Problem 175 - Elementary Algebra

Question

Chi has \(\$ 11.30\) in dimes and quarters. The number of dimes is three more than three times the number of quarters. How many of each are there?

Step-by-Step Solution

Answer

Chi has 20 quarters and 63 dimes.

Step 1: Define the variables

Let the number of quarters be denoted by q. Since the number of dimes is three more than three times the number of quarters, let the number of dimes be denoted by 3q + 3.

Step 2: Write the value equations

The value of the quarters can be written as 0.25q and the value of the dimes can be written as 0.10(3q + 3). Given that the total value is $11.30, we form the equation: 0.25q + 0.10(3q + 3) = 11.30

Step 3: Simplify and solve the equation

First, distribute the 0.10: 0.25q + 0.30q + 0.30 = 11.30 Combine like terms: 0.55q + 0.30 = 11.30 Subtract 0.30 from both sides to isolate the terms with q: 0.55q = 11.00 Divide both sides by 0.55 to solve for q: q = 20

Step 4: Find the number of dimes

Use the value of q to find the number of dimes: Dimes = 3q + 3 = 3(20) + 3 = 60 + 3 = 63

Step 5: Verify the solution

Check the total value: Value from quarters = 20 * 0.25 = 5.00 Value from dimes = 63 * 0.10 = 6.30 Total value = 5.00 + 6.30 = 11.30. The amounts match the given condition.

Key Concepts

System of Equations

When solving problems involving multiple variables, we often use a system of equations. This means we have more than one equation involving the same set of variables. For example, in our problem, we used two pieces of information:

• The number of dimes is three more than three times the number of quarters
• The total value of the dimes and quarters is $11.30

By converting these into equations, we can solve for the unknown values step-by-step. This method allows us to break down complex problems into manageable parts and find accurate solutions efficiently.

Word Problems

Word problems often describe a situation in words that need to be translated into mathematical expressions or equations. The first step is to define what we're solving for—in this case, the number of quarters and dimes. Then, identify and translate relationships and values described in the problem into algebraic equations. Always begin by:

• Defining your variables
• Expressing relationships between the variables using equations
• Double-checking the problem statement to ensure all important details are included

This method can make even complicated scenarios easier to understand and solve.

Value Equations

Value equations represent the total value of items or scenarios described in a problem. In our example, we had:

• The value of the quarters as 0.25q
• The value of the dimes as 0.10(3q + 3)

By knowing the total value, we can set up an equation such as:

<>0.25q + 0.10(3q + 3) = 11.30ewline Aligning values and prices into a single equation allows us to solve for unknown variables accurately, facilitating the problem-solving process.