Joe has a ring weighing 16 grams made of an alloy of 20% silver and the rest gold. He decides to melt down the rings and add enough silver to reduce the gold content to 69% . How many grams of silver should he add?
considering just the mass of the silver,
(.20)(16)+x = (.31)(16+x)
x =2.55
so, add 2.55g of silver
To find out how many grams of silver Joe should add, we can first calculate the amount of gold and silver in the initial ring.
Given that the ring weighs 16 grams and is made of an alloy of 20% silver, we can calculate the weight of silver and gold in the ring.
Weight of silver in the ring = 20% of 16 grams = (20/100) * 16 grams = 3.2 grams
Weight of gold in the ring = Total weight of the ring - Weight of silver in the ring = 16 grams - 3.2 grams = 12.8 grams
Now, Joe wants to reduce the gold content to 69% by adding silver. Let's assume he adds 'x' grams of silver.
The new weight of silver after adding 'x' grams = Total weight of silver before + Weight of silver added = 3.2 grams + x grams
The new weight of gold = Total weight of gold before = 12.8 grams
According to the given condition, the gold content after adding silver should be 69%.
Gold content after adding silver = (Weight of gold / Total weight of the alloy) * 100
69% = (12.8 grams / (12.8 grams + 3.2 grams + x grams)) * 100
To find X, we can rearrange the equation:
69/100 = 12.8 / (12.8 + 3.2 + x)
6.9 = 12.8 / (16 + x)
6.9 * (16 + x) = 12.8
110.4 + 6.9x = 12.8
6.9x = 12.8 - 110.4
6.9x = -97.6
x = -97.6 / 6.9
x ≈ -14.15
We can't have negative grams of silver, which means that Joe doesn't need to add any silver.