Question

Design an algorithm to convert the change given in quarters, dimes, nickels, and pennies into pennies.

Short answer

Convert each coin type to pennies using their values and sum them.

Step-by-step solution

Understand the Coin Values

Each type of coin has a specific value in cents: quarters are worth 25 cents, dimes are worth 10 cents, nickels are worth 5 cents, and pennies are worth 1 cent.

Write the Variables

Define the variables for each type of coin. Let's denote the number of quarters as \( q \), the number of dimes as \( d \), the number of nickels as \( n \), and the number of pennies as \( p \).

Define the Conversion Formula

The total change in pennies can be represented as the sum of pennies from each type of coin. The formula is: \( T = 25q + 10d + 5n + p \), where \( T \) is the total number of pennies.

Algorithm Design

Create an algorithm that uses the formula to calculate the total number of pennies. The algorithm steps are: 1. Input the number of quarters \( q \).2. Input the number of dimes \( d \).3. Input the number of nickels \( n \).4. Input the number of pennies \( p \).5. Compute \( T = 25q + 10d + 5n + p \).6. Output the total pennies \( T \).

Key concepts

Coin Values

When dealing with coins, it's important to know their values to make accurate calculations. Quarters, one of the largest common coins in the US currency system, are worth 25 cents each. Dimes come next, valued at 10 cents, and nickels are worth 5 cents. Lastly, pennies are valued at 1 cent each. Knowing the value of each coin is essential for both everyday transactions and creating algorithms that involve financial computations.

Data Types

In programming and algorithm design, data types help define what kind of data we are working with. For our coin conversion algorithm, we deal with integer data types. These are used because the number of coins and their respective values are whole numbers.

• Integers are the simplest and most common data types for such calculations. They help in maintaining precision during arithmetic operations, as they do not deal with fractional values.
• Additionally, clearly defining our variables, such as using letters like \( q \) for quarters and \( p \) for pennies, helps clarify their roles in arithmetic expressions.

Arithmetic Operations

Arithmetic operations involve basic mathematical functions such as addition, subtraction, multiplication, and division. In the context of our exercise, we primarily use multiplication and addition.
The conversion of coins into pennies requires determining how each type of coin contributes to the total value in terms of pennies.
For example, if you have 2 quarters, you multiply the number of quarters by 25 as each quarter is equivalent to 25 pennies. The complete arithmetic operation for all coins is carried out through our conversion formula: \[ T = 25q + 10d + 5n + p \]
where each term in the formula represents the process of converting each type of coin to pennies, followed by summing up these results to get the total in pennies.

Conversion Formula

The conversion formula is central to solving the exercise of converting various coins into pennies. It encapsulates the entire logic required for the conversion in a simple mathematical expression. The formula is:
\[ T = 25q + 10d + 5n + p \]
Breaking down the formula:

• \( 25q \) converts quarters to pennies.
• \( 10d \) converts dimes to pennies.
• \( 5n \) converts nickels to pennies.
• \( p \) already represents pennies.

The sum of these calculated values gives \( T \), the total number of pennies. This formula forms the core of the algorithm and streamlines the conversion process by providing a clear and concise method to perform the calculation.