Applications of linear equations

Question

 For thousands of years gold has been considered one of the earths most precious metals. 100% pure gold is 24 karat gold which is too soft to be made into jewelry. Most gold  jewelry is 14 karat gold approximately 58% goal. If 18 karat gold is 75% gold and 12 karat gold is 50% gold how much of each should be used to make a 14 karat gold bracelet weighing 100 g. 

 Question how many grams of 18 karat gold needs to be used to make the bracelet. Answer is in grams

Answer 1

The number of karats is TMI. The percentages of pure gold are important.

The question is, “How many grams of 75% gold must be mixed with 50% gold to make 100 grams of 58% gold?”

There are 100 grams of 58% gold; how many grams of pure gold is that? 58 grams.

Now, you have a standard mixture problem.

         x = grams of 75% gold; solve for x
                 mixed with (100-x) grams of 50% gold

Now, the number of grams of pure gold in x grams of 75% gold is 0.75x.
The number of grams of pure gold in (100-x) grams of 50% gold is 0.50(100-x).

Together, there is a total of 58 grams of pure gold (in a mixture weighing 100 grams).

         0.75x + 0.50(100-x) = 58
         75x + 50(100-x) = 5800                    [multiply by 100]
          75x + 5000 - 50x = 5800

             25x = 800
               x = 32 grams
Check:
Does a mixture of 32 grams of a 75% gold (24 karat) with 68 grams of 50% gold (12 karat) make 100 grams of 58% gold (14 karat)?

              Does 32 + 68 = 100? yes

             Does 32*0.75 + 68*0.50 = 100*0.58 ?
                     24 + 34 = 58 ?

                    58 = 58 ? yes

Answer 2

Let X be the amount of 18 karat gold and Y be the amount of 12 karat gold.

 

14 karat = 14/24 = 7/12 = 0.58333333... = 58.3%

 

X + Y = 100

 

The amount of gold is  0.75*X + 0.50 * Y = 7/12*100

 

                                  0.75*X + 0.50 * Y = 700/12

 

                                  75X + 50Y = 70000/12  <-- multiplies everything by 100 to clear the decimals

 

                                                  = 35000/6

 

                                                  = 17500/3

 

Solving the first equation for X, X = 100 -Y

 

               75(100-Y) + 50Y = 17500/3

 

             7500 - 75Y + 50Y = 17500/3

 

               7500  - 25Y = 17500/3

 

             22500 - 75Y = 17500

 

             -75Y = 17500 - 22500

 

            - 75Y = -5000

 

            Y = -5000/-75 = 66 and 2/3 = 66.66666.... = 66.6

 

X = 100 - Y = 100 - 66 and 2/3 = 33 1/3 = 33.33333.... = 33.3

 

CHECK:

 

   0.75 * 33.3 + 0.5 * 66.6  = 58.3333333333333333333 = 58 and 1/3

 

So X=33. 3 grams of 18 karat gold
Y = 66. 6 grams of 12 karat gold