Question

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

Short answer

Convert to pennies using \( 25q + 10d + 5n + p \).

Step-by-step solution

Understand the Problem

To convert the coins into pennies, you need to know the value of each coin in terms of pennies. A quarter is worth 25 pennies, a dime is worth 10 pennies, a nickel is worth 5 pennies, and a penny is worth 1 penny.

Define the Inputs

Consider the number of quarters, dimes, nickels, and pennies you have. Let's denote these numbers as \( q \), \( d \), \( n \), and \( p \) respectively.

Calculate the Total Penny Value for Each Coin Type

Calculate the total value of the quarters in pennies by multiplying the number of quarters by 25: \( 25q \). Similarly, calculate the total value of dimes with \( 10d \), nickels with \( 5n \), and pennies with \( 1p \).

Sum the Penny Values

Add all the results from the previous step to get the total amount in pennies: \( 25q + 10d + 5n + 1p \).

Formulate the Algorithm

Create an algorithm that takes \( q \), \( d \), \( n \), and \( p \) as inputs and returns the total pennies calculated as \( 25q + 10d + 5n + 1p \).

Test the Algorithm

Test the algorithm with different sets of quarters, dimes, nickels, and pennies to ensure it outputs the correct total in pennies.

Key concepts

Problem Analysis

When faced with any algorithm design task, the first crucial step is a thorough problem analysis. This involves understanding exactly what the problem is demanding and identifying the necessary components to solve it. In this instance, the goal is to convert a collection of coins - specifically quarters, dimes, nickels, and pennies - into a total value represented solely in pennies.

To tackle this, we must first identify the values of each type of coin:

• A quarter equals 25 pennies.
• A dime equals 10 pennies.
• A nickel equals 5 pennies.
• A penny, unsurprisingly, equals 1 penny.

This foundational understanding is the bedrock of the conversion process and serves as our basis for calculating the entire value of our coin collection in terms of pennies.

By clearly defining these values, we can set the stage for algorithmically processing any given quantity of these coin types. Understanding these core values helps in breaking down more complex operations into smaller, manageable units, making problem-solving efficient and straightforward.

Coin Conversion

Coin conversion is a straightforward yet essential process in this problem-solving exercise. By converting various types of coins into pennies, we standardize their value, thus making calculations easier and universal. This process is akin to translating everything into a common language, in this case, that language is the penny.

The conversion process involves multiplying each type of coin by its value in pennies:

• Quarters are converted by multiplying their quantity by 25, so for any given number of quarters, you calculate the total value as \( 25q \).
• For dimes, multiply the quantity by 10, giving \( 10d \).
• Nickels’ total value is found by multiplying by 5, yielding \( 5n \).
• Finally, pennies are already in their basic unit, so their total value remains \( 1p \).

These calculations transform diverse coin amounts into a simple cumulative penny amount, ready to be summed up for a complete value assessment.

Step-by-step Solution

Achieving a successful algorithm design relies on a methodical approach. Here's the step-by-step breakdown that underlies the solution for converting coins into pennies:

**Step 1: Understand the Problem**
Recognizing the conversion values for each coin type is fundamental. Always start by confirming these values, ensuring accuracy as you proceed.

**Step 2: Define the Inputs**
Identify the variables involved. Here, the variables are the quantities of each coin type: quarters \( q \), dimes \( d \), nickels \( n \), and pennies \( p \). Clarifying these inputs is critical for accurate calculation.

**Step 3: Calculate the Total Penny Value for Each Coin Type**
Multiply the quantity of each coin type by their respective penny value:

• Quarters: \( 25q \)
• Dimes: \( 10d \)
• Nickels: \( 5n \)
• Pennies: \( 1p \)

**Step 4: Sum the Penny Values**
Add up these multiplied values to get the total amount in pennies: \( 25q + 10d + 5n + 1p \).

**Step 5: Formulate the Algorithm**
Craft an algorithm using the formula from Step 4. This algorithm should accept \( q \), \( d \), \( n \), and \( p \) as inputs and output the total in pennies.

**Step 6: Test the Algorithm**
Run tests with various quantities of coins. Confirm the algorithm consistently produces the correct total in pennies.

This deliberately laid out solution path not only ensures accurate results but also heightens understanding for any learner tackling similar algorithmic challenges.