A gold density is 19 times greater than the water and
for copper it is 9 times. At what ratio you can mix gold
and copper to get 15 times denser than water.
Ans: 3 : 2
To find the ratio at which you can mix gold and copper to get a mixture that is 15 times denser than water, you can use the concept of weighted averages.
Let's assume we mix x parts of gold and y parts of copper.
The density of the gold-copper mixture can be calculated using the formula:
Mixture Density = (x * Gold Density + y * Copper Density) / (x + y)
Given that the density of gold is 19 times greater than water and the density of copper is 9 times greater than water, we can write the following equations:
Gold Density = 19 * Water Density
Copper Density = 9 * Water Density
Substituting the above equations into the mixture density formula:
Mixture Density = (x * 19 * Water Density + y * 9 * Water Density) / (x + y)
Now, we need to find the values of x and y that will make the mixture density 15 times greater than water density:
Mixture Density = 15 * Water Density
We can set up the following equation:
(x * 19 * Water Density + y * 9 * Water Density) / (x + y) = 15 * Water Density
We can simplify this equation further:
19x + 9y = 15(x + y)
19x + 9y = 15x + 15y
4x = 6y
2x = 3y
x/y = 3/2
Therefore, the ratio at which you can mix gold and copper to get a mixture that is 15 times denser than water is 3:2.
To find the ratio in which gold and copper should be mixed to get a density 15 times greater than water, we first need to establish the density of the mixture.
Let's denote the density of gold as Dg and the density of copper as Dc. We are given that Dg = 19 times the density of water and Dc = 9 times the density of water.
Since the density is directly proportional to the ratio of the mass and volume of a substance, we can compare the densities of gold, copper, and water by using their ratios of mass or volume.
Let's assume we have x units of gold and y units of copper. The total mass of the mixture will be the sum of the masses of gold and copper, and the total volume of the mixture will be the sum of the volumes of gold and copper. The density of the mixture (Dm) can be calculated as:
Dm = (mass of gold + mass of copper) / (volume of gold + volume of copper)
Now let's substitute the known values into the equation:
Dg = 19 * density of water
Dc = 9 * density of water
We can simplify the equation to express it in terms of the ratio of gold to copper:
Dm = (mass of gold + mass of copper) / (x * m/v + y * m/v) where m/v is the density of water
Since Dm should be 15 times the density of water, we can write:
15 * density of water = (mass of gold + mass of copper) / (x * density of water + y * density of water)
density of water cancels out on both sides of the equation:
15 = (mass of gold + mass of copper) / (x + y)
Let's denote the mass ratio of gold to copper as mG and mC. We have:
mG = mass of gold / mass of copper
mC = mass of copper / mass of copper = 1
We can express the mass of gold and copper in terms of mG and mC:
mass of gold = mG * mass of copper
mass of copper = mC * mass of copper
Now substitute these values into the previous equation:
15 = (mG * mass of copper + mass of copper) / (x + y)
Since we want a ratio of gold to copper, we can express mG as:
mG = x / y
Now we have:
15 = ((x / y) * mass of copper + mass of copper) / (x + y)
Simplifying further:
15 = (x + y) * mass of copper / (x + y)
Notice that (x + y) cancels out on both sides of the equation:
15 * (x + y) = mass of copper
We can find the ratio of gold to copper by dividing the mass of gold by the mass of copper:
x / y = mass of gold / mass of copper = mG
Recall the definition of mG:
mG = x / y
Therefore, we have:
mG = 15 * (x + y) / (x + y)
To simplify the expression, we can cancel out the common factor of (x + y) on both numerator and denominator:
mG = 15
This means that the ratio of gold to copper should be 15:1.
Since we are asked for the ratio in terms of x and y, where x is the number of units of gold and y is the number of units of copper, we have:
x = 15
y = 1
Therefore, the ratio in which gold and copper should be mixed to obtain a density 15 times greater than water is 15:1, or simply 15 units of gold to 1 unit of copper.