Problem 25 - Intermediate Algebra

Question

When a child breaks open her piggy bank, she finds a total of 64 coins, consisting of nickels, dimes, and quarters. The total value of the coins is \(\$ 6 .\) If the nickels were dimes, and the dimes were nickels, the value of the coins would be S5. How many nickels, dimes, and quarters were in the piggy bank?

Step-by-Step Solution

Answer

There are 20 nickels, 40 dimes, and 4 quarters.

Step 1: Define Variables

Let \( n \) represent the number of nickels, \( d \) represent the number of dimes, and \( q \) represent the number of quarters. We know that the total number of coins is 64, hence the equation: \( n + d + q = 64 \).

Step 2: Equation for Total Value

The total value of the coins is \( \$6 \). Therefore, the value equation for the nickels, dimes, and quarters is \( 0.05n + 0.10d + 0.25q = 6 \).

Step 3: Value Swap Condition

If the nickels were swapped for dimes and the dimes for nickels, the total value would be \( \$5 \). The equation representing this scenario would be \( 0.10n + 0.05d + 0.25q = 5 \).

Step 4: Solve Equations Simultaneously

We now have three equations:1. \( n + d + q = 64 \)2. \( 5n + 10d + 25q = 600 \) (from multiplying by 100 to eliminate decimals)3. \( 10n + 5d + 25q = 500 \) (also multiplied by 100)Subtract equation 3 from equation 2:\( (5n + 10d + 25q) - (10n + 5d + 25q) = 600 - 500 \)This simplifies to: \(-5n + 5d = 100\), or \(d = n + 20\).

Step 5: Substitute Back

Substitute \(d = n + 20\) into equation 1:\( n + (n + 20) + q = 64 \) This simplifies to \(2n + q = 44\).

Step 6: Solve for Quarters

Now substitute \(d = n + 20\) and \(2n + q = 44\) into the first equation:\[ n + (n + 20) + q = 64 \]\[ 2n + q = 44 \]From these, we know \(q = 44 - 2n\).

Step 7: Substitution in the Value Equation

We further substitute into \(5n + 10(n + 20) + 25(44 - 2n) = 600\):Multiply and simplify:\(5n + 10n + 200 + 1100 - 50n = 600\)\(-35n + 1300 = 600\)Solve for \(n\):\(-35n = -700\)\(n = 20\).

Step 8: Solve for Dimes and Quarters

Using \(n = 20\), substitute back:If \(d = n + 20\), then \(d = 40\).Then \(q = 44 - 2n = 44 - 40 = 4\).

Step 9: Solution Verification

Check:\(20 + 40 + 4 = 64\) coins\(5(20) + 10(40) + 25(4) = 600\) cents i.e., \(\\(6\)Swap condition:\(10(20) + 5(40) + 25(4) = 500\) cents i.e., \(\\)5\)All conditions satisfied.

Key Concepts

Understanding Algebraic Equations

Algebraic equations are tools that allow us to represent real-life situations using mathematical language. In this piggy bank problem, we use algebraic equations to express the relationships between the different types of coins and their values.

Here's a brief overview of how we utilize algebraic equations in the problem:

• We define variables for each type of coin: \( n \) for nickels, \( d \) for dimes, and \( q \) for quarters. This is a crucial first step as it transforms the problem into mathematical terms.
• The first equation, \( n + d + q = 64 \), simply states that the total number of coins is 64. Every algebraic solution begins with clearly defined constraints.
• The second equation, \( 0.05n + 0.10d + 0.25q = 6 \), translates the coin values into their summed monetary value of $6. This uses the monetary values of nickels (5 cents), dimes (10 cents), and quarters (25 cents).

Breaking down the problem in this way helps us to see how each algebraic equation plays a role in comprehensively solving the problem.

Navigating Systems of Equations

Solving systems of equations is at the heart of unraveling this piggy bank puzzle. A system of equations is a set of equations with multiple variables, where solutions are the values of these variables that satisfy all equations simultaneously.

In this exercise, we constructed a system of three equations:

• The total number of coins equation: \( n + d + q = 64 \).
• The original value equation: \( 5n + 10d + 25q = 600 \) (multiplied by 100 to avoid decimals).
• The swapped value equation: \( 10n + 5d + 25q = 500 \), also multiplied by 100 for similar reasons.

Systems of equations allow us to leverage simultaneous information about relationships among variables to pinpoint accurate solutions.
The process of isolating variables, substituting them back into equations, and systematically eliminating variables is a methodical process that provides clarity and progress toward a solution.

Effective Problem-Solving Strategies

Approaching problems logically plays an essential role in effective problem-solving, especially with complex situations like coin word problems. Here are some sensible strategies that can be applied:

• **Define the Problem:** Clearly identify what you need to solve. Here, it's the count of nickels, dimes, and quarters.
• **Set Up Equations:** Use the information given to build accurate equations. Ensure that all conditions of the problem are considered.
• **Elimination and Substitution:** Utilize strategies like elimination of one variable through the substitution method to simplify complex systems.
• **Verification:** Double-check your results by substituting them back into original equations to confirm all conditions are met.

By applying these strategies, the process becomes more manageable. Solving any complex algebraic problem involves repeatedly breaking it down into its simplest parts and ensuring the logic flows from beginning to end.