Problem 171 - Elementary Algebra

Question

In a cash drawer there is \(\$ 125\) in \(\$ 5\) and \(\$ 10\) bills. The number of \(\$ 10\) bills is twice the number of \(\$ 5\) bills. How many of each are in the drawer?

Step-by-Step Solution

Answer

There are 5 \$5 bills and 10 \$10 bills in the drawer.

Step 1 - Define Variables

Let the number of \$5 bills be \(x\), and the number of \$10 bills be \(y\).

Step 2 - Set Up Equations

There are two pieces of information given: \Total amount of money is \$125, and the number of \$10 bills is twice the number of \$5 bills. This gives us two equations: \(5x + 10y = 125\) and \(y = 2x\).

Step 3 - Substitute in the Equation

Substitute the expression \(y = 2x\) into the first equation: \[5x + 10(2x) = 125\]

Step 4 - Simplify and Solve

Simplify the equation: \[5x + 20x = 125\] \ Combine like terms: \[25x = 125\] \ Solve for \(x\): \[x = 5\].

Step 5 - Find the Value of \(y\)

Use the expression \(y = 2x\) to find \(y\): \[y = 2(5) = 10\]

Step 6 - Verify the Solution

Verify that the values of \(x\) and \(y\) satisfy the original equation: \For \(x = 5\) and \(y = 10\): \[5(5) + 10(10) = 25 + 100 = 125\] \ This verifies that the solution is correct.

Key Concepts

linear equations

Linear equations are equations of the first degree, meaning they have no exponents higher than one. In the equation, each term is either a constant or the product of a constant and a single variable. For example, the equations given in this exercise are linear: \ The first equation, \(5x + 10y = 125\), represents the total amount of money in the cash drawer in terms of the number of \$5\ and \$10\ bills. \ The second equation, \(y = 2x\), describes the relationship between the number of \$5\ and \$10\ bills, where the number of \$10\ bills is twice that of the \$5\ bills.
Linear equations often appear in real-life problems involving finances, distances, or time. These equations are useful because they can help solve for unknown variables to find a specific solution.

substitution method

The substitution method is a technique used to solve systems of equations. It involves solving one of the equations for one variable and substituting that expression into the other equation. In this exercise, we first define our variables and set up our equations: \ \ Let \(x\) be the number of \$5\ bills and \(y\) be the number of \$10\ bills.
Then, we have: \ \ \[5x + 10y = 125\] \ \[y = 2x\]
We solve the second equation for \(y\): \ \[y = 2x\]
Next, we substitute the expression \(2x\) for \(y\) in the first equation: \ \[5x + 10(2x) = 125\]
This substitution helps us to solve the equations step-by-step by focusing on one variable at a time. The method efficiently reduces the system to a single-variable equation, making it easier to handle and solve.

solving equations

Solving equations is the process of finding the values for the variables that make the equation true. In this exercise, after substituting \(2x\) for \(y\), we simplify to get: \ \ \[5x + 20x = 125\]
Combine like terms to get: \ \[25x = 125\]
To find the value of \(x\), we divide both sides by 25: \ \[x = 5\]
Once we find \(x\), we use it in the second equation to find \(y\): \ \[y = 2x = 2(5) = 10\]
Verifying the solution by substituting both values back into the original equations ensures they are correct. For \(x = 5\) and \(y = 10\): \ \[5(5) + 10(10) = 25 + 100 = 125\]
This confirms the solution is accurate.

variable definition

Variable definition is the first step in solving word problems with equations. It involves assigning symbols to represent unknown quantities. In this exercise, we define: \ \ - \(x\): the number of \$5\ bills \ - \(y\): the number of \$10\ bills
Defining variables provides a clear way to translate the problem into mathematical equations. Accurate definitions are crucial – they keep the problem structured and guide the formation of equations to solve the problem efficiently. Properly defined variables also help in substituting values and solving equations straightforwardly. Translating the word problem into defined variables and equations is the foundation of finding the correct solution.