Problem 32 - Prealgebra

Question

Coin Problem Bob has \(\$ 1.65\) in dimes and nickels. If he has 9 more nickels than dimes, how many of each coin does he have?

Step-by-Step Solution

Answer

Bob has 8 dimes and 17 nickels.

Step 1: Understanding Coin Values

Let the number of dimes be denoted as \( d \). Each dime is worth \( \\(0.10 \), so the total value of the dimes is \( 0.10d \). Let the number of nickels be denoted as \( n \). Each nickel is worth \( \\)0.05 \), so the total value of the nickels is \( 0.05n \).

Step 2: Equation for Total Value

According to the problem, the total value of the coins is \( \$1.65 \). Thus, we have the equation: \[ 0.10d + 0.05n = 1.65 \]

Step 3: Equation for Coin Count Difference

The problem states that Bob has 9 more nickels than dimes. Thus, we have \[ n = d + 9 \]

Step 4: Substituting the Coin Count Equation

Substitute \( n = d + 9 \) into the total value equation: \[ 0.10d + 0.05(d + 9) = 1.65 \]

Step 5: Simplifying the Equation

Expand and simplify the equation: \[ 0.10d + 0.05d + 0.45 = 1.65 \] Combine like terms to get: \[ 0.15d + 0.45 = 1.65 \]

Step 6: Solving for Dimes

Subtract \( 0.45 \) from both sides of the equation: \[ 0.15d = 1.20 \] Divide both sides by \( 0.15 \) to solve for \( d \): \[ d = \frac{1.20}{0.15} = 8 \] So there are 8 dimes.

Step 7: Solving for Nickels

Use the equation \( n = d + 9 \) and substitute \( d = 8 \): \[ n = 8 + 9 = 17 \] So there are 17 nickels.

Key Concepts

Coin Value Calculation

Understanding the value of each coin type is essential when solving coin value problems. In this scenario, we have two types of coins: nickels and dimes. In the United States:

• A dime is worth \(0.10\) dollars, so for each dime, you multiply the number of dimes \(d\) by \(0.10\).
• A nickel is worth \(0.05\) dollars, so for each nickel, you multiply the number of nickels \(n\) by \(0.05\).

The total value of all coins is found by adding the value contributed by each type of coin. Here, the equation becomes: \[ 0.10d + 0.05n = 1.65 \] This equation represents the combined value of both nickels and dimes equating to the total amount Bob has, which is \(1.65\) dollars.

Systems of Equations

Systems of equations are a powerful tool used when you have more than one condition to satisfy. In the coin problem, we use two equations:

• The first equation comes from the total value of the coins: \[ 0.10d + 0.05n = 1.65 \]
• The second equation is from the condition about the number of coins: \[ n = d + 9 \]

Understanding systems of equations involves using methods like substitution or elimination to resolve equations that interact. In this context, substitution is a great method because the second equation \( n = d + 9 \) easily lets us express \(n\) in terms of \(d\). By substituting this relationship into the first equation, we simplify the problem to one variable, making it manageable to solve.

Word Problems

Word problems can sometimes seem intimidating, but a structured approach makes them much more approachable. Start by identifying the key information and translating it into mathematical statements. For example:- Bob has \(1.65\) dollars in coins: This suggests a total value equation.- He has 9 more nickels than dimes: This gives a relationship between the numbers of nickels and dimes.Next, convert these verbal cues into mathematical equations. Then solve these equations by applying algebraic techniques. Through this method of understanding and breaking down the problem, you translate the narrative of the word problem into solvable mathematical equations, just like we've done with Bob's coin value problem. This approach simplifies solving complex word problems by consistently following logical steps based on the given information.