Problem 172 - Elementary Algebra

Question

John has \(\$ 175\) in \(\$ 5\) and \(\$ 10\) bills in his drawer. The number of \(\$ 5\) bills is three times the number of \(\$ 10\) bills. How many of each are in the drawer?

Step-by-Step Solution

Answer

There are 7 \)10 bills and 21 $5 bills in the drawer.

Step 1 - Define Variables

Let the number of \(10 bills be denoted as x. Since the number of \)5 bills is three times the number of \(10 bills, we denote the number of \)5 bills as 3x.

Step 2 - Set Up the Equation

The total value of the \(10 bills is \)10x. The total value of the \(5 bills is \)5 \times 3x = \(15x. Since the total amount of money is \)175, the equation will be: \( 10x + 15x = 175 \).

Step 3 - Combine Like Terms

Combine the terms on the left-hand side of the equation: \( 25x = 175 \).

Step 4 - Solve for x

Divide both sides of the equation by 25 to isolate x: \( x = 175 / 25 = 7 \).

Step 5 - Find the Number of \(5 Bills

We know that the number of \)5 bills is three times the number of \(10 bills. Since there are 7 \)10 bills, the number of \(5 bills is \) 3 \times 7 = 21 \(.

Key Concepts

Defining Variables

Defining variables is the first crucial step in solving any word problem involving algebra. Variables act as placeholders for unknown values that we need to determine. In our exercise, let's define the number of \(\text{\textdollar 10}\) bills as \(x\). Since the number of \(\text{\textdollar 5}\) bills is given to be three times the number of \(\text{\textdollar 10}\) bills, we can denote it as \(3x\).

By assigning these variables, we create a mathematical model that represents the scenario described in the problem. This step sets the foundation for the entire solution process. It is essential to clearly state what each variable represents to avoid confusion later on.

Setting Up Equations

Once we have defined our variables, the next step is to set up an equation. This involves translating the words of the problem into a mathematical statement.

In our exercise, we know the total amount of money John has is \(\text{\textdollar 175}\). The value of each \(\text{\textdollar 10}\) bill is \(\text{\textdollar 10}\) and since there are \(x\) \(\text{\textdollar 10}\) bills, the total value of the \(\text{\textdollar 10}\) bills is \(10x\).

Similarly, each \(\text{\textdollar 5}\) bill is worth \(\text{\textdollar 5}\), and with \(3x\) \(\text{\textdollar 5}\) bills, the total value of the \(\text{\textdollar 5}\) bills is \(5 \times 3x = 15x\).

Therefore, the equation representing the total amount of money becomes:
\[ 10x + 15x = 175 \]

Combining Like Terms

Combining like terms is an important step in simplifying equations. It makes solving the equation more manageable by reducing it to its simplest form.

In the equation \[10x + 15x = 175\], both terms on the left-hand side contain the variable \(x\). We can combine them by adding their coefficients:
\[10x + 15x = 25x\].

So, the equation simplifies to \[25x = 175\]. Combining like terms helps us to directly focus on isolating the variable in the next step.

Solving Equations

After combining like terms, the last step is to solve the equation for the variable. This often involves isolating the variable on one side of the equation.

Our simplified equation is \[25x = 175\]. To solve for \(x\), divide both sides by 25:
\[ x = \frac{175}{25} = 7 \].

This means there are 7 \(\text{\textdollar 10}\) bills. To find the number of \(\text{\textdollar 5}\) bills, we use our relationship \(3x\):
\[3x = 3 \times 7 = 21\].

Therefore, John has 7 \(\text{\textdollar 10}\) bills and 21 \(\text{\textdollar 5}\) bills. Solving equations efficiently requires using systematic steps and understanding how to manipulate and isolate variables.