Math word problem

Question

Deep thought granola is 25% nuts and dried fruit. Oat cream granola is 10% nuts and dried fruit. How much of deep thought and how much of oat dream should be mixed to form a 20lb batch of granola that is 19% nuts and dried fruit? 

Answer 1

Lets use 3 variables,

 

X represents Lb. of Deep Thought Granola = unknown quantity

Y represents Lb. of Oat Cream Granola      = unknown quantity

Z represents Lb. of the final mixture         = 20 Lb

 

  We know the % nuts and fruit in each.

       DTG has 25% or 0.25 Fruit and Nut

       OCG has 10% or 0.10 Fruit and Nut

       Mixture has 19% or 0.19 Fruit and Nut

 

Now we can make our equations.

 Since we know the final weight of the mixture we can use our above variables.

   Lb. of X + Lb. of Y = Lb. of Z which we know is 20

      

       X + Y = Z = 20

       X + Y = 20

 

Second equation;

  (Lb. of X)(% fruit and nut X as a decimal) + (Lb. of Y)(% f & nut Y as a decimal) = (Lb. of Z)(% f & nut Z as a decimal)

 

            The equation looks like this    X(0.25) + Y(0.10) = Z(0.19)

                                                          0.25X + 0.10Y = 0.19Z = 0.19(20) = 3.8 sincewe know the value of Z 

 

we now have two equations and two unknowns so we can solve. First we will solve the easier equation in terms of X or Y

(it makes no difference which) I will solve for Y in terms of X. Then we will replace Y in the second equation and solve for X. 

 

                X + Y = 20                                                                 0.25X + 0.10Y = 3.8

                Y = 20 - X        -------Substitute for Y----->             0.25X + 0.10(20 - X) = 3.8

                                                                                                     0.25X + 2 - 0.10X = 3.8

                                                                                                     0.15X = 1.8

                                                                                                           X = 1.8/0.15

                 Y = 20 - X      <-------Substitute for X------------             X = 12

                 Y = 20 - 12

                 Y = 8

 

X = 12 = Lb. of DTG

Y = 8 = Lb. of OCG

 

One las thing we can do to check our work is plug everything back in to our other equation to make sure it adds up

 

     0.25X + 0.10Y = 0.19Z

     0.25(12)+ 0.10(8) = 0.19(20)

            3      +      0.8   =     3.8

                             3.8   =     3.8            looks good to me!

                                                    

 

   

Answer 2

DT = lbs.deepthought and lbs.Oat Cream =OC

 

.25(DT) + .10(OC)= .19(20lbs)

 

DT + OC= 20  So OC =20-DT  Substituting 20-DT for OC we get

 

.25 DT+ .10(20-DT)=.19(20)  Distributing the .10 we get

.25 DT + 2.0 - .10 DT = 3.8

collecting terms give us   .15DT= 1.8 therefore DT = (1.8/.15) lbs

SO DT= 12 lbs and OC=20-12= 8 lbs