college algebra

Question

A jeweler has five rings, each weighing 16 g, made of an alloy of 10% silver and 90% gold. She decides to melt down the rings and add enough silver to reduce the gold content to 80%. How much silver should she add? 

Answer 1

So, we have 16(5) = 80 g total with 0.1(80) = 8 g being silver.

Let s = the amount of silver added.

We want the silver to be 100 - 80 = 20% of the new total amount.

8 + s = 0.2(80 + s)

8 + s = 16 + 0.2s

0.8s = 8

s = 10 g

Let's check.

8 + 10 = 18

0.2(80 + 10) = 0.2(90) = 18

Check!

Answer 2

Ah, mixture problems! The best way to complete these problems is by creating a table:

% silver	Grams of metal	Total
10%	16 g * 5 = 80gram	.1(80)
100%	x	1(x)
20%	80 + x	.2(80 + x)

Now to explain the table. Our 5 rings originally have 10% silver (we really don't care much about the gold content). Each ring weighs 16 grams so she has a total of 80 grams. She is going to add pure silver (100% silver) until the mixture reaches 20% silver content (found by subtracting the 80% gold content from 100%). We are trying to find how much of the 100% we need so we put x in that column. Our total column is found by multiplying the rows across, making sure to change the percentages to decimals. Now to find an equation to solve we are going to add the top two rows of the total column and set it equal to the bottom row:

.1(80) + 1(x) = .2(80 + x)

8 + x = 16 + .2x

-8 -8

x = 8 + .2x

-.2x -.2x

0.8x = 8

/.8 /.8

x = 10 grams

So we now know she needs to add 10 grams of pure silver. I hope this explanation was thorough and answered all of your questions. If you have any additional questions don't hesitate to reach out. Learn on! :)

Answer 3

5 * 16g = 80 g

10% of 80g = 8 g of silver and therefore 90% of 80g = 72g = (80-8) g of gold.

X is the amount of silver added

Then there is (X+80)g of the alloy which STILL contains 72 g of gold, since only silver was added.

72/ (x+80) = 0.8 <--- it must contain 80% gold

72 = 0.8 ( x+80)

72 = 0.8 x + 64

8 = 0.8x

80 = 8x

x = 10

So adding 10 grams of silver will do the trick... let's see.

Adding 10 grams of silver means there is going to be 90 grams of total metal alloy:

72 grams of gold and 8+10 = 18 grams of silver

72/90 = 8/10 = 4/5 = 0.8 = 80% , so yes it is correct

10 grams of silver will reduce the amount of gold to 80%