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Chapter 41: Problem 1198

What quantities of silver 60 per cent and 82 per cent pure must be mixed together to give 12 ounces of silver 70 per cent pure?

Short Answer

Expert verified
To obtain 12 ounces of 70% pure silver, mix 6.55 ounces of 60% pure silver and 5.45 ounces of 82% pure silver.

Step by step solution

01

Define the variables

Let x be the number of ounces of 60% silver, and y be the number of ounces of 82% silver. Both x and y contribute to the total weight (12 ounces) and the total purity (70%).
02

Create a system of linear equations

Since the total weight of the mixture is 12 ounces, we get the following equation: \(x + y = 12\) And for the total purity, 60% of the 60% silver mixture plus 82% of the 82% silver mixture equals 70% of the 12 ounces of final silver mixture: \(0.60x + 0.82y = 0.70(12)\)
03

Solve the system of linear equations

We can solve the system of linear equations using any method, such as substitution or elimination method. We will use the substitution method in this case. Solve the first equation for one of the variables, say x: \(x = 12 - y\) Now substitute this into the second equation to get: \(0.60(12 - y) + 0.82y = 0.70(12)\)
04

Simplify and solve for y

Simplify the equation and solve for y: \(7.2 - 0.60y + 0.82y = 8.4\) \(0.22y = 1.2\) \(y = \frac{1.2}{0.22}\) \(y = 5.45\)
05

Find x

Now that we know y, we can substitute it back into the equation for x: \(x = 12 - y\) \(x = 12 - 5.45\) \(x = 6.55\)
06

State the answer

The quantities required are 6.55 ounces of 60% silver and 5.45 ounces of 82% silver to give 12 ounces of 70% pure silver.

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