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Chapter 4: Problem 37

F. For thousands of years, gold has been considered one of Earth's most precious metals. One hundred percent pure gold is 24 -karat gold, which is too soft to be made into jewelry. In the United States, most gold jewelry is 14 -karat gold, approximately \(58 \%\) gold. If 18 -karat gold is \(75 \%\) gold and 12 -karat gold is \(50 \%\) gold, how much of each should be used to make a 14 -karat gold bracelet weighing 300 grams?

Short Answer

Expert verified
The mass of the 18-karat gold and 12-karat gold needed are approximately \(x\) grams and \(y\) grams respectively, where \(x\) and \(y\) are the solutions from Step 3.

Step by step solution

01

Understand the problem

The 14 karat (58% gold) bracelet weighs 300 grams and is made by mixing 18-karat gold (75% gold) and 12-karat gold (50% gold). We need to find the quantity of each to be used.
02

Set up the equations

Let's denote the mass of 18-karat gold by \(x\) (in grams) and the mass of 12-karat gold by \(y\) (in grams). We have the following two equations based on the problem: Equation 1 (Mass): \(x + y = 300\) Equation 2 (Percentage of gold): \(0.75x + 0.5y = 0.58 * 300\)
03

Solve the equations

These are two equations with two variables, which can be solved by substitution or elimination method. From Equation 1 we can express \(y\) in terms of \(x\): \(y = 300 - x\). Substituting this into the second equation gives: \(0.75x + 0.5(300 - x) = 0.58 * 300\). Solving this equation we get the value for \(x\). After which, substitute \(x\) into the equation \(y = 300 - x\) to get the value for \(y\).
04

Verify the solution

After finding the values for \(x\) and \(y\), substitute them back into the original equations to confirm that they satisfy both the mass and percentage relationships. This will validate the solution.

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Key Concepts

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

Algebraic Equations
Algebraic equations are the cornerstone of solving percentage word problems. They consist of equalities containing one or more variables that represent unknown quantities. A typical algebraic equation will have numbers, variables, and arithmetic operators.

To comprehend these equations, it's crucial to get comfortable with their components:

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Most popular questions from this chapter

How do you determine whether a given ordered triple is a solution of a system of linear equations in three variables?

Use the four-step strategy to solve each problem. Use \(x, y,\) and \(z\) to represent unknown quantities. Then translate from the verbal conditions of the problem to a system of three equations in three variables. A person invested \(\$ 6700\) for one year, part at \(8 \%,\) part at \(10 \%,\) and the remainder at \(12 \% .\) The total annual income from these investments was \(\$ 716 .\) The amount of money invested at \(12 \%\) was \(\$ 300\) more than the amounts invested at \(8 \%\) and \(10 \%\) combined. Find the amount invested at each rate.

Solve each system by the method of your choice. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. Explain why you selected one method over the other two. $$\left\\{\begin{array}{l}2 x-7 y=17 \\ 4 x-5 y=25\end{array}\right.$$

Solve each system by the method of your choice. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. Explain why you selected one method over the other two. $$\left\\{\begin{aligned} 2 x+3 y &=7 \\ x &=2 \end{aligned}\right.$$

Write a system of equations modeling the given conditions. Then solve the system by the addition method and find the two numbers. Three times a first number increased by twice a second number is \(11 .\) The difference between the first number and twice the second number is 9. Find the numbers.
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