increased by three
                       • Three added to twice
                         a number


                                                3
     Given an expression in words, provide a
mathematical expression that would be equivalent:

• Four less than three   • 3x – 4
  times a number
• Five more than twice   • 5 + 2x
  a number
• Three times the sum    • 3(x + 8)
  of a number and 8
• Twice the difference
  of a number and 4      • 2(x – 4)


                                                4
Consider this:           • 2(x – 4)
Two times a number
  minus 4
Which expression does
  that represent?        • 2x – 4
What if I read it with
  inflection and
  pauses?
Be careful!!!

                                      5
      Given a sentence in words, provide a
 mathematical equation that would be equivalent:

Three times the difference between a
 number and two is four less than six times
 the number
     3(x – 2) = 6x – 4
When four is added to a number, the sum
 will be twenty
     x +4 = 20

                                                   6
  Steps for solving a word problem
1. Read and reread the problem until you
   understand it (not until you know how to solve
   it, but until you understand what the story is.)
2. Take notes. (they only have to make sense to
   you, but they should include all the important
   information in the problem)
3. Define the variable (and any other unknown
   quanities.
4. Write an equation.
5. Solve the Equation.
6. Label your answer.
7. Ask yourself, Does my answer make sense?
                                                      7
3.1 Changing App Probs into Equations

In this section, you are learning how to pick
  out the single variable and how to define
  your unknown quantities.




                                                8
  Steps for solving a word problem
1. Read and reread the problem until you
   understand it (not until you know how to solve
   it, but until you understand what the story is.)
2. Take notes. (they only have to make sense to
   you, but they should include all the important
   information in the problem)
3. Define the variable (and any other unknown
   quanities.
4. Write an equation.
5. Solve the Equation.
6. Label your answer.
7. Ask yourself, Does my answer make sense?           9
 3.2 Solving Application Problems
Two subtracted from 4 times a number is 10.
 Find the number.
                n = number
                4n – 2 = 10
                     +2 +2
                  4n = 12
                   4    4
                    n=3

                                          10
 3.2 Solving Application Problems
The sum of two numbers is 26. Find the two
 numbers if the larger number is 2 less than
 three times the smaller number.




                                           11
  NOTES: sum is 26
  Larger is 3(sm) -2
 ?find both numbers?

 3n-2 =Larger number
n     =Smaller number
 smaller + larger = 26
    n + (3n – 2)= 26
      4n – 2 = 26
        4n = 28
          n=7
smaller is 7; larger is 19
                             12
 3.2 Solving Application Problems
A company currently produces 1200 widgets
  per year. They plan to increase production
  by 550 widgets each year until they reach
  their goal of 4500 widgets produced in a
  year. How many years will it take them to
  reach their goal?




                                           13
1200/yr starting point
     550/yr increase
     4500/year goal
?how long will it take?
 y = number of years
 starting + inc = goal
 1200 + 550y = 4500
      550y = 3300
           y=6
  it will take 6 years.

                          14
 3.2 Solving Application Problems
Joe is moving and needs to rent a truck. It
  will cost him a flat fee of $60/day plus
  $0.40/mile. He has $92 budgeted for the
  truck rental. How many miles can he drive
  without going over the amount he has
  budgeted?



                                          15
       $60/day
      $0.40/mile
       $92 total
 ?how many miles?
      m = miles
  60 + 0.40 m = 92
     0.40 m = 32
        m = 80
he can drive 80 miles.

                         16
 3.2 Solving Application Problems
If I went out to eat and spent exactly $20
   including tax (7.5%), how much was the
   pretax price of my meal?




                                             17
           spent 20
      including 0.075 tax
        ?pretax price?

       p = pretax price
   price + tax = total spent
       p + 0.075p = 20
         1.075p = 20
          p = 18.60
my meal was $18.60 before tax.
                                 18
 3.2 Solving Application Problems
I received a raise in 2007 of 18% over what I
   made in 2006. If I made $43,000 in 2007,
   what did I make in 2006?




                                            19
    18% raise (2007)
     43,000 in 2007
    ?make in 2006?

    m = 2006 salary
06salary + raise = 07salary
  m + 0.18m = 43,000
    1.18m = 43,000
    m = 36,440.68
I made $36,441 in 2006
                              20
 3.2 Solving Application Problems
I am considering two sales jobs: one where I
   would earn $450 base and 3%
   commission on my sales. The second I
   would earn 0 base but 10% commission
   on sales. What would my weekly sales
   need to be in order for my earnings to be
   equal?


                                           21
450 base + 3% com
0 base + 10% com
?sales to be equal?
               w=weekly sales
            450 + 0.03w = 0+0.10w
                  450=0.07w
                    6428=w
 if I had weekly sales of $6428, my earnings
                 would be equal.

                                           22
      3.3 Geometric Problems
I plan to build a rectangular patio where the
   length is 8 feet more than the width. The
   perimeter of the patio will be 56 feet. Find
   the length and width. P=2L + 2W




                  w

        w+8
                                                  23
Length is 8 more than width
    Perimeter is 56 feet
      ?find L and W?
      W + 8 = length
         W = width
       2L + 2W = P
    2(w + 8) + 2w = 56
    2w + 16 + 2w = 56
       4w + 16 = 56
          4w = 40
           w = 10
 Width =10 ft; length = 18 ft   24
         3.3 Geometric Problems
A triangle has two congruent angles. The
  third angle is 30°greater than the two
  equal angles. Find all three angles.
(angles of a triangle add up to 180°)


          2

     1        3


                                           25
      There are two = angles
3rd angle is 30 more than other two
       ?find all three angles?
              Angle1=x
              Angle2=x
           Angle3=30 + x
       Angles add up to 180
       x + x + (30 + x) = 180
            3x + 30 = 180
              3x = 150
               x = 50°
        The angles measure
          50°, 50° and 80°            26
     3.3 Geometric Problems
In a trapezoid, the bottom angles are 15
   degrees less than twice the top angles.
   Find all the angles.
(in a quadrilateral, the angles add to 360°)




                                               27
Bottom angles are 15 less than twice the top
          Quad adds up to 360°
         Bottom angle = 2x - 15
         Bottom angle = 2x - 15
              Top angle = x
              Top angle = x
         Sum of all angles = 360
        x+x+(2x-15)+(2x-15)=360
              6x – 30 =360
                 6x = 390
                  X=65°
  The angles are 65°, 65°, 115° and 115°       28
     3.4 Motion, Money, Mixture Problems
  A family went canoeing. The parents are in
    one canoe and the children in another.
    Both canoes start at the same time from
    the same point and travel in the same
    direction. The parents paddle at 2 miles per
    hour and the children at 4 miles per hour.
    How long until the canoes will be 5 miles
    apart?

parents 2 mph   5 miles

 kids   4 mph                                  29
  3.4 Motion, Money, Mixture Problems
?how long until they are 5 miles apart
      R         x      T = D
(P) 2 mph X          T = 2T
(K) 4 mph X          T = 4T
                           5 miles

4T – 2T = 5
2T = 5
T = 2.5 hours

 It will take 2 ½ hours for them to be 5 miles apart.
                                                        30
3.4 Motion, Money, Mixture Problems
My husband and I go running at the same
 time but starting from different locations
 and running toward each other. We are 6
 miles apart when we start . My husband
 runs 0.5 mph faster than I run. After twenty
 minutes, we meet. How fast were we each
 running?
   Me                Hubby
          20 min
   R                 R + 0.5


           6 miles                         31
?how fast is each running?
       R         x       T = D
    R mph        X     1/3 = (1/3)R
    R+0.5mph X         1/3 = (1/3)(R+0.5)
                                6 miles
My distance + his distance = 6 miles
(1/3) R + (1/3) (R + 0.5) = 6
       R + R + 0.5 = 18 (after clearing fractions)
       2R + 0.5 = 18
       2R = 17.5
       R = 8.75 mph (my rate)
    r+ 0.5 = 9.25 (hubby’s rate)
                                                     32
 3.4 Motion, Money, Mixture Problems

I have $15,000 to invest. The two
   investments I am considering are:
-a mortgage investment that has an 11%
   return
-a cd investment that has a 5% return
How much should I put into each investment
   if I want to earn $1500 interest in one
   year?

                                         33
?how much should I put into each investment?

      P X          R x          T =       I
(M) X              0.11         1         0.11x
(CD) 15000-x       0.05         1   0.05(15000-x)
                                          $1500
Int(m) + int(cd) = 1500
0.11x + 0.05(15000-x) = 1500
11x + 5(15000-x) = 150000        (after clearing dec)
11x + 75000 – 5x = 150000
6x + 75000 = 150000
6x = 75000
x = 12,500
 $12,500 goes into the mortgage and $2,500 into the CD
                                                         34
3.4 Motion, Money, Mixture Problems
The community playhouse is putting on a
 play. Tickets are $15 for adults and $8 for
 students. They have sold 361 tickets for a
 total of $4841. How many adult tickets and
 how many student tickets have they sold?




                                           35
?how many adult tickets and student tickets were
    sold?
price x number sold = income from tickets
(A) 15     361-x        15(361-x)
(S) 8      x                  8x
                              $4841
15(361-x) + 8x = 4841
5415 – 15x + 8x = 4841
5415 – 7x = 4841
-7x = -574
X= 82 student tickets sold
361-x = 279 adult tickets sold
                                                   36
 3.4 Motion, Money, Mixture Problems
A candy store sells licorice sticks for
  $4.50/lb and licorice balls for $2.75/lb. If
  they want to sell a new mix of the two for
  $3.75/lb, how many pounds of licorice
  balls will they need to put with 5 pounds of
  licorice sticks?




                                             37
?how many pounds of licorice balls will they put
  with 5 pounds of licorice sticks?
  price     x     quantity     =    value
(st) 4.50               5           4.5(5)
(bl) 2.75               x           2.75x
(mx)3.75                5+x         3.75(5+x)
4.5(5) + 2.75x = 3.75(5+x)
450(5) + 275x = 375(5+x)
2250+275x = 1875 + 375x
2250 = 1875 + 100x
375 = 100x
3.75 = x
3.75 pounds of licorice balls would be needed.
                                                   38
 3.4 Motion, Money, Mixture Problems

A Chemistry teacher is preparing for
  tomorrow’s lab when he realizes that he
  needs a 15% base solution but only has
  on hand the same type base solution in
  10% and 25%. How much of the 10%
  solution will he need to add to 6 liters of
  the 25% solution in order to make a 15%
  base solution?

                                                39
?how much 10% solution shall be added to the 6
  liters of 25% solution to create 15%?
  strength x quantity (L) =         total
0.10                    x                 0.10x
0.25                    6                 0.25(6)
0.15                    x+6            0.15(x+6)
0.10x + 0.25(6) = 0.15(x+6)
10x + 25(6) = 15(x+6)
10x + 150 = 15x + 90
150 = 5x + 90
60 = 5x
12=x so 12 liters of 10% solution
                                                    40

				
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