The amounts by weight of gold, silver and lead in three alloys of these metals are in the ratio 1:5:3 in the first alloy, 2:3:4 in the second and 5:2:2 in the third. How many kg of the first alloy must be used to obtain 10kg of an alloy containing equal amounts by weight of gold, silver and lead?
If there are x,y,z kg of the alloys, then we want
x+5y+3z = 10
2x+3y+4z = 10
5x+2y+2z = 10
Just solve those to find x,y,z. Since the x value is from the 1st alloy, in the ratios 1:5:3, that is how many kg of that alloy we need.
I don't think they should be equal to 10.
kg of 1st alloy + kg of 2nd alloy + kg of 3rd alloy = 10 kg
I got an answer, my answer is 2kg.
Hmmm. If 2 kg of alloy 1 are used, then that means there are
2 kg of gold
10kg of silver
6 kg of lead
You are already over 10 kg of the mixture. Add in to that any amount of the other alloys and things are even worse.
The question is how many kg of the first alloy must be used...
So my answer is 2kg of first alloy, 4kg of 2nd alloy and 4kg of 3rd alloy resulting to 10kg.
Hmmm. I see I was mistaken.
You are correct. My equations add up the wrong stuff. I see I should have checked my answer.
To solve this problem, we need to determine the amount of each metal in the first alloy and then use that information to calculate the amount of the first alloy needed.
Let's represent the amounts of gold, silver, and lead in the first alloy as x, 5x, and 3x respectively. Since the ratio of the amounts by weight of gold, silver, and lead in the first alloy is 1:5:3, we'll assign x as the common factor.
Now, let's calculate the total amount of metal (gold + silver + lead) in the first alloy:
Total weight = x + 5x + 3x = 9x
Since we want to obtain an alloy containing equal amounts of gold, silver, and lead, we know that each metal will have a weight of 10kg/3 = 3.33kg in the final alloy.
Now, let's set up an equation to solve for x:
9x = 3.33 kg
Dividing both sides of the equation by 9, we get:
x = 3.33 kg / 9 = 0.37 kg
This tells us that in the first alloy, the weight of gold is 0.37 kg, the weight of silver is 5 * 0.37 kg = 1.85 kg, and the weight of lead is 3 * 0.37 kg = 1.11 kg.
To obtain an alloy with equal amounts of gold, silver, and lead, we need to find the amount of the first alloy that contains 3.33 kg of each metal. Since the first alloy has gold, silver, and lead in a ratio of 1:5:3, we need to find the weight of the first alloy that contains 3.33 kg of gold.
We can set up another equation to solve for the weight of the first alloy (y):
0.37 kg/y = 3.33 kg/1
We can cross-multiply and solve for y:
0.37y = 3.33
Dividing both sides by 0.37, we get:
y = 3.33 / 0.37 = 9 kg
Therefore, we need 9 kg of the first alloy to obtain 10 kg of an alloy containing equal amounts by weight of gold, silver, and lead.