May you please help me with word problems?

Question

A candy shop manager mixes M&M's worth $2.00 per pound with trail mix worth $1.50 per pound to make 50 pounds of party mix worth $1.80 per pound. How many pounds of each she should use?

Can you please explain how you got the answer and it will be nice if you can show the process. Thank you.

Answer 1

Hi Allison!  To start word problems, one could simplify the matter: what if the mix were half-and-half with 25 pounds each?  You'd get the average between $1.50 & $2 ==> $1.75 per pound.  So, to reach $1.80 / pound mix, we'd need a little more of the M&M's ... the next logical ratio to try is 30 / 20:

30x $2 + 20x $1.50= $60+$30= $90.  Divide $90 by 50 pounds & we have the $1.80 / pound mix.

STRATEGY: make the problem super-simple to begin, and then work in the complication ... Regards :)  

 

Answer 2

You have to solve a system of two linear equations. Let's put x - number of pounds for M&M, y - number of pounds for trail. Your equations are (according to the text):

                            (2x + 1.5 y)/50 = 1.8

                             x+y = 50

Multiply firsty equation by 50 and the second one by 2. You will have

                           2x + 1.5y = 90

                           2x +2y = 100

Then subtract first equation from the second one. We will obtail

                          0.5y = 10  or y = 20 and x = 50-20=30.

                                        Answer: x = 30, y = 20

Answer 3

Hi, Allison.

I thought I'd throw in my two cents here too.

When I teach these type of problems (mixture story problems), I tell my students to look for two ideas: Quantity and Value.  These two ideas will create two equations for the story.

Quantity - answers the question, "How many do I have?"
In this problem we have 50 pounds of M&M's and trail mix combined.  So choose a letter to represent each snack (x can be the M&M's and y can be the trail mix).  

x + y = 50  ... see how easy it is to create the Quantity sentence?

Value - places value on each item in the mix
In this problem M&M's are worth $2.00 per pound, and the trail mix is worth $1.50 per pound.  So to start off, we will write 2.00 x  +   1.50 y  = something... the something is the party mix amount multiplied by its value of $1.80 per pound.  So here's my Value sentence:

2.00 x  +   1.50 y  =  1.80 (50)
2.00 x  +   1.50 y  =  90

Now that we have a Quantity sentence and a Value sentence, we can proceed with solving our system of equations.

       x +   y = 50
2.00 x + 1.50 y = 90

If we multiply the bottom one by 10, we can "get rid of the decimals," and we will get:
20x + 15y = 900

Okay, so now we have:

    x +    y = 50
20x + 15y = 900

From here, we may use one of three methods to find our answers: graphing (not too desirable - too difficult to maintain accuracy), substitution, and elimination.  My favorite method is elimination, but I can show you both substitution and elimination here.

Substitution

Solve the first equation for one of the variables... let's say we're solving for x:
x + y = 50
   - y       - y                             .
x       =  - y + 50

Now, put the expression we have for x into the other equation:

  20   x         +  15y = 900
20( - y + 50)  + 15y = 900
-20y  + 1000  + 15y  = 900
-5y  + 1000   =   900
      - 1000      - 1000
-5y      =  -100

-5y     =   -100
 -5             -5 

y = 20 (Trail Mix = 20 pounds)
x = 30 because 20 + 30 = 50

So, the person in the story should use 30 pounds of M&M's and 20 pounds of Trail Mix.

Elimination

   x +   y  =  50
20x + 15y = 900

Elimination is cool because you get to clear out one of the variables and work out an "easier" problem.
If we wish to eliminate the x's, multiply the top equation by negative 20:

-20x   - 20y   =   -1000
 20x   + 15y   =     900
           - 5y    =   -100

           - 5y    =   -100
             -5            -5

y = 20 (again)
And so x = 30 (again)

It's up to you which way you prefer to solve systems of equations (unless the directions on your homework or test tell you otherwise).

Hope this long-winded explanation made some sense to you.