Problem 246 - Elementary Algebra

Question

In the following exercises, translate to a system of equations and solve. Sherri saves nickels and dimes in a coin purse for her daughter. The total value of the coins in the purse is \(\$ 0.95\). The number of nickels is two less than five times the number of dimes. How many nickels and how many dimes are in the coin purse?

Step-by-Step Solution

Answer

Sherri has 3 dimes and 13 nickels.

Step 1: Define Variables

Let the number of dimes be represented by the variable \(d\). Let the number of nickels be represented by the variable \(n\).

Step 2: Write the Value Equation

Each dime is worth \(0.10\) dollars and each nickel is worth \(0.05\) dollars. The total value of the coins is \(0.95\) dollars. Hence, the value equation is: \[ 0.10d + 0.05n = 0.95 \]

Step 3: Write the Quantity Relationship

The problem states that the number of nickels is two less than five times the number of dimes. Therefore, the quantity relationship is: \[ n = 5d - 2 \]

Step 4: Substitute to Solve for One Variable

Substitute \(n\) in the value equation with \(5d - 2\): \[ 0.10d + 0.05(5d - 2) = 0.95 \]

Step 5: Simplify and Solve for \(d\)

Expand and combine like terms: \[ 0.10d + 0.25d - 0.10 = 0.95 \] Combine: \[ 0.35d - 0.10 = 0.95 \] Add 0.10 to both sides: \[ 0.35d = 1.05 \] Divide both sides by 0.35: \[ d = 3 \]

Step 6: Solve for \(n\)

Use the relationship \(n = 5d - 2\) with \(d = 3\): \[ n = 5(3) - 2 \] \[ n = 15 - 2 \] \[ n = 13 \]

Step 7: Verify the Solution

Check the solution in the original value equation: \[ 0.10(3) + 0.05(13) = 0.30 + 0.65 = 0.95 \] The verification confirms that the solution is correct.

Key Concepts

coin problems

Coin problems are a common type of algebraic word problem where you need to determine the number of coins of different denominations. These problems often involve combining the values of coins to match a given total value. In our example, Sherri has a mix of nickels and dimes with a total value of 95 cents. Identifying the relationship between the number of each type of coin and their total value is the first step in solving these types of problems. By clearly defining variables for each coin type, we can set up equations that represent the problem's conditions, simplifying the solution process.

algebraic equations

Algebraic equations are mathematical statements that show the equality between two expressions by using variables and constants. In the coin problem, we deal with two types of equations:

• Value Equation: Represents the total value of all coins.
• Quantity Relationship: Shows the relationship between the number of nickels and dimes.

Our value equation is: \(0.10d + 0.05n = 0.95\), while our quantity relationship is: \(n = 5d - 2\).
Combining these equations helps solve for the unknown variables (number of dimes and nickels). We use substitution or elimination methods to find the values of these variables. In our example, substitution allows us to replace n in the value equation with an expression involving d, leading to a solvable equation.

substitution method

The substitution method is a technique used to solve systems of equations. Here’s a step-by-step guide to using the substitution method for our coin problem:

• First, solve one of the equations for one variable in terms of the other. In our case, solve the quantity relationship for n: \(n = 5d - 2\).
• Next, substitute this expression into the other equation. Replace n in the value equation with \(5d - 2\) to get: \(0.10d + 0.05(5d - 2) = 0.95\).
• Simplify and solve this new equation for the remaining variable, d.
• Once you find d, substitute it back into the quantity relationship to find n.

Using substitution simplifies the problem, making it easier to solve the system of equations.

problem-solving steps

Solving a coin problem involves several clear and structured steps. Let’s outline them:

• Define Variables: Assign symbols to represent unknown quantities, such as d for dimes and n for nickels.
• Write Equations: Establish equations based on the total value of the coins and any given relationships. For our problem, these equations are \(0.10d + 0.05n = 0.95 \) and \( n = 5d - 2 \).
• Substitute: Replace one variable in the equation with its equivalent from another equation.
• Solve for One Variable: Simplify the equation to find the value of one variable.
• Solve for the Other Variable: Substitute the found value back into one of the original equations to find the second variable.
• Verify: Check your solution by substituting the values back into the original equations to ensure they satisfy both equations.

Following these steps ensures a systematic approach to solving any coin problem.