A jeweler needs to mix an alloy with a 16 % gold content and...

A jeweler needs to mix an alloy with a $16 \%$ gold content and an alloy with a $28 \%$ gold content to obtain 32 ounces of a new alloy with a $25 \%$ gold content. How many ounces of each of the original alloys must be used?

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Key Concepts

Mixture Problems

Mixture problems involve combining substances with different properties to achieve a desired overall composition. They typically require setting up relationships between the quantities and the concentrations of the substances in order to determine how much of each component is needed.

Weighted Average

The weighted average concept is used when the overall average of a combined mixture is calculated by taking into account the individual contributions of its components, each multiplied by its respective fraction of the total. This concept ensures that the proportion of each component is appropriately reflected in the final result.

Percentage Composition

Percentage composition is the representation of a specific component's proportion within a mixture or alloy, usually expressed as a percent. This concept is crucial in determining how the properties of individual ingredients contribute to the properties of the final mixture.

System of Linear Equations

Many mixture problems are solved by forming a system of linear equations that represent the relationships between the amounts of each component and the overall desired characteristics of the mixture. Solving these equations allows for precise determination of the required amounts of each component.

Transcript

00:01 So a jeweler needs to mix an alloy of 16 % gold content and an alloy with the 28 % gold content to obtain a 32 ounce of a new alloy with 25 % gold content, how many ounce of each of the original alloys must be used.

00:26 So basically you always create your equations or your variables based on the question itself, a main ounce of each of the original aloids must be used.

00:42 So you say let the 16 % code content be x.

00:52 Okay, so you say let the 15 % gold allow be equal to x and let the first one not 15, it's actually 16.

01:24 Okay, so the 16 % let it appropriately.

01:37 60 % and let there.

01:43 The other one is 28%.

01:47 The other one is 28 % gold and equals to y.

01:58 Now we can then start modeling our functions or equations.

02:07 The first one is that if we mix our x plus our y, we should be able to get roundabout, around about 32 hours.

02:23 The total that we want to get is 32 hours.

02:27 So this becomes our first equation.

02:32 Then the second one has got the percentages.

02:40 So it says if we want 16, we have 16 % 0 .16 of x.