Problem 88 - Intermediate Algebra

Question

Answer the question with an algebraic expression. Jesse has \(n\) nickels, \(d\) dimes, and \(q\) quarters in his bank. How much money, in cents, does he have in his bank?

Step-by-Step Solution

Answer

Jesse has \( 5n + 10d + 25q \) cents in his bank.

Step 1: Calculate the value of the nickels

Each nickel is worth 5 cents. Therefore, the value of the nickels Jesse has can be calculated by multiplying the number of nickels, \( n \), by 5. This gives us a total value of:\[ 5n \text{ cents} \]

Step 2: Calculate the value of the dimes

Each dime is worth 10 cents. To find the total value of the dimes, multiply the number of dimes, \( d \), by 10. This results in:\[ 10d \text{ cents} \]

Step 3: Calculate the value of the quarters

Each quarter is worth 25 cents. Multiply the number of quarters, \( q \), by 25 to find the total value of the quarters:\[ 25q \text{ cents} \]

Step 4: Add up the total value of all coins

To find the total amount of money Jesse has, add the value of the nickels, dimes, and quarters together:\[ 5n + 10d + 25q \text{ cents} \]

Step 5: Conclusion: Write the final algebraic expression

The total amount of money Jesse has in his bank, in cents, can be represented by the algebraic expression:\[ 5n + 10d + 25q \]

Key Concepts

Understanding Basic Algebra

Algebra might seem intimidating at first, but it's just about using letters and numbers to solve problems. In basic algebra, we use letters like \( n \), \( d \), and \( q \) to represent unknown values or quantities in equations. These letters are called variables.

Think of variables as placeholders that can be replaced with real numbers. For example, if \( n \) represents the number of nickels, then \( n \) could be 1, 2, or any whole number. The aim in algebra is to create expressions and equations using these variables to describe real-world situations.

Expressions like \( 5n \) mean you multiply the number represented by \( n \) by 5. It shows the value of nickels in this problem. This manipulation of variables helps us calculate values, prove relationships, and solve problems efficiently.

Approaches to Problem Solving

Problem solving is a key skill in math and involves understanding and analyzing given information. It's about breaking down a problem into smaller parts and tackling each step systematically.

In this coin problem, the key is to understand what each variable stands for. Decipher the problem's questions or requirements and identify what needs to be calculated. Here, we are asked to find the total money in cents that Jesse has.

It's helpful to outline steps to approach the solution:

• Identify the known values: the value of each coin type (nickel, dime, quarter).
• Use algebraic methods to relate variables: define the equations that describe the total value for each coin type.
• Perform required calculations: use the equations to add up the total value.

Work through each piece individually before assembling the whole answer. This organized method simplifies complex problems.

Coin Value Calculations Explained

Calculations in this problem focus on understanding the worth of each type of coin. Coins have standardized values:

• Nickels are worth 5 cents.
• Dimes are worth 10 cents.
• Quarters are worth 25 cents.

To find the total value of each coin type, multiply the number of each coin by its individual value. So, if Jesse has \( n \) nickels, \( d \) dimes, and \( q \) quarters, you calculate accordingly:

• Nickels: \( 5n \) cents.
• Dimes: \( 10d \) cents.
• Quarters: \( 25q \) cents.

Combine these amounts to get the total value of Jesse's coins, expressed as \( 5n + 10d + 25q \). This expression accurately represents Jesse's savings in cents, showcasing algebra's power in simplifying real-world calculations.