Problem 75 - Elementary Algebra

Question

Mary has a jar with one-dollar bills, half-dollar coins, and quarters valuing \(\$ 14.00 .\) She has twice as many quarters than she \text { does half-dollar coins and the same amount of half-dollar coins as one-dollar bills. How many of each does she have? }

Step-by-Step Solution

Answer

Mary has 7 half-dollar coins, 7 one-dollar bills, and 14 quarters.

Step 1: Define the Variables

Let's define the variables for the number of each type of currency Mary has. Let \( x \) represent the number of half-dollar coins. Since there are as many one-dollar bills as half-dollar coins, the number of one-dollar bills is also \( x \). Mary has twice as many quarters as half-dollar coins, so she has \( 2x \) quarters.

Step 2: Express the Total Value

Write an equation for the total value of the money in the jar. The value of the one-dollar bills is \( 1x \) dollars, the half-dollar coins contribute \( 0.5x \) dollars to the total, and the quarters contribute \( 0.25(2x) \) dollars. The total value equation is: \[ x + 0.5x + 0.25(2x) = 14 \]

Step 3: Simplify the Equation

Simplify the equation from Step 2. Start by expanding the terms: \[ x + 0.5x + 0.5x = 14 \] Combine the like terms: \[ (1 + 0.5 + 0.5)x = 14 \] Thus, the equation becomes: \[ 2x = 14 \]

Step 4: Solve for x

Solve the simplified equation for \( x \). Divide both sides by 2: \[ x = \frac{14}{2} \] Thus, \( x = 7 \).

Step 5: Find the Number of Each Currency

Use the value of \( x \) to determine the number of each type of currency. Since \( x = 7 \), Mary has 7 half-dollar coins, 7 one-dollar bills, and \( 2x = 2 \times 7 = 14 \) quarters.

Key Concepts

Solving Equations

In algebra, solving equations involves finding the value that makes an equation true. To solve the problem with Mary's jar of currency, we start by setting up an equation based on the information provided. We use the total value of the currencies in the jar, which is $14, and express it using variables for each type of currency.

After defining the variables, we translate the word problem into a mathematical equation. For example, we express the total value as a sum of the value of one-dollar bills, half-dollar coins, and quarters. By translating the relationships given in the problem into algebraic expressions, we receive an equation that we can solve.

• The equation combines these expressions to create a single equation representing the total value of the jar.
• We often simplify the equation by combining like terms, which can help make the solving process straightforward. In this example, it allows us to solve for a single variable, leading us to find the precise number of each currency type that Mary has.

Variables in Algebra

Variables are symbols used to represent unknown values in algebraic expressions or equations. In solving word problems, variables help translate words into mathematical language. In Mary's currency jar scenario, we define variables to represent how many of each type of currency she has.

In this exercise, we started by letting \( x \) represent the number of half-dollar coins. Given that the number of one-dollar bills is equal to the number of half-dollar coins, the variable \( x \) also represents one-dollar bills. Additionally, since Mary has twice as many quarters as half-dollar coins, we use \( 2x \) for quarters.

• Variables act as placeholders in equations, making it easier to solve for unknown quantities.
• Choosing appropriate variables allows complex worded problems to be structured into solvable mathematical statements.
• Correct variable definition is crucial for setting up an accurate equation that reflects the core problem.

Currency Conversion

Currency conversion involves translating different currencies into their monetary equivalents. In algebraic word problems that deal with currency, this often means converting counts of coins or bills into their respective dollar values to solve a problem.

For Mary's problem, currency value is crucial as each type of coin contributes a specific amount to the total money in the jar. Hence, we consider:

• One-dollar bills have a value of \(1, resulting in a contribution of \( 1x \) dollars if there are \( x \) one-dollar bills.
• Half-dollar coins have a value of \)0.50 or 50 cents each, contributing \( 0.5x \) dollars when there are \( x \) such coins.
• Quarters are worth 25 cents each, leading to a total contribution of \( 0.25(2x) \) when there are \( 2x \) quarters.

By converting the counts of each type of currency into their dollar equivalents, we can accurately set up and solve the equation that leads us to the solution.