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Open-Ended Template

 

Names:

Gerald Burkepile, Heather Godine, Carol Wengerd, Eugene Wengerd

 

Grade Level:

 9-12

 

Content Area:

Numbers/Computation

 

PA standard addressed:

2.2.11.A         

Develop and use computation concepts, operations and procedures with real numbers in problem-solving situations.

 

NCTM standard addressed:

Understand meanings of operations and how they relate to one another.

 

Problem Name:

A Working Dilemma

 

Problem:

Sally works at Pi’s Grocery and earns $6.70 per hour. She receives time and a half for any time worked over 8 hours in a single day. Twenty-four percent of her gross pay (total dollars earned) is deducted for taxes.

 

During one week Sally worked the following hours:

Day

Hours Worked

Sunday

6.5

Monday

0

Tuesday

0

Wednesday

8.5

Thursday

10

Friday

7.5

Saturday

 

 

Since Sally’s car payment is due, she needs to receive a net pay (total dollars earned after taxes are deducted) of at least $200 for the week. To the nearest whole hour, what is the minimum number of hours Sally would need to work on Saturday?

 

Directions:

For full credit, you must do the following:

  1. Show all the steps you used to solve the problem.  If you used a calculator or did some of the work  in your head, you must write a description of the steps that you followed.

AND

  1. Write an explanation stating the mathematical reason(s) why  you chose each of 

      the steps.

 

 

Solutions:  G. Burkepile

 

If G is the gross pay before taxes, then G - .24G = $200 yields the net pay after taxes. This yields .76G = $200 or dividing yields G = $263.16.

 

She earned:          Sunday -  6.5 x $6.70 = $43.55

                               Monday -                                                                               overtime pay

                               Tuesday –                                                                              $6.70 x 1.5 = $10.05 per hour

                               Wednesday -  8 x $6.70 = $53.60

                                                     .5 x 10.05 = $5.03

                               Thursday -   8 x $6.70 = $53.60

                                                      2 x 10.05 = $20.10

                                Friday -  7.5 x $6.70 = $50.25

                                Saturday -  

                                Total Gross pay =   $226.13

 

Since she needs to earn $263.16, and has already earned $226.13, the difference will be the gross amount that she has to earn on Saturday. 

                                                                               $263.16

-226.13

                $37.03

 

Dividing the needed gross pay by her hourly wages will yield the number of hours needed to work on Saturday.

 

                                                 $37.03/$6.70 = 5.52 hours

Since she needs to work more than 5 hours to earn the desired $200, to the nearest hours worked on Saturday, she will have to work 6 hours.

 

 

 

For $200 there is (.24)($200) or $48 for taxes.

Thus she will need to gross at least $248, but would need almost $12 more tax for the additional $48. Since $48 is almost $50, .24 x $50 = $12.

That then means she has to gross an additional $12 or around $260 alltogether.

 

For $260, dividing by the hourly rate will yield the total number of regular hours she needs to work for the week.

                   $260/$6.70 =  38.81 regular hours

 

She has worked 6.5 + 8 + 8 + 7.5 = 30 regular hours

                         And  .5 + 2 = 2.5 overtime hours or  2.5 x 1.5 = 3.75 regular hours

Thus she has worked a total of 30 + 3.75 = 33.75 regular hours already.

 

Therefore since she needs 38.81 and has worked 33.75, the difference will give the number of hours needed for Saturday.  38.81 – 33.75 = 5.06 hours needed.

 

Since she needs to work more than 5 hours to earn the desired $200, to the nearest hours worked on Saturday, she will have to work 6 hours

 

 

 

 

 

 

Solution #2

H. Godine

 

·        Calculate the total number of regular hours worked:

5.6 + 8 + 8 + 7.5 = 30

·        Multiply the number of regular hours worked by the salary per hour to find the gross pay for the regular hours.

$6.70(30) = $201.00

·        Calculate the total number of time and a half hours worked:

0.5 + 2 = 2.5

·        Calculate the rate of pay for time-and-a-half.

$6.70(1.5) = $10.05

·        Multiply the number of time-and-a-half hours worked by the time-and-a-half salary per hours to find the gross pay for the time-and-a-half hours.

$10.05(2.5) = $25.13

·        Add the gross pay for time-and-a-half hours and regular hours to find the total gross pay.

$201.00+$25.13 = $226.13

·        Multiply the total gross pay by 0.24 to find the amount deducted for taxes.

$226.13(0.24) = $54.27

·        Subtract the amount deducted for taxes to find the net pay earned.

$226.13-$54.27 = $171.86

·        Subtract the net pay earned from $200 to determine the net pay that needs to be earned on Saturday.

$200.00-$171.86 = $28.14

·        Guess and test to determine how many hours Sally would need to work on Saturday to meet her goal.

4 hours:      4($6.70) = $26.80  (Not enough even before taxes are deducted)

5 hours:      5($6.70) = $33.50

                        $33.50(0.24) = $8.04

                        $33.50 - $8.04 = $25.46 (Not enough after taxes are deducted)

6 hours       6($6.70) = $40.20

                        $40.20(0.24) = $9.65

                        $40.20-$9.65 = $30.55 (Enough to meet her goal)

·        By guess and test, 6 hours would be enough to meet her goal.

 

 

 

 

 

Solution 3

Solution C. Wengerd

 

 

            Sally needs to calculate the number of regular hours and the number of  overtime hours already worked.

                                    Regular       Overtime

            Sunday   6.5                  0

            Wednesday         8                    .5

            Thursday             8                     2

            Friday                7.5                   0

                 Totals             30                2.5

 

To calculate her gross pay she needs to multiply the number of regular hours worked times $6.70.  So she needs to take 30 times 6.70 which gives $201.

 

Then she needs to figure her overtime pay.  She needs to multiply 6.70 times 1.5 times the number of overtime hours worked.  So  (2.5)(6.70)(1.5) which gives $25.13.

 

She needs to add her regular pay and her overtime pay.  $201 + $25.13 = $226.13.  This is the amount of money she has earned so far. 

 

Tax is deducted at the rate of 24%.  Since 100% - 24% = 76%, she actually receives 76% of what she earned.  This means she needs to multiply  .76 times 226.13.

              (.76)(226.13) = $171.86

She will receive $171.86  for the hours she has already worked.

 

Now, we know how much she has earned already, and she will get 76% of  $6.70 for every hour she works on Saturday.  She needs a total of $200.  If X is the number of hours she needs to work on  Saturday, then the equation is:

             171.86 + 6.70(.76X) = 200.

                                         X= 5.52

 

5 hours will not earn her quite enough money, so she will need to work 6 hours on Saturday.

 

The answer is 6 hours.

           

 

 

 

 

Solution 4: E. Wengerd

 

 

               Regular Hours                                     Overtime Hours

6.5+8+8+7.5=30                                                        .5+2=2.5

Multiply Regular hours by $6.70.               Multiply $6.70 by 1.5=$10.50

             $30(6.70)                                                    2.5($10.50)

                 $201                                                             $25.13

                         Add regular pay and overtime pay

                                      $201 + $25.13

                                           $ 226.13  Gross Pay for 4 days

                                 Find deduction for taxes

                                           .24(226.13)

                                               $54.27

                                 Find Net Pay for 4 days

                                            226.13 – 54.27

                                               $171.86       Net Pay for 4 days

 

                       total needed – already earned

                               200      -     171.86       

                                 needs  28.14  net pay to get to 200 for the week

 

          X represents gross pay for Saturday

                    X - .24X = 28.14

                          .76X   =   28.14

                                X =  37.02

 

  Her hours needed for Saturday  $37.02/$6.70.

                                                          5.5 hours

 

Since the problem requires a solution to the nearest hour, she must work 6 hours.

 

So, the solution is 6 hours.

 

 

 

 

 

Problem Name:                      Grade Level:                          Content Area:

A Working Dilemma                 9-12                                        Numbers/Computation

 

Specific Rubric:

5

A
  • Correct answer (6 hours)
  • All work is shown.
  • Explanation includes how and why the steps were performed.

4

A
  • Correct answer (6 hours)
  • All work is shown.
  • Some explanation is included.

3

A
  • Correct answer (6 hours)
  • All work is shown.
  • No explanation is included.

 

B
  • Correct answer (6 hours)
  • Some work is shown.
  • Some explanation is included.

 

C
  • Incorrect answer due to a calculation error.
  • Some work is shown.
  • Some explanation is included

2

A
  • Correct answer (6 hours)
  • Some work is shown.
  • No explanation is included.

 

B
  • Incorrect answer due to multiple calculation errors.
  • Some work is shown.
  • Some or no explanation is included.

 

C
  • Incorrect answer due to a procedural error.
  • Some work is shown.
  • Some or no explanation is included.

1

A
  • Correct answer (6 hours)
  • No work is shown.
  • No explanation is included.

 

B
  • Incorrect answer due to multiple procedural or calculation errors.
  • At least one procedure that could lead to the solution is included.
  • Some or no explanation is included.

0

A
  • Blank -OR-
  • Off task response -OR-
  • Does not meet requirements for a 1.

 

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