Problem 161 - Elementary Algebra

Question

Jaime has \(\$ 2.60\) in dimes and nickels. The number of dimes is 14 more than the number of nickels. How many of each coin does he have?

Step-by-Step Solution

Answer

Jaime has 8 nickels and 22 dimes.

Step 1: Define Variables

Let the number of nickels be represented by the variable \( n \). Since the number of dimes is 14 more than the number of nickels, we can represent the number of dimes as \( n + 14 \).

Step 2: Set Up the Equations

The value of a nickel is \( 0.05 \) dollars and the value of a dime is \( 0.10 \) dollars. The total value of the nickels and dimes together is given as \( 2.60 \) dollars. Thus, we can create the equation: \[ 0.05n + 0.10(n + 14) = 2.60 \]

Step 3: Simplify the Equation

Distribute and simplify the equation: \[ 0.05n + 0.10n + 1.40 = 2.60 \] Combine like terms: \[ 0.15n + 1.40 = 2.60 \]

Step 4: Solve for \( n \)

Isolate the variable \( n \) by subtracting 1.40 from both sides: \[ 0.15n = 1.20 \] Divide both sides by 0.15: \[ n = \frac{1.20}{0.15} = 8 \]

Step 5: Find the Number of Dimes

Since the number of dimes is 14 more than the number of nickels, we can find the number of dimes by adding 14 to the number of nickels: \[ n + 14 = 8 + 14 = 22 \]

Key Concepts

Defining Variables

When solving word problems, it's essential to define variables because it translates the problem into mathematical terms.
Let's take Jaime's coin problem as an example. We start by defining a variable for the number of nickels, let’s call it \( n \).
Then, according to the problem, the number of dimes is 14 more than the number of nickels. So, if there are \( n \) nickels, the number of dimes can be represented as \( n + 14 \).
Defining variables this way helps to break down the problem into manageable pieces. After defining the variables, the next step is to use them to set up equations.

Setting Up Equations

Setting up equations is a critical step in solving word problems because it allows us to use algebra to find the unknowns.
Once you've defined your variables, you can convert the given information into an algebraic equation.
In our coin problem, we know the value of a nickel is 0.05 dollars and the value of a dime is 0.10 dollars. The total value of the coins is 2.60 dollars. This can be expressed as:

• Value of nickels: \( 0.05n \)
• Value of dimes: \( 0.10(n + 14) \)

Combining these, we set up the equation:
\( 0.05n + 0.10(n + 14) = 2.60 \)
This equation represents the total value of the nickels and dimes. From here, we can use algebraic methods to find the value of \( n \).

Solving Linear Equations

Once we have our equation set up, solving it involves isolating the variable.
Let's solve the equation step by step:
1. Distribute and combine like terms:
\( 0.05n + 0.10n + 1.40 = 2.60 \)
Combine the terms involving \( n \):
\( 0.15n + 1.40 = 2.60 \)
2. Isolate the variable by subtracting 1.40 from both sides:
\( 0.15n = 1.20 \)
3. Finally, solve for \( n \) by dividing both sides by 0.15:
\( n = \frac{1.20}{0.15} = 8 \)
So, \( n = 8 \), which means Jaime has 8 nickels.
Since the number of dimes is 14 more than the number of nickels, we find the number of dimes by adding 14 to the number of nickels: \( 8 + 14 = 22 \)
Hence, Jaime has 8 nickels and 22 dimes.
By following these steps, solving linear equations becomes a straightforward process.