word problems

Question

Kathryn and Kim are selling cheesecakes for a school fundraiser. Customers can buy pecan
cheesecakes and apple cheesecakes. Kathryn sold 12 pecan cheesecakes and 13 apple
cheesecakes for a total of $266. Kim sold 6 pecan cheesecakes and 2 apple cheesecakes for a
total of $70. Find the cost each of one pecan cheesecake and one apple cheesecake.

Answer 1

The key to solving word (story) problems is to correctly translate the words into concise, precise math expressions, with variables and operations.

 

After reading and understanding the problem, put it into your own words.  How's this:

   12 pecan cheesecakes and 13 apple cheesecakes sold for $266

    6 pecan cheesecakes and 2 apple cheesecakes sold for $70

 

Assign variables:

    P = cost of one pecan cheesecake

    A = cost of one apple cheesecake

 

Note that (cost per cheesecake)*(number of cheesecakes) = (cost)  because

          cheesecakes cancel out, leaving only cost

 

Now, the easy math:

    12P + 13A = 266    (eq1)

      6P +  2A = 70      (eq2)

 

Either solve for a variable in one equation and substitute it into the other equation or use the elimination method (add or subtract equations).  Let's use elimination --

   12P + 13A = 266   (eq1)

   12P +  4A  = 140    (2*eq2)

-----------------------   (elimination:  subtract equations)

              9A  = 126

                A  = $14            (divide both sides by 9) (eq3)

 

   6P + 2A  = 70      (eq2)

           2A  = 28      (2*eq3)

  ------------------        (elimination subtract equations)

   6P          =  42

        P = $7             (divide both sides by 6)

 

Checking (very important):

  Is  12(7) + 13(14)  =  266   ?

        84    +  182   =   266   ?

                266  =  266   ? yes

  Is   6(7) + 2(14)  =  70   ?

         42   +  28    =  70   ?

                 70  -  70   ? yes

Answer 2

Answer: x (pecan)=$7 y (apple)=$14 1. Write out the word problem so you can understand what is going on Kat sold 12 pecan + 13 apple = $266 Kim sold 6 pecan + 2 apple = $70 2. To find the cost of one pecan and one apple cheesecake, write the above statements in an expression. Kat: 12x+13y=266 Kim: 6x+2y=70 3. Cancel out either the x or y. Your choice, but we will do x for simplicity. Multiply Kim equation by -2 to cancel out x and solve for y. 12x + 13y =266 -2(6x +2y=70) 12x + 13y = 266 -12x -4y = -140 4. Solve 9y=126 y= 126/9 y=14 5. Use this to solve for x. We can use the Kim's equation again because it will be the most simple. 6x +2(14)=70 6x+28=70 6x=42 x=7 6. Check your answer. x=7 y=14 in the original equations Kat: 12x+13y=266 12(7) + 13(14) = 266 84+ 182=266 :) Kim: 6x+2y=70 6(7) + 2(14) = 70 42+28= 70 :) If you need any further help on understanding the steps more in depth, I do offer online tutoring here on Wyzant. Hope this helps.