Question

Gold used to make jewerly is often a blend of gold, silver, and copper. Consider three alloys of these metals. The first alloy is 75% gold, 5% silver, and 20% copper. The second alloy is 75%gold, 12.5% silver, and 12.5% copper. The third alloy is 37.5% gold and 62.5% silver. If 100 g of the first alloy costs $2350.80, 100 g of the second alloy costs $2388.00, and 100 g of thethird alloy costs $1475.00, how much does each metal cost?Gold costs $'9. (Round to the nearest cent as needed.)Silver costs $19. (Round to the nearest cent as needed.)Copper costs $19 (Round to the nearest cent as needed.)

Answer 1

Okay, here we have this:

We obtain the following system of equations:

Let's take x as gr of gold, y as gr of silver and z as gr of copper:

75x+5y+20z=2350.80 (1)

75x+12.5y+12.5z=2388 (2)

37.5x+62.5y=1475 (3)

Now, let's solve the system:

Let's clear x in (3):

X=(1475-62.5y)/37.5

x=(118-5y)/3 (4)

Let's replace (4) in (1) and (2):

75((118-5y)/3)+5y+20z=2350.80

75((118-5y)/3)+12.5y+12.5z=2388

-120y+20z+2950=2350.8 (4)

-112.5y+12.5z+2950=2388 (5)

Let's clear y in (4):

y=-((-20z-599.2)/120) (6)

Let's replace (6) in (5):

-112.5(-((-20z-599.2)/120))+12.5z+2950=2388

-6.25z+2388.25=2388 (7)

Let's clear z in (7):

z=1/25=0.04

Let's replace z=0.04 in (6):

y=-((-20z-599.2)/120)

y=-((-20*0.04-599.2)/120)

y=-((-0.8-599.2)/120)

y=-(-600/120)

y=-(-5)

y=5

Let's replace y=5 in (4):

x=(118-5y)/3

x=(118-5*5)/3

x=(118-25)/3

x=(93)/3

x=31

Finally we obtain that the Gold costs $31 per gram, Silver cost $5 per gram, and Copper cost $0.04 per gram.