Problem 177 - Elementary Algebra

Question

Mukul has \(\$ 3.75\) in quarters, dimes and nickels in his pocket. He has five more dimes than quarters and nine more nickels than quarters. How many of each coin are in his pocket?

Step-by-Step Solution

Answer

Mukul has 7 quarters, 12 dimes, and 16 nickels.

Step 1: Define Variables

Let the number of quarters be denoted as \(q\). According to the problem, the number of dimes is \(q + 5\) and the number of nickels is \(q + 9\).

Step 2: Write the Value Equations

Each quarter is worth \(0.25\), each dime is worth \(0.10\), and each nickel is worth \(0.05\). Form the total value equation as follows:\[ 0.25q + 0.10(q + 5) + 0.05(q + 9) = 3.75 \]

Step 3: Distribute and Combine Like Terms

Distribute each term and combine like terms:\[ 0.25q + 0.10q + 0.50 + 0.05q + 0.45 = 3.75 \]Combine like terms:\[ 0.40q + 0.95 = 3.75 \]

Step 4: Solve for \(q\)

Subtract 0.95 from both sides to isolate term with \(q\):\[ 0.40q + 0.95 - 0.95 = 3.75 - 0.95 \]\[ 0.40q = 2.80 \]Divide by 0.40:\[ q = \frac{2.80}{0.40} \]\[ q = 7 \]

Step 5: Find the Number of Dimes and Nickels

Use the value of \(q\) to find the number of dimes and nickels:\[ q + 5 = 7 + 5 = 12 \]\[ q + 9 = 7 + 9 = 16 \]

Step 6: Write the Final Answer

Mukul has 7 quarters, 12 dimes, and 16 nickels in his pocket.

Key Concepts

system of equations

To solve the problem of counting Mukul's coins, we use a system of equations. A system of equations is a set of equations with multiple variables that you solve simultaneously. This method helps us handle different conditions given in the problem without ignoring any part of the information. In Mukul's case, the conditions are about the number of each type of coin he has and their total value. By forming equations based on these conditions, we can find the exact numbers of quarters, dimes, and nickels. This systematic approach is essential when dealing with multiple unknowns in algebra problems.

variable definition

The first step in solving this problem is defining our variables. We start by letting the number of quarters be represented by the variable \(q\). This simplifies the problem because we now have a single variable instead of talking about quarters directly.
According to the problem, the number of dimes Mukul has is five more than the number of quarters, so we represent dimes as \(q + 5\). Similarly, since he has nine more nickels than quarters, we represent the number of nickels as \(q + 9\).
By defining these variables, we streamline the process of writing our equations and make it easier to manipulate and solve them.

value equations

Next, we need to convert the information about the total value of the coins into an equation. Each type of coin has a specific value:

• Each quarter is worth \(0.25\).
• Each dime is worth \(0.10\).
• Each nickel is worth \(0.05\).

Using these values, we write an equation for the total value of the coins:
\(0.25q + 0.10(q + 5) + 0.05(q + 9) = 3.75\)
This equation includes the value of all quarters, dimes, and nickels combined and equates to the total amount Mukul has in his pocket. Writing such value equations helps us quantitatively equate the monetary values to numbers of coins.

solving linear equations

With our value equation set up, we solve for \(q\). This involves multiple steps:
1. Distribute and combine like terms:
\(0.25q + 0.10q + 0.50 + 0.05q + 0.45 = 3.75\)
Combine like terms to get:
\(0.40q + 0.95 = 3.75\)
2. Isolate the term with the variable by subtracting 0.95 from both sides:
\(0.40q = 2.80\)
3. Divide by 0.40:
\(q = \frac{2.80}{0.40}\)
\(q = 7\)
So, Mukul has 7 quarters. Using this value, we determine the number of dimes and nickels:
\(q + 5 = 7 + 5 = 12\) dimes.
\(q + 9 = 7 + 9 = 16\) nickels.
This step-by-step approach ensures that we systematically isolate the variable, simplify the equation, and find the solution.