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Chapter 4: Problem 34

A karat describes the percent gold in an alloy (a mixture of metals). $$ \begin{array}{|c|c|} \hline \text { Name of alloy } & \text { Percent gold } \\ \hline \text { 10-karat gold } & 41.7 \% \\ \text { 14-karat gold } & 58.3 \% \\ \text { 18-karat gold } & 75 \% \\ \text { 20-karat gold } & 83.3 \% \\ \text { 24-karat gold } & 100 \% \\ \hline \end{array} $$ If 9 oz of 14-karat gold jewelry is melted down, find the amount of 20 -karat gold to add to the melted jewelry to create a new alloy that is \(18-k\) arat gold. Find the amount of the new alloy. Round to the nearest tenth.

Short Answer

Expert verified
18.1 oz of 20-karat gold are needed, resulting in a total of 27.1 oz of 18-karat gold.

Step by step solution

01

- Define Variables

Let the amount of 20-karat gold to add be denoted as \( x \) oz. The final amount of the new alloy will be the sum of 9 oz and \( x \) oz, i.e., \( 9 + x \) oz.
02

- Write the Percentages as Decimals

Convert the percentages to decimals: \[ 14\text{-karat: } \frac{58.3}{100} = 0.583 \text{ gold content} 20\text{-karat: } \frac{83.3}{100} = 0.833 \text{ gold content} 18\text{-karat: } \frac{75}{100} = 0.75 \text{ gold content} \]
03

- Set Up the Equation

Set up the equation based on the gold content: \(0.583(9) + 0.833(x) = 0.75(9 + x)\)
04

- Solve for x

Distribute and combine like terms to solve for \( x \): \(5.247 + 0.833x = 6.75 + 0.75x 0.083x = 1.503 x = \frac{1.503}{0.083} \approx 18.1\)
05

- Calculate the Amount of New Alloy

The total amount of the new alloy is the sum of the original 9 oz plus the added amount \( x \): \( 9 + 18.1 = 27.1 \text{ oz} \)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Karat Gold Percentages
Gold alloys are created by mixing pure gold with other metals, hence the term 'karat' to describe their purity.
The amount of gold in an alloy is expressed in karats, where 24 karat is pure gold, or 100%.
Lower karat values indicate a lower percentage of gold and a higher percentage of other metals.
Algebraic Equation Solving Simplified
Solving algebraic equations means finding the value of unknown variables that make the equation true.
Here, we define the variable to represent the amount of gold we need to add: let x be the amount of 20-karat gold to add.
Then we set up and solve the equation based on the percentages of gold in each alloy.
Gold Alloy Mixture Calculation
To create a desired alloy, we combine different gold alloys and use their percentages to figure out the right mixture.
In this problem, we need to add 20-karat gold to 14-karat gold to make 18-karat gold.
We translate these into decimal form and set up an equation to find how much 20-karat gold to add and calculate the final amount of the new alloy.

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Most popular questions from this chapter

A karat describes the percent gold in an alloy (a mixture of metals). $$ \begin{array}{|c|c|} \hline \text { Name of alloy } & \text { Percent gold } \\ \hline \text { 10-karat gold } & 41.7 \% \\ \text { 14-karat gold } & 58.3 \% \\ \text { 18-karat gold } & 75 \% \\ \text { 20-karat gold } & 83.3 \% \\ \text { 24-karat gold } & 100 \% \\ \hline \end{array} $$ Find the amount of 10-karat gold and 20 -karat gold to combine to make 3 g of 14-karat gold. Round to the nearest hundredth.

The cost of playground grass seed mix is $$\$ 2.16$$ per pound, and the cost of shade grass seed mix is $$\$ 3.60$$ per pound. If a total of \(500 \mathrm{lb}\) of grass seed is sold for $$\$ 1728$$, find the amount of playground grass seed mix and the amount of shade grass seed mix sold.

A campsite is \(6 \mathrm{mi}\) from a trailhead. One group of hikers begins at the trailhead, hikes on the trail to this campsite, and spends the night. At 8 a.m. the next morning, they start hiking on the trail again at a speed of 3 mi per hour. At 10 a.m. the same day, another group of hikers begins the trail at the trailhead, hiking at a speed of 5 mi per hour. Find the time and distance hiked by the second group when they catch up with the first group. (Assume that the groups do not take breaks or stop.)

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