Problem 9 - Algebra for College Students

Question

Otis has a box of coins that he uses when he plays poker with his friends. The box contains 44 coins, consisting of pennies, dimes, and quarters. The number of pennies is equal to the number of dimes. The total value is \(\$ 4.37 .\) How many of each denomination of coin does he have in the box?

Step-by-Step Solution

Answer

17 pennies, 17 dimes, 10 quarters

Step 1 - Define Variables

Let the number of pennies be denoted by \(p\). Since the number of pennies is equal to the number of dimes, the number of dimes is also \(p\). Let the number of quarters be denoted by \(q\).

Step 2 - Set Up System of Equations

We are given two crucial pieces of information: the total number of coins and the total value of the coins. Set up the following equations: \[ p + p + q = 44 \] which simplifies to \[ 2p + q = 44 \] And for the total value: \[ 0.01p + 0.10p + 0.25q = 4.37 \] Which simplifies to \[ 0.11p + 0.25q = 4.37 \]

Step 3 - Solve for One Variable

From the equation \( 2p + q = 44 \), solve for \(q\): \[ q = 44 - 2p \]

Step 4 - Substitute and Solve for \(p\)

Substitute \( q = 44 - 2p \) into the value equation: \[ 0.11p + 0.25(44 - 2p) = 4.37 \] Simplify and solve for \(p\): \[ 0.11p + 11 - 0.50p = 4.37 \] \[ -0.39p + 11 = 4.37 \] \[ -0.39p = -6.63 \] \[ p = 17 \]

Step 5 - Solve for \(q\)

Substitute \( p = 17 \) back into the equation \( q = 44 - 2p \): \[ q = 44 - 2(17) \] \[ q = 44 - 34 \] \[ q = 10 \]

Step 6 - Verify the Solution

Now verify by calculating the total value: \[ 17 \text{ pennies} + 17 \text{ dimes} + 10 \text{ quarters} \] Check the total number of coins: \[ 17 + 17 + 10 = 44 \] Check the total value: \[ 0.01(17) + 0.10(17) + 0.25(10) = 0.17 + 1.7 + 2.5 = 4.37 \]

Key Concepts

System of Equations

A system of equations is a collection of two or more equations with the same set of variables. In the case of Otis and his coin collection, we write two equations based on the problem's requirements: the total number of coins and the total value of the coins. For this problem:

• First equation: \(2p + q = 44\) (since the number of pennies is equal to the number of dimes, and their combined count plus quarters equals 44)
• Second equation: \(0.11p + 0.25q = 4.37\) (to represent the total value of the coins in dollars)

Each equation helps us understand a different aspect of the problem and the relationship between the variables. Solving these equations simultaneously will give us the number of each type of coin that Otis has.

Solving Linear Equations

A linear equation is an equation that forms a straight line when graphed. It can be written in the form \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) and \(y\) are variables. In our problem, we solve the linear equations to find the values of the variables.

To solve the system, we first rearrange one of the equations to express one variable in terms of another. For example, from \(2p + q = 44\), we can solve for \(q\) as follows:

• \(q = 44 - 2p\)

Then, we substitute this expression into the other equation \(0.11p + 0.25q = 4.37\). This way, we turn a system of two linear equations into a single equation with one variable, making it easier to solve.

Algebraic Substitution

Algebraic substitution is a key method for solving systems of equations. Once we have an expression for one variable, we substitute it into another equation to simplify the problem. For instance, substituting \( q = 44 - 2p \) into \( 0.11p + 0.25q = 4.37 \) gives us:

• \(0.11p + 0.25(44 - 2p) = 4.37\)
• Substituting the value: \(0.11p + 11 - 0.50p = 4.37\)
• Simplifying: \( -0.39p + 11 = 4.37\)
• Finally solving for \( p\): \(p = 17\)

This technique reduces the complexity by focusing on one variable at a time. After finding \( p \), we can use the earlier substituted equation to find \( q \), making the process of solving for all unknowns straightforward.